{'fraction_non_alphanumeric': 0.033628350455130684, 'fraction_numerical': 0.012983154312261968, 'mean_word_length': 4.508617482951023, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 5, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 20, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Topics in quantum information and the theory of open quantum systems

This thesis examines seven topics in the areas of deterministic open-quantum-system dynamics, quantum measurements, and quantum error correction (QEC). The first topic concerns weak measurements and their universality as a means of generating quantum operations. It is shown that every generalized measurement can be implemented as a sequence of weak (infinitesimal) measurements. The second topic is an application of this result to the theory of entanglement. Necessary and sufficient differential conditions for entanglement monotones are derived and are used to find a new entanglement monotone for three-qubit states. The third topic is a study of the performance of different master equations for the description of non-Markovian dynamics. The system studied is a qubit coupled to a spin bath via the Ising interaction. The fourth topic investigates continuous QEC in the presence of non-Markovian noise. It is shown that due to the existence of a Zeno regime in non-Markovian dynamics, the performance of continuous QEC exhibits a quadratic improvement for a sufficiently high time resolution of the error-correcting operations. The fifth topic studies the conditions for correctability of subsystem codes under continuous dynamics. Necessary and sufficient conditions on the Lindbladian and the system-environment Hamiltonian are derived. The sixth topic examines the robustness of operator QEC codes against initialization errors. A new measure of fidelity for encoded information is introduced and is used to show that operator codes are robust against imperfect initialization without the need for restriction of the standard operator QEC conditions. The last topic concerns holonomic quantum computation (HQC) and stabilizer codes. A fault-tolerant scheme for HQC is presented, proving the scalability of the holonomic approach.

First and foremost, I would like to express my gratitude to Todd A. Brun for his guidance and support throughout the years of our work together. He has been a great advisor and mentor! I am deeply indebted to him for introducing me to the field of quantum information science and helping me advance in it. From him I learned not only how to do research, but also numerous other skills important for the career of a scientist, such as writing, giving presentations, or communicating professionally. I highly appreciate the fact that he was always supportive of any research direction I wanted to undertake, never exerted pressure on my work, and was available to give me advice or encouragement every time I needed them. This provided for me the optimal environment to develop and made my work with him a wonderful experience. for example some transformation can be thought of as resulting from measurements whose outcomes are discarded. This description, however, can also be understood as originating from unitary evolution of a system which includes the measurement apparatus and all systems on which the outcome has been imprinted.
Including the environment in the description is generally difficult due to the large number of environment degrees of freedom. This is why it is useful to have a description which involves only the effective evolution of the reduced density operator of the system. When the system and the environment are initially uncorrelated, the effective evolution of the density operator of the system can be described by a completely positive trace-preserving (CPTP) linear map [93]. CPTP maps are widely used in quantum information science for describing noise processes and operations on quantum states [115]. They do not, however, describe the most general form of transformation of the state of an open system, since the initial state of the system and the environment can be correlated in a way which gives rise to non-CP transformations. Furthermore, the effective transformation by itself does not provide direct insights into the process that drives the transformation. For the latter one needs a description in terms of a generator of the evolution, similar to the way the Schrödinger equation describes the evolution of a closed system as generated by a Hamiltonian. The main difficulty in obtaining such a description for open systems is that the evolution of the reduced density matrix of the system is subject to non-trivial memory effects arising from the interaction with the environment [30].
In the limit where the memory of the environment is short-lived, the evolution of an open system can be described [30] by a time-local semi-group master equation in the Lindblad form [106]. Such an equation can be thought of as corresponding to a sequence of weak (infinitesimal) CPTP maps. When the memory of the environment cannot be ignored and the effective transformation on the initial state (which is not necessarily CP) is reversible, the evolution can be described by a time-local master equation, known as the time-convolutionless (TCL) master equation [143,142]. In contrast to the Lindblad equation, this equation does not describe completely positive evolution.
The most general continuous deterministic evolution of an open quantum system is described by the Nakajima-Zwanzig (NZ) equation [112,181]. This equation involves convolution in time. Both the TCL and NZ equations are quite complicated to obtain from first principles and are usually used for perturbative descriptions. Somewhere in between the Lindblad equation and the TCL or NZ equations are the phenomenological post-Markovian master equations such as the one proposed in Ref. [139].
In this thesis, we will examine the deterministic evolution of open quantum systems both from the point of view of the full evolution of the system and the environment and from the point of view of the reduced dynamics of the system. We will study the performance of different master equation for the description of the non-Markovian evolution of a qubit coupled to a spin bath [97], compare Markovian and non-Markovian models in light of continuous quantum error correction [120], and study the conditions for preservation of encoded information under Markovian evolution of the system and general Hamiltonian evolution of the system and the environment [122].

Quantum measurements
In addition to deterministic transformations, the state of an open quantum system can also undergo stochastic transformations. These are transformations for which the state may change in a number of different ways with non-unit probability. Since according to the postulates of quantum mechanics the only non-deterministic transformations are those that result from measurements [165,108], stochastic transformations most generally result from measurements applied on the system and its environment. Just like deterministic transformations, stochastic transformations need not be completely 4 positive. If the system of interest is initially entangled with its environment and after some joint unitary evolution of the system and the environment a measurement is performed on the environment, the effective transformation on the system resulting from this measurement need not be CP.
The class of completely positive stochastic operations are commonly referred to as generalized measurements [93]. Although this class includes standard projective measurements [165,108] as well as other operations whose outcomes reveal information about the state, not all operations in this category reveal information. Some operations simply consist of deterministic operations applied with probabilities that do not depend on the state, i.e., they amount to trivial measurements.
In recent years, a special type of generalized quantum measurements, the so called weak measurements [4,5,99,127,6], have become of significant interest. A measurement is called weak if all of its outcomes result in small (infinitesimal) changes to the state. Weak measurements have been studied both in the abstract, and as a means of understanding systems with continuous monitoring. In the latter case, we can think of the evolution as the limit of a sequence of weak measurements, which gives rise to continuous stochastic evolutions called quantum trajectories (see, e.g., [41,48,61,64,52,65,129]). Such evolutions have been used also as models of decoherence (see, e.g., [32]). Weak measurements have found applications in feedback quantum control schemes [54] such as state preparation [74,153,154,169,46] or continuous quantum error correction [7,135].
In this thesis, we look at weak measurements as a means of generating quantum transformations [118]. We show that any generalized measurement can be implemented as a sequence of weak measurements, which allows us to use the tools of differential calculus in studies concerning measurement-driven transformations. We apply this result to the theory of entanglement, deriving necessary and sufficient conditions for a function on quantum states to be an entanglement monotone [119]. We use these conditions to find a new entanglement monotone for three-qubit pure states, a subject of previously unsuccessful inquiries [63]. We also discuss the use of weak measurements for continuous quantum error correction.

Quantum error correction
Whether deterministic or stochastic, the evolution of a system coupled to its environment is generally irreversible. This is because the environment, by definition, is outside of the experimenter's control. As irreversible transformations involve loss of information, they could be detrimental to a quantum information scheme unless an error-correcting method is employed.
A common form of error correction involves encoding the Hilbert space of a single information unit, say a qubit, in a subspace of the Hilbert space of a larger number of qubits [144,150,22,89]. The encoding is such that if a single qubit in the code undergoes an error, the original state can be recovered by applying an appropriate operation. Clearly, there is a chance that more than one qubit undergoes an error, but according to the theory of fault tolerance [145,53,89,2,85,91,68,67,131] this problem can be dealt with by the use of fault-tolerant techniques and concatenation.
Error correction encompasses a wide variety of methods, each suitable for different types of noise, different tasks, or using different resources. Examples include passive error-correction methods which protect against correlated errors, such as decoherencefree subspaces [55,176,105,103] and subsystems [90,51,83,174], the standard active methods [144,150,22,89] which are suitable for fault-tolerant computation [67], entanglement assisted quantum codes [34,35] useful in quantum communication, or linear quantum error-correction codes [141] that correct non-completely positive errors. Recently, a general formalism called operator quantum error correction (OQEC) [94,95,24] was introduced, which unified in a common framework all previously proposed error-correction methods. This formalism employs the most general encoding 6 of information-encoding in subsystems [88,163]. OQEC was generalized to include entanglement-assisted error correction resulting in the most general quantum errorcorrection formalism presently known [73,62].
In the standard formulation of error correction, noise and the error-correcting operations are usually represented by discrete transformations [94,95,24]. In practice, however, these transformations result from continuous processes. The more general situation where both the noise and the error-correcting processes are assumed to be continuous, is the subject of continuous quantum error correction [125,136,7,135]. In the paradigm of continuous error correction, error-correcting operations are generated by weak measurements, weak unitary operations or weak completely positive maps.
This approach often leads to a better performance in the setting of continuous decoherence than that involving discrete operations. In this thesis, we will discuss topics concerning both the discrete formalism and the continuous one. The topics we study include continuous quantum error correction for non-Markovian decoherence [120], conditions for correctability of operator codes under continuous decoherence [122], the performance of OQEC under imperfect encoding [117], as well as fault-tolerant computation based on holonomic operations [121].

Outline
This work examines seven topics in the areas of deterministic open-quantum-system dynamics, quantum measurements, and quantum error correction. Some of the topics concern all of these three themes, while others concern only two or only one. As each of the main results has a significance of its own, each topics has been presented as a separate study in one of the following chapters. The topics are ordered in view of the background material they introduce and the logical relation between them.
We first study the theme of weak measurements and their applications to the theory of entanglement. In Chapter 2, we show that every generalized quantum measurement can be generated as a sequence of weak measurements [118], which allows us to think of measurements in quantum mechanics as generated by continuous stochastic processes.
In the case of two-outcome measurements, the measurement procedure has the structure of a random walk along a curve in state space, with the measurement ending when one of the end points is reached. In the continuous limit, this procedure corresponds to a quantum feedback control scheme for which the type of measurement is continuously adjusted depending on the measurement record. This result presents not only a practical prescription for the implementation of any generalized measurement, but also reveals a rich mathematical structure, somewhat similar to that of Lie algebras, which allows us to study the transformations caused by measurements by looking only at the properties of infinitesimal stochastic generators. The result suggests the possibility of constructing a unified theory of quantum measurement protocols. The universality of weak measurements allows us to look at LOCC as the class of transformations generated by infinitesimal local operations. We show that a necessary and sufficient condition for a function of the state to be an entanglement monotone under local operations that do not involve information loss is that the function be a monotone under infinitesimal local operations. We then derive necessary and sufficient differential conditions for a function of the state to be an entanglement monotone [119].
We first derive two conditions for local operations without information loss, and then show that they can be extended to more general operations by adding the requirement of convexity. We then demonstrate that a number of known entanglement monotones 8 satisfy these differential criteria. We use the differential conditions to construct a new polynomial entanglement monotone for three-qubit pure states.
In Chapter 4, we extend the scope of our studies to include the deterministic dynamics of open quantum systems. We study the analytically solvable Ising model of a single qubit system coupled to a spin bath for a case for which the Markovian approximation of short bath-correlation times cannot be applied [97]. In Chapter 5, we investigate further the difference between Markovian and non-Markovian decoherence-this time, form the point of view of its implications for the performance of continuous quantum error correction. We study the performance of a quantum-jump error correction model in the case where each qubit in a codeword is subject to a general Hamiltonian interaction with an independent bath [120]. We first consider the scheme in the case of a trivial single-qubit code, which provides useful insights into the workings of continuous error correction and the difference between Markovian and non-Markovian decoherence. We then study the model of a bit-flip code with each qubit coupled to an independent bath qubit and subject to continuous correction, and find its solution. We show that for sufficiently large error-correction rates, the encoded state approximately follows an evolution of the type of a single 9 decohering qubit, but with an effectively decreased coupling constant. The factor by which the coupling constant is decreased scales quadratically with the error-correction rate. This is compared to the case of Markovian noise, where the decoherence rate is effectively decreased by a factor which scales only linearly with the rate of error correction. The quadratic enhancement depends on the existence of a Zeno regime in the Hamiltonian evolution which is absent in purely Markovian dynamics. We analyze the range of validity of this result and identify two relevant time scales. Finally, we extend the result to more general codes and argue that there the performance of continuous error correction will exhibit the same qualitative characteristics. In the appendix of this chapter, we discuss another application of weak measurements-we show how the quantum-jump error-correction scheme can be implemented using weak measurements and weak unitary operations.
In Chapter 6, we study the conditions under which a quantum code is perfectly correctable during a time interval of continuous decoherence for the most general type of encoding-encoding in subsystems. We study the case of Markovian decoherence as well as the general case of Hamiltonian evolution of the system and the environment, and derive necessary and sufficient conditions on the Lindbladian and the systemenvironment Hamiltonian [122], respectively. Our approach is based on a result obtained in Ref. [96] according to which a subsystem is correctable if and only if it is unitarily recoverable. The conditions we derive can be thought of as generalizations of the previously derived conditions for decoherence-free subsystems to the case where the subsystem is time-dependent. As a special case we consider conditions for unitary correctability. In the case of Hamiltonian evolution, the conditions for unitary correctability concern only the effect of the Hamiltonian on the system, whereas the conditions for general correctability concern the entire system-environment Hamiltonian. We also derive conditions on the Hamiltonian which depend on the initial state of the environment. We discuss possible implications of our results for approximate quantum error correction.
Chapter 7 also concerns subsystem codes. Here we study the performance of operator quantum error correction (OQEC) in the case of imperfect encoding [117]. In the OQEC, the notion of correctability is defined under the assumption that states are perfectly initialized inside a particular subspace, a factor of which (a subsystem) contains the protected information. It was believed that in the case of imperfect initialization, OQEC codes would require more restrictive than the standard conditions if they are to protect encoded information from subsequent errors. In this chapter, we examine this requirement by looking at the errors on the encoded state. In order to quantitatively analyze the errors in an OQEC code, we introduce a measure of the fidelity between the encoded information in two states for the case of subsystem encoding. A major part of the chapter concerns the definition of the measure and the derivation of its properties. In contrast to what was previously believed, we obtain that more restrictive conditions are not necessary neither for DFSs nor for general OQEC codes.
This is because the effective noise that can arise inside the code as a result of imperfect initialization is such that it can only increase the fidelity of an imperfectly encoded state with a perfectly encoded one.
In Chapter 8, we present a scheme for fault-tolerant holonomic computation on stabilizer codes [121]. Chapter 9 summarizes the results and discusses problems for future research.

12
Chapter 2: Generating quantum measurements using weak measurements

Preliminaries
In the original formulation of measurement in quantum mechanics, measurement outcomes are identified with a set of orthogonal projection operators, which can be thought of as corresponding to the eigenspaces of a Hermitian operator, or observable [165,108].
After a measurement, the state is projected into one of the subspaces with a probability given by the square of the amplitude of the state's component in that subspace.
In recent years a more general notion of measurement has become common: the so called generalized measurement which corresponds to a positive operator valued measure (POVM) [93]. This formulation can include many phenomena not captured by projective measurements: detectors with non-unit efficiency, measurement outcomes that include additional randomness, measurements that give incomplete information, and many others. Generalized measurements have found numerous applications in the rapidly-growing field of quantum information processing [115]. Some examples include protocols for unambiguous state discrimination [126] and optimal entanglement manipulation [114,76].
Upon measurement, a system with density matrix ρ undergoes a random transformation ρ → ρ j = M j ρM † j /p j , with probability p j = Tr(M j ρM † j ), where the index j labels the possible outcomes of the measurement. Eq. (1) is not the most general stochastic operation that can be applied to a state. For example, one can consider the transformation where p j = Tr( i M ij ρM † ij ) is the probability for the j th outcome (see Chapter 3).
The letter can be thought of as resulting from a measurement of the type (1) with measurement operators M ij of which only the information about the index j labeling the outcome is retained. In this thesis, when we talk about generalized measurements, we will refer to the transformation (1).
The transformation (1)  In this chapter we show that any generalized measurement can be implemented as a sequence of weak measurements and present an explicit form of the decomposition.
The main result was first presented in Ref. [118]. We call a measurement weak if all outcomes result in very small changes to the state. (There are other definitions of weak measurements that include the possibility of large changes to the state with low probability; we will not be considering measurements of this type.) Therefore, a weak measurement is one whose operators can be written as where q j ∈ C, 0 ≤ |q j | ≤ 1, and ε is an operator with small norm ε ≪ 1.

Decomposing projective measurements
It has been shown that any projective measurement can be implemented as a sequence of weak measurements; and by using an additional ancilla system and a joint unitary transformation, it is possible to implement any generalized measurement using weak measurements [21]. This procedure, however, does not decompose the operation on the original system into weak operations, since it uses operations acting on a larger Hilbert space-that of the system plus the ancilla. If we wish to study the behavior of a function-for instance, an entanglement monotone-defined on a space of a particular dimension, it complicates matters to add and remove ancillas. We will show that an ancilla is not needed, and give an explicit construction of the weak measurement operators for any generalized measurement that we wish to decompose.
It is easy to show that a measurement with any number of outcomes can be performed as a sequence of measurements with two outcomes. Therefore, for simplicity, we will restrict our considerations to two-outcome measurements. To give the idea of the construction, we first show how every projective measurement can be implemented as a sequence of weak generalized measurements. In this case the measurement operators P 1 and P 2 are orthogonal projectors whose sum P 1 + P 2 = I is the identity. We introduce the operators Note that P 2 (x)+P 2 (−x) = I and therefore P (x) and P (−x) describe a measurement.
If x = ǫ, where |ǫ| ≪ 1, the measurement is weak. Consider the effect of the operators P (x) on a pure state |ψ . The state can be written as |ψ = P 1 |ψ +P 2 |ψ = √ p 1 |ψ 1 + √ p 2 |ψ 2 , where |ψ 1,2 = P 1,2 |ψ / √ p 1,2 are the two possible outcomes of the projective measurement and p 1,2 = ψ|P 1,2 |ψ are the corresponding probabilities. If x is positive (negative), the operator P (x) increases (decreases) the ratio √ p 2 / √ p 1 of the |ψ 2 and |ψ 1 components of the state. By applying the same operator P (ǫ) many times in a row for some fixed ǫ, the ratio can be made arbitrarily large or small depending on the sign of ǫ, and hence the state can be transformed arbitrarily close to |ψ 1 or |ψ 2 .
The ratio of the p 1 and p 2 is the only parameter needed to describe the state, since Also note that P (−x)P (x) = (1 − tanh 2 (x)) 1/2 I/2 is proportional to the identity.
If we apply the same measurement P (±ǫ) twice and two opposite outcomes occur, the system returns to its previous state. Thus we see that the transformation of the state under many repetitions of the measurement P (±ǫ) follows a random walk along a curve |ψ(x) in state space. The position on this curve can be parameterized by x = ln p 1 /p 2 . Then |ψ(x) can be written as p 1 (x)|ψ 1 + p 2 (x)|ψ 2 , where The measurement given by the operators P (±ǫ) changes x by x → x ± ǫ, with probabilities p ± (x) = (1 ± tanh(ǫ)(p 1 (x) − p 2 (x)))/2. We continue this random walk until |x| ≥ X, for some X which is sufficiently large that |ψ(X) ≈ |ψ 1 and |ψ(−X) ≈ |ψ 2 to whatever precision we desire. What are the respective probabilities of these two outcomes? 16 Define p(x) to be the probability that the walk will end at X (rather than −X) given that it began at x. This must satisfy p( Substituting our expressions for the probabilities, this becomes If we go to the infinitesimal limit ǫ → dx, this becomes a continuous differential equation with boundary conditions p(X) = 1, p(−X) = 0. The solution to this equation is In the limit where X is large, tanh(X) → 1, so The probabilities of the outcomes for the sequence of weak measurements are exactly the same as those for a single projective measurement. Note that this is also true for a walk with a step size that is not infinitesimal, since the solution p(x) satisfies (5) for an arbitrarily large ǫ.
Alternatively, instead of looking at the state of the system during the process, we could look at an operator that effectively describes the system's transformation to the current state. This has the advantage that it is state-independent, and will lead the way to decompositions of generalized measurements; it also becomes obvious that the procedure works for mixed states, too.
We think of the measurement process as a random walk along a curve P (x) in operator space, given by Eq. (4), which satisfies It can be verified that P (x)P (y) ∝ P (x + y), where the constant of proportionality is (cosh(x + y)/2 cosh(x) cosh(y)) 1/2 . Due to normalization of the state, operators which differ by an overall factor are equivalent in their effects on the state. Thus, the random walk driven by weak measurement operators P (±ǫ) has a step size |ǫ|.

Decomposing generalized measurements
Next we consider measurements where the measurement operators M 1 and M 2 are positive but not projectors. We use the well known fact that a generalized measurement can be implemented as joint unitary operation on the system and an ancilla, followed by a projective measurement on the ancilla [115]. (One can think of this as an indirect measurement; one lets the system interact with the ancilla, and then measures the ancilla.) Later we will show that the ancilla is not needed. We consider two-outcome measurements and two-level ancillas. In this case M 1 and M 2 commute, and hence can be simultaneously diagonalized.
Let the system and ancilla initially be in a state ρ ⊗ |0 0|. Consider the unitary operation where X = σ x and Z = σ z are Pauli matrices acting on the ancilla bit. By applying U (0) to the extended system we transform it to: Then a projective measurement on the ancilla in the computational basis would yield one of the possible generalized measurement outcomes for the system. We can perform the projective measurement on the ancilla as a sequence of weak measurements by the procedure we described earlier. We will then prove that for this process, there exists a corresponding sequence of generalized measurements with the same effect acting solely on the system. To prove this, we first show that at any stage of the measurement process, the state of the extended system can be transformed into the form ρ(x)⊗|0 0| is the main result of this chapter. From the construction above, one can find the explicit form of the weak measurement operators: These expressions can be simplified further. The current state of the system at any point during the procedure can be written as where The weak measurement operators can be written as where the weights C ± are chosen to ensure that these operators form a generalized measurement: Note that this procedure works even if the step of the random walk is not small, since P (x)P (y) ∝ P (x + y) for arbitrary values of x and y. So it is not surprising that the effective operator which gives the state of the system at the point In the limit when ǫ → 0, the evolution under the described procedure can be described by a continuous stochastic equation. We can introduce a time step δt and a rate γ = ǫ 2 /δt.
Then we can define a mean-zero Wiener process δW as follows: where M [δx] is the mean of δx, The probabilities p ± (x) can be written in the form where Q(x) denotes the expectation value of the operator Note that M [(δW ) 2 ] = δt+O(δt 2 ). Expanding the change of a state |ψ upon the measurement M (x, ±ǫ) up to second order in δW and taking the limit δW → 0 averaging over many steps, we obtain the following coupled stochastic differential equations: This process corresponds to a continuous measurement of an observable Q which is continuously changed depending on the value of x. In other words, it is a feedbackcontrol scheme where depending on the measurement record, the type of measurement is continuously adjusted.

22
Finally, consider the most general type of two-outcome generalized measurement, with the only restriction being M † 1 M 1 + M † 2 M 2 = I. By polar decomposition the measurement operators can be written where V 1,2 are appropriate unitary operators. One can think of these unitaries as causing an additional disturbance to the state of the system, in addition to the reduction due to the measurement. The operators (M † 1,2 M 1,2 ) 1/2 are positive, and they form a measurement. We could then measure M 1 and M 2 by first measuring these positive operators by a sequence of weak measurements, and then performing either V 1 or V 2 , depending on the outcome.
However, we can also decompose this measurement directly into a sequence of weak measurements. Let the weak measurement operators for (M † 1,2 M 1,2 ) 1/2 be M p (x, ±ǫ).
Let V (x) be any continuous unitary operator function satisfying V (0) = I and V (±x) → V 1,2 as x → ∞. We then define By construction M (x, ±y) are measurement operators. Since V (x) is continuous, if y = ǫ, where ǫ ≪ 1, the measurements are weak. The measurement procedure is analogous to the previous cases and follows a random walk along the curve In summary, we have shown that for every two-outcome measurement described by operators M 1 and M 2 acting on a Hilbert space of dimension d, there exists a 23 continuous two-parameter family of operators M (x, y) over the same Hilbert space with the following properties: We have presented an explicit solution for M (x, y) in terms of M 1 and M 2 . The measurement is implemented as a random walk on the curve M (0, x) by consecutive application of the measurements M (x, ±ǫ), which depend on the current value of the parameter x. In the case where |ǫ| ≪ 1, the measurements driving the random walk are weak. Since any measurement can be decomposed into two-outcome measurements, weak measurements are universal.

Measurements with multiple outcomes
Even though two-outcome measurements can be used to construct any multi-outcome measurement, it is interesting whether a direct decomposition similar to the one we presented can be obtained for measurements with multiple outcomes as well. In Ref. [159] it was shown that such a decomposition exists. For a measurement with n positive operators L j , j = 1, ..., n, n j=1 L 2 j = I, the effective measurement operator M (x) describing the state during the procedure is given by [159] M where Here the parameter s is chosen such that n j=1 s j = 1, s ∈ [0, 1], i.e., it describes a simplex. The system of stochastic equations describing the process in the case when the measurement operators L j are commuting, can be written as where a(s) is the square root of g(s), and we have assumed Einstein's summation convention.
The decomposition can be easily generalized to the case of non-positive measurement operators in a way similar to the one we described for the two-outcome case-by inserting suitable weak unitaries between the weak measurements.

Summary and outlook
The The results of Bennett et al. [17,20,22] and Nielsen [114], among many others [160,76,72,77,162], have given us a nearly complete theory of entanglement for bipartite systems in pure states. Unfortunately, great difficulties have been encountered in trying to extend these results both to mixed states and to states with more than two subsystems (multipartite systems). The reasons for this are many; but one reason is that the set LOCC is complicated and difficult to describe mathematically [21].
One mathematical tool which has proven very useful is that of the entanglement monotone: a function of the state which is invariant under local unitary transformations and always decreases (or increases) on average after any local operation. These functions were described by Vidal [161], and large classes of them have been enumerated since then.
We will consider those protocols in LOCC that preserve pure states as the set of operations generated by infinitesimal local operations: operations which can be performed locally and which leave the state little changed including infinitesimal local unitaries and weak generalized measurements. In Bennett et al. [21] it was shown that infinitesimal local operations can be used to perform any local operation with the additional use of local ancillary systems-extra systems residing in the local laboratories, which can be coupled to the subsystems for a time and later discarded. As we saw in the previous section, any local generalized measurement can be implemented as a sequence of weak measurements without the use of ancillas. This implies that a necessary and sufficient condition for a function of the state to be a monotone under local operations that preserve pure states is the function to be a monotone under infinitesimal local operations.
In this chapter we derive differential conditions for a function of the state to be an entanglement monotone by considering the change of the function on average under infinitesimal local operations up to the lowest order in the infinitesimal parameter.
We thus obtain conditions that involve at most second derivatives of the function.
We then prove that these conditions are both necessary and sufficient. We show that the conditions are satisfied by a number of known entanglement monotones and we use them to construct a new polynomial entanglement monotone for three-qubit pure states.
We hope that this approach will provide a new window with which to study LOCC, and perhaps avoid some of the difficulties in the theory of multipartite and mixed-state entanglement. By looking only at the differential behavior of entanglement monotones, we avoid concerns about the global structure of LOCC or the class of separable operations.
In Section 3. In Section 3.5, we look at some known monotones-the norm of the state, the local purity, and the entropy of entanglement-and show that they obey the differential criteria. In Section 3.6, we use the differential conditions to construct a new polynomial entanglement monotone for three-qubit pure states which depends on the invariant identified by Kempe [82]. In Section 3.7 we conclude. In the Appendix (Section 3.8), we show that higher derivatives of the function are not needed to prove monotonicity. 30

LOCC
An operation (or protocol) in LOCC consists of a sequence of local operations with classical communication between them. Initially, we will consider only those local operations that preserve pure states: unitaries, in which the state is transformed and generalized measurements, in which the state randomly changes as in Eq. (1), where the index j labels the possible outcomes of the measurement. Note that we can think of a unitary as being a special case of a generalized measurement with only one possible outcome. One can think of this class of operations as being limited to those which do not discard information. Later, we will relax this assumption to consider general operations, which can take pure states to mixed states. Such operations do involve loss of information. Examples include performing a measurement without retaining the result, performing an unknown unitary chosen at random, or entangling the system with an ancilla which is subsequently discarded.
The requirement that an operation be local means that the operators U or M j must have a tensor-product structure U ≡ U ⊗I, M j ≡ M j ⊗I, where they act as the identity on all except one of the subsystems. The ability to use classical communication implies that the choice of later local operations can depend arbitrarily on the outcomes of all earlier measurements. One can think of an LOCC operation as consisting of a series of "rounds." In each round, a single local operation is performed by one of the local parties; if it is a measurement, the outcome is communicated to all parties, who then agree on the next local operation.

Entanglement monotones
For the purposes of this study, we define an entanglement monotone to be a realvalued function of the state with the following properties: if we start with the system in a state ρ and perform a local operation which leaves the system in one of the states ρ 1 , · · · , ρ n with probabilities p 1 , . . . , p n , then the value of the function must not increase on average: Furthermore, we can start with a state selected randomly from an ensemble {ρ k , p k }.
If we dismiss the information about which particular state we are given (which can be done locally), the function of the resultant state must not exceed the average of the function we would have if we keep this information: Some functions may obey a stronger form of monotonicity, in which the function cannot increase for any outcome: but this is not the most common situation. Some monotones may be defined only for pure states, or may only be monotonic for pure states. In the latter case, monotonicity is defined as non-increase on average under local operations that do not involve information loss.

Infinitesimal operations
We call an operation infinitesimal if all outcomes result in only very small changes to the state. That is, if after an operation the system can be left in states ρ 1 , · · · , ρ n , we must have For a unitary, this means that where ε is a Hermitian operator with small norm, ||ε|| ≪ 1, ε = ε † . For a generalized measurement, every measurement operator M j can be written as in Eq. (3), where 0 ≤ q j ≤ 1 and ε j is an operator with small norm ||ε j || ≪ 1.

Local operations from infinitesimal local operations
In this section we show how any local operation that preserves pure states can be performed as a sequence of infinitesimal local operations. The operations that preserve pure states are unitary transformations and generalized measurements.

Unitary transformations
Every local unitary operator has the representation where H is a local hermitian operator. We can write and define for a suitably large value of n. Thus, in the limit n → ∞, any local unitary operation can be thought of as an infinite sequence of infinitesimal local unitary operations driven by operators of the form where ε is a small ( ε ≪ 1) local hermitian operator.

Generalized measurements
As was shown in Chapter 2, any measurement can be generated by a sequence of weak measurements. Since a measurement with any number of outcomes can be implemented as a sequence of two-outcome measurements, it suffices to consider generalized measurements with two outcomes. The form of the weak operators needed to generate any measurement (Eq. (20)) is From these expressions it is easy to see that if |ǫ| ≪ 1, we have M (x, ǫ) = 1/2(I + O(ǫ)), i.e., the coefficients q j in Eq. (3) are q 1 = q 2 = 1 √ Clearly, the fact that infinitesimal local operations are part of the set of LO means that an entanglement monotone must be a monotone under infinitesimal local operations. The discussion in this section implies that if a function is a monotone under infinitesimal local unitaries and generalized measurements, it is a monotone under all local unitaries and generalized measurements (the operations that do not involve information loss and preserve pure states). Based on this result, in the next section we derive necessary and sufficient conditions for a function to be an entanglement monotone.

Differential conditions for entanglement monotones
Let us now consider the change in the state under an infinitesimal local operation.
Without loss of generality, we assume that the operation is performed on Alice's subsystem. In this case, it is convenient to write the density matrix of the system as where the set {|i A } and the set {|j BC... } are arbitrary orthonormal bases for subsystem A and the rest of the system, respectively. Any function of the state f (ρ) can be thought of as a function of the coefficients in the above decomposition:

Local unitary invariance
Unitary operations are invertible, and therefore the monotonicity condition reduces to an invariance condition for LU transformations. Under local unitary operations on subsystem A the components of ρ transform as follows: where U ik are the components of the local unitary operator in the basis {|i A }. We consider infinitesimal local unitary operations: where ε is a local hermitian operator acting on subsystem A, and Up to first order in ε the coefficients ρ ijlm transform as Requiring LU-invariance of f (ρ), we obtain that the function must satisfy Analogous equations must be satisfied for arbitrary hermitian operators ε acting on the other parties' subsystems. In a more compact form, the condition can be written as

36
where ε is an arbitrary local hermitian operator.

Non-increase under infinitesimal local measurements
As mentioned earlier, a measurement with any number of outcomes can be implemented as a sequence of measurements with two outcomes, and a general measurement can be done as a measurement with positive operators, followed by a unitary conditioned on the outcome; therefore, it suffices to impose the monotonicity condition for two-outcome measurements with positive measurement operators. Consider local measurements on subsystem A with two measurement outcomes, given by operators Without loss of generality, we assume where ε is again a small local hermitian operator acting on A (in the previous section we saw that any two-outcome measurement with positive operators can be generated by weak measurements of this type). Upon measurement, the state undergoes one of two possible transformations with probabilities p 1,2 = Tr M 1,2 2 ρ . Since ε is small, we can expand The condition for non-increase on average of the function f under infinitesimal local measurements is Expanding (64) in powers of ε up to second order, we obtain where {ε, ρ} is the anti-commutator of ε and ρ. The inequality must be satisfied for an arbitrary local hermitian operator ε.
So long as (65) is satisfied by a strict inequality, it is obvious that we need not consider higher-order terms in ε. But what about the case when the condition is satisfied by equality? In the appendix we will show that even in the case of equality, (65) is still the necessary and sufficient condition for monotonicity under local generalized measurements. There we also prove the sufficiency of the LU-invariance condition (59).
This allows us to state the following We point out that from the condition of LU invariance applied up to second-order in ε, one obtains Therefore, in the case when both Eq. (59) and Eq. (65) are satisfied, condition (65) can be written equivalently in the form Tr properties of the entropy of entanglement will be discussed further in Section 3.5).
One generalization of the entropy of entanglement to mixed states is the entanglement of formation [22], which is defined as the minimum of i p i S A (ρ i ) over all ensembles of bipartite pure states {ρ i , p i } realizing the mixed state: ρ = i p i ρ i . This quantity is a mixed-state entanglement monotone. As a function of ρ, it has a much more complicated form than the above expression for the entropy of entanglement. In fact, there is no known analytic expression for the entanglement of formation in general.
The problem of extending pure-state entanglement monotones to mixed states is an important one, since every mixed-state entanglement monotone can be thought of as an extension of a pure-state entanglement monotone. Note, however, that a purestate entanglement monotone may have many different mixed-state generalizations.
The relation between the entanglement of formation and the entropy of entanglement presents one way to perform such an extension (convex-roof extension).
By making the substitution ρ ijℓm = α ij α * ℓm into (59) and (65), we can (after considerable algebra) derive alternative forms of the differential conditions for functions of the state vector: i,j,k,l,m,n Here ε is a local hermitian operator acting on subsystem A. Analogous conditions must be satisfied for ε acting on the other parties' subsystems.

Monotonicity under operations with information loss
Besides monotonicity under local unitaries and generalized measurements, an entanglement monotone for mixed states should also satisfy monotonicity under local operations which involve loss of information. The most general transformation that involves loss of information has the form where is the probability for outcome k. The operators {M k,j } must satisfy We can see that this includes unitary transformations, generalized measurements, and completely positive trace-preserving maps as special cases.
It occasionally makes sense to consider even more general transformations, where the operators need not sum to the identity: This corresponds to a situation where only certain outcomes are retained, and others are discarded; the probabilities add up to less than 1 due to these discarded outcomes.
We say such a transformation involves postselection.
With or without postselection, we are concerned with the case where all operations are done locally, so that all the operators {M k,j } act on a single subsystem. Every such transformation can be implemented as a sequence of local generalized measurements (possibly discarding some of the outcomes) and local completely positive maps. In operator-sum representation [93], a completely positive map can be written Therefore, in addition to (59) and (65) we must impose the condition for all sets of local operators {M k } satisfying (76).
Suppose the parties are supplied with a state ρ k taken from an ensemble {ρ k , p k }.
Discarding the information of the actual state amounts to the transformation As pointed out in [161], discarding information should not increase the entanglement of the system on average. Therefore, for any ensemble {ρ k , p k }, an entanglement monotone on mixed states should be convex: Condition (79), together with condition (65) for monotonicity under local generalized measurements, implies monotonicity under local completely positive maps: It is easy to see that if this inequality holds without postselection, it must also hold with postselection.
It follows that a function of the density matrix is an entanglement monotone for mixed states if and only if it is (1) a convex function on the set of density matrices and (2) a monotone under local unitaries and generalized measurements. Fortunately, there are also simple differential conditions for convexity. A necessary and sufficient condition for a twice-differentiable function of multiple variables to be convex on a convex set is that its Hessian matrix be positive on the interior of the convex set (in this case, the set of density matrices). Therefore, in addition to (59) and (65) we add the differential condition Tr which must be satisfied at every ρ on the interior of the set of density matrices for an arbitrary traceless hermitian matrix σ.

Examples
In this section we demonstrate how conditions (59), (65) and (81) can be used to verify if a function is an entanglement monotone. We show this for three well known entanglement monotones: the norm of the state of the system, the trace of the square of the reduced density matrix of any subsystem, and the entropy of entanglement. In the next section we will use some of the observations made here to construct a new polynomial entanglement monotone for three-qubit pure states. 43

Norm of the state
The most trivial example is the norm or the trace of the density matrix of the system: Clearly I 1 is a monotone under LOCC, since all operations that we consider either preserve or decrease the trace. But for the purpose of demonstration, let us verify that I 1 satisfies the differential conditions.
The LU-invariance condition (59) reads Tr The second equality follows from the cyclic invariance of the trace.
Since the trace is linear, the second term in condition (65) vanishes, and we consider only the first term: Tr The condition is satisfied with equality, again due to the cyclic invariance of the trace, implying that the norm remains invariant under local measurements. The convexity condition (81) is also satisfied by equality.

Local purity
The second example is the purity of the reduced density matrix: where ρ A is the reduced density matrix of subsystem A (which in general need not be a one-party subsystem). Note that this is an increasing entanglement monotone for pure states-the purity of the local reduced density matrix can only increase under LOCC.
It has been shown in [33] that every m-th degree polynomial of the components of the density matrix ρ can be written as an expectation value of an observable O on m copies of ρ: Here we have where the components of C are Therefore Tr where by O A we denote the partial trace of an operator O over all subsystems except A.
If ε does not act on subsystem A, then [ε, ρ] A = 0 and the above expression vanishes.
If ε acts on A, we can use the fact that for pure states where B denotes the subsystem complementary to A. Then we can apply the same argument as before for the function Tr ρ 2 B . Therefore I 2 does not decrease on average under local generalized measurements, and is an entanglement monotone for pure states.
What about mixed states? For increasing entanglement monotones the convexity condition (81) becomes a concavity condition-the direction of the inequality is inverted. In the case of I 2 , however, we have Tr i.e., the function is convex. This means that Tr{ρ 2 A } is not a good measure of entanglement for mixed states. Indeed, when extended to mixed states, I 2 cannot distinguish between entanglement and classical disorder. 46

Entropy of entanglement
Finally consider the von Neumann entropy of entanglement: Expanding around ρ A = I, we get The LU-invariance follows from the fact that every term in this expansion satisfies (59). If we substitute the n-th term in the condition, we obtain Consider the case when ε does not act on A. Substituting S A in (65), we get If ρ −1 A does not exist, it is only on a subset of measure zero-where one or more of the eigenvalues of ρ A vanish. Therefore, we can always find an arbitrarily close vicinity in the parameters describing ρ A , where ρ −1 A is regular and where (65) is satisfied. Since the condition is continuous, it cannot be violated on this special subset.
If ε acts on A, we can use an equivalent definition of the entropy of entanglement: and apply the same arguments. Therefore S A is an entanglement monotone for pure states.
The convexity condition is not satisfied, since This reflects the fact that the entropy of entanglement, like I 2 , does not distinguish between entanglement and classical randomness. 48

A new entanglement monotone
It has been shown [63] that the set of all entanglement monotones for a multipartite pure state uniquely determine the orbit of the state under the action of the group of local unitary transformations. For three-qubit pure states the orbit is uniquely determined by 5 independent continuous invariants (not counting the norm) and one discrete invariant [1,44]. Therefore, for pure states of three qubits there must exist five independent continuous entanglement monotones that are functions of the five independent continuous invariants.
Any polynomial invariant in the amplitudes of a state is a sum of homogenous polynomials of the form [155] P στ ··· (|ψ ) = α i 1 j 1 k 1 ... α * i 1 j σ(1) k τ (1) ... · · · α injnkn... α * inj σ(n) k τ (n) ... , where σ, τ, . . . are permutations of (1,2,. . . ,n), and repeated indices indicate summation. A set of five independent polynomial invariants for three-qubit pure states is [155] I 1 = P e, (12) In the last expression ǫ ij is the antisymmetric tensor in 2 dimensions. The first three invariants are the local purities of subsystems C, B and A, I 4 is the invariant identified by Kempe [82] and I 5 is (up to a factor) the square of the 3-tangle identified by Coffman, Kundu and Wootters [45]. According to [63] the four known independent continuous entanglement monotones that do not require maximization over a multi-dimensional space are and any fifth independent entanglement monotone must depend on I 4 . Numerical evidence suggested that the tenth order polynomial σ ABC = 3 − (I 1 + I 2 + I 3 )I 4 might be such an entanglement monotone. However, no rigorous proof of monotonicity was given. Here, we will use conditions (59) and (65) to construct a different independent entanglement monotone, which is of sixth order in the amplitudes of the state and their complex conjugates.
Observe that in (101) (65), since when ε acts on subsystem A, we can again consider the expression for the function only in terms of ρ BC... and the first term would vanish according to (90).
We will aim at constructing a polynomial function of three-qubit pure states ρ which has the same form when expressed in terms of ρ AB , ρ AC , or ρ BC , in order to avoid the necessity for separate proofs of monotonicity under measurements on the different subsystems. It has been shown in [155] that For local measurements on subsystem C it is convenient to use the first of the above expressions for I 4 . The terms Tr ρ 3 A and Tr ρ 3 B are entanglement monotones by themselves. This can be easily seen by plugging them in condition (65): These terms, however, are not independent of the invariants I 2 and I 3 . The term which is independent of the other polynomial invariants is Tr {ρ AB (ρ A ⊗ ρ B )}. When we plug this term into condition (65) we obtain an expression which is not manifestly positive or negative. Is it possible to construct a function dependent on this term, which similarly to Tr ρ 3

A,B
would yield a trace of a manifestly positive operator when substituted in (65)?
It is easy to see that if the function has the form Tr X 3 , where the operator X(ρ AB ) is a positive operator linearly dependent on ρ AB , it will be an increasing monotone under local measurements on C (for simplicity we assume X(0) = 0): Since we want the function to depend on Tr . This is clearly positive for positive ρ AB . Expanding the trace, we obtain: One can show that Tr We also have that Tr ρ 3 AB = Tr ρ 3 C . Using this and (111), we obtain

52
This expression is an increasing monotone under local measurements on C. If we add to it 3Tr ρ 2 AB = 3Tr ρ 2 C , it becomes invariant under permutations of the subsystems.
Since Tr ρ 2 C is an increasing entanglement monotone, the whole expression will be a monotone under operations on any subsystem. We can define the closely related This is a decreasing entanglement monotone that vanishes for product states, which is more standard for a measure of entanglement. It depends on the invariant identified by Kempe and is therefore independent of the other known monotones for three-qubit pure states.

Summary and outlook
We have derived differential conditions for a twice-differentiable function on quantum states to be an entanglement monotone. There are two such conditions for purestate entanglement monotones-invariance under local unitaries and diminishing under local measurements-plus a third condition (overall convexity of the function) for mixed-state entanglement monotones. We have shown that these conditions are both necessary and sufficient. We then verified that the conditions are satisfied by a number of known entanglement monotones and we used them to construct a new polynomial entanglement monotone for three-qubit pure states.
It is our hope that this approach to the study of entanglement may circumvent some of the difficulties that arise due the mathematically complicated nature of LOCC. It may be possible to find new classes of entanglement monotones, for both pure and mixed states, and to look for functions with particularly desirable properties (such as additivity).

53
There may also be other areas of quantum information theory where it will prove advantageous to consider general quantum operations as continuous processes. This seems a very promising new direction for research.

Appendix: Proof of sufficiency
The LU-invariance condition can be written as where we define with ε being a local hermitian operator. This condition has to be satisfied for every ρ and every ε. By expanding up to first order in ε we obtained condition (59), which is equivalent to This is a linear form of the components of ε and the requirement that it vanishes for every ε implies that ∂F (ρ, ε) ∂ε ij ε=0 = 0.
This has to be satisfied for every ρ. Consider the first derivative of F (ρ, ε) with respect to ε ij , taken at an arbitrary point ε 0 . We have But from the form of F (ρ, ε) one can see that i.e., the first derivatives of F (ρ, ε) with respect to the components of ε vanish identically. This means that F (ρ, ε) = F (ρ, 0) = 0 for every ε and condition (59) is sufficient.
The condition for non-increase on average under local generalized measurements (64) can be written as where The operators M 1 and M 2 in terms of ε are given by (60), and the probabilities p 1 and p 2 are defined as before. As we have argued in Section 3.3, it is sufficient that this condition is satisfied for infinitesimal ε. By expanding the condition up to second order in ε we obtained condition (65), which is equivalent to Clearly, if this condition is satisfied by a strict inequality, it is sufficient, since corrections of higher order in ε can be made arbitrarily smaller in magnitude by taking ε We will now show that if (128)  Consider the two terms of G(ρ, ε) that depend on ε: They differ only by the sign of ε, i.e. G 1 (ρ, ε) = G 2 (ρ, −ε), and therefore If (128) is satisfied in an open vicinity of ρ, we have in this vicinity. Consider the second derivatives of G(ρ, ε) with respect to the components of ε, taken at a point ε 0 : From the expression for G 1 (ρ, ε) one can see that ε occurs in G 1 (ρ, ε) only in the where So we can write where and Then one can verify that Similarly where Note that ∂ε ′ pq /∂ε ij and ∂ε ′′ pq /∂ε ij have no dependence on ε. Nor do p ′ and p ′′ . Therefore we obtain We assumed that (132) is satisfied in an open vicinity of ρ. If ρ ′ and ρ ′′ are within this vicinity, the above expression will vanish. But from (137) and (142) we see that as ε 0 tends to zero, the quantities ρ ′ − ρ and ρ ′′ − ρ also tend to zero. Therefore there exists an open vicinity of ε 0 = 0, such that for every ε 0 in this vicinity, the corresponding ρ ′ and ρ ′′ will be within the vicinity of ρ for which (132) is satisfied and This means that higher derivatives of G(ρ, ε) with respect to the components of ε taken at points in this vicinity will vanish, in particular derivatives taken at ε = 0. So higher order corrections in ε to G(ρ, ε) will also vanish. Therefore G(ρ, ε) = 0 in the vicinity of ρ for which we assumed that (65) is satisfied with equality, which implies that condition (65)

Preliminaries
As we pointed out in Chapter 1, a major conceptual as well as technical difficulty in the practical implementation of quantum information processing schemes is the unavoidable interaction of quantum systems with their environment. This interaction can destroy quantum superpositions and lead to an irreversible loss of information, a process known as decoherence. Understanding the dynamics of open quantum systems is therefore of considerable importance. The Schrödinger equation, which describes the evolution of closed systems, is generally inapplicable to open systems, unless one includes the environment in the description. This is, however, generally difficult, due to the large number of environment degrees of freedom. An alternative is to develop a description for the evolution of only the subsystem of interest. A multitude of different approaches have been developed in this direction, exact as well as approximate [8,30].
Typically the exact approaches are of limited practical usefulness as they are either phenomenological or involve complicated integro-differential equations. The various 59 approximations lead to regions of validity that have some overlap. Such techniques have been studied for many different models, but their performance in general, is not fully understood.
In this work we consider an exactly solvable model of a single qubit coupled to an environment of qubits. We are motivated by the physical importance of such spin bath models [132] in the description of decoherence in solid state quantum information processors, such as systems based on the nuclear spin of donors in semiconductors [81,166], or on the electron spin in quantum dots [107]. Rather than trying to accurately model decoherence due to the spin bath in such systems (as in, e.g., Refs. [149,170]), our goal in this work is to compare the performance of different master equations which have been proposed in the literature. Because the model we consider is exactly solvable, we are able to accurately assess the performance of the approximation techniques that we study. In particular, we study the Born-Markov and Born master equations, and the perturbation expansions of the Nakajima-Zwanzig (NZ) [112,181] and the timeconvolutionless (TCL) master equations [143,142] up to fourth order in the coupling constant. We also study the post-Markovian (PM) master equation proposed in [139].
The dynamics of the system qubit in the model we study is highly non-Markovian and hence we do not expect the traditional Markovian master equations commonly used, e.g., in quantum optics [42] and nuclear magnetic resonance [147], to be accurate. This is typical of spin baths, and was noted, e.g., by Breuer et al. [29]. As we will see in Chapter 5, the non-Markovian character of the dynamics can be used to our advantage in error-correction schemes, hence understanding these models is of special significance.

The work by Breuer et al. (as well as by other authors in a number of subsequent
publications [124,36,71,175,40,75]) is conceptually close to ours in that in both cases an analytically solvable spin-bath model is considered and the analytical solution for the open system dynamics is compared to approximations. However, there are also important differences, namely, in Ref. [29] a so-called spin-star system was studied, where the system spin has equal couplings to all the bath spins, and these are of the XY exchange-type. In contrast, in our model the system spin interacts via Ising couplings with the bath spins, and we allow for arbitrary coupling constants. As a result there are also important differences in the dynamics. For example, unlike the model in Ref.
[29], for our model we find that the odd order terms in the perturbation expansions of Nakajima-Zwanzig and time-convolutionless master equations are non-vanishing. This reflects the fact that there is a coupling between the x and y components of the Bloch vector which is absent in [29]. In view of the non-Markovian behavior of our model, we also discuss the relation between a representation of the analytical solution of our model in terms of completely positive maps, and the Markovian limit obtained via a coarse-graining method introduced in [104], and the performance of the post-Markovian master equation [139].
This chapter is organized as follows. In Section 4.2, we present the model, derive the exact solution and discuss its behavior in the limit of small times and large number of bath spins, and in the cases of discontinuous spectral density co-domain and alternating sign of the system-bath coupling constants. In

Exact dynamics 4.2.1 The model
We consider a single spin-1 2 system (i.e., a qubit with a two-dimensional Hilbert space H S ) interacting with a bath of N spin-1 2 particles (described by an N -fold tensor product of two-dimensional Hilbert spaces denoted H B ). The observables describing the spin of a spin-1 2 particle in each of the three spatial directions are described by the Pauli operators We model the interaction between the system qubit and the bath by the Ising Hamil- where g n are dimensionless real-valued coupling constants in the interval [−1, 1] (n labels the different qubits in the bath), and α > 0 is a parameter having the dimension of frequency (we work in units in whichh = 1), which describes the coupling strength and will be used below in conjunction with time (αt) for perturbation expansions.
The system and bath Hamiltonians are and For definiteness, we restrict the frequencies ω 0 and Ω n to the interval [−1, 1], in inverse time units. Even though the units of time can be arbitrary, by doing so we do not lose generality, since we will be working in the interaction picture where only the frequencies Ω n appear in relation to the state of the bath [Eq. (158)]. Since the ratios of these frequencies and the temperature of the bath occur in the equations, only their values relative to the temperature are of interest. Therefore, henceforth we will omit the units of frequency and temperature and will treat these quantities as dimensionless.
The interaction picture is defined as the transformation of any operator where does not alter the present analysis.] All the quantities discussed in the rest of this article are assumed to be in the interaction picture.
The dynamics can be described using the superoperator notation for the Liouville operator where ρ(t) is the density matrix for the total system in the Hilbert space The dynamics is governed by the von Neumann equation and the formal solution of this equation can be written as follows: The state of the system is given by the reduced density operator where Tr B denotes a partial trace taken over the bath Hilbert space H B . This can also be written in terms of the Bloch sphere vector where σ ≡ (σ x , σ y , σ z ) is the vector of Pauli matrices. In the basis of σ z eigenstates this is equivalent to We assume that the initial state is a product state, i.e., and that the bath is initially in the Gibbs thermal state at a temperature T where k is the Boltzmann constant. Since ρ B commutes with the interaction Hamiltonian H I , the bath state is stationary throughout the dynamics: ρ B (t) = ρ B . Finally, the bath spectral density function is defined as usual as

Exact solution for the evolution of the system qubit
We first shift the system Hamiltonian in the following way: As a consequence the interaction Hamiltonian is modified from Eq. (147) to where This shift is performed because now Tr B [H I , ρ(0)] = 0, or equivalently This property will simplify our calculations later when we consider approximation techniques in Section 4.3. Now, we derive the exact solution for the reduced density operator ρ S corresponding to the system. We do this in two different ways. The Kraus operator sum representation is a standard description of the dynamics of a system initially decoupled from its environment and it will also be helpful in studying the coarse-graining approach to the quantum semigroup master equation. The second method is computationally more effective and is helpful in obtaining analytical expressions for N ≫ 1.

Exact Solution in the Kraus Representation
In the Kraus representation the system state at any given time can be written as where the Kraus operators satisfy ij K ij (t) † K ij (t) = I S [93]. These operators can be expressed easily in the eigenbasis of the initial state of the bath density operator as where the bath density operator at the initial time is ρ B (0) = i λ i |i i|. For the Gibbs thermal state chosen here, the eigenbasis is the N -fold tensor product of the σ z basis. In this basis where β = 1/kT . Here is the energy of each eigenstate |l , where l = l 1 l 2 . . . l n is the binary expansion of the integer l, and the partition function is Z = l exp(−βE l ). Therefore, the Kraus operators become and λ i = exp(−βE i )/Z. Substituting this expression for K ij into Eq. (164) and writing the system state in the Bloch vector form given in Eq.
The equations (170) are the exact solution to the system dynamics of the above spin bath model. We see that the evolution of the Bloch vector is a linear combination of rotations around the z axis. This evolution reflects the symmetry of the interaction Hamiltonian which is diagonal in the z basis. By inverting Eqs. (170) for v x (0) or v y (0), we see that the Kraus map is irreversible when C(t) 2 + S(t) 2 = 0. This will become important below, when we discuss the validity of the time-convolutionless approximation.

Alternative Exact Solution
Another way to derive the exact solution which is computationally more useful is the following. Since all σ z n commute, the initial bath density matrix factors and can be written as where and −1 ≤ β n ≤ 1. Using this, we obtain an expression for θ defined in Eq. (160) The evolution of the system density matrix in the interaction picture is In terms of the system density matrix elements in the computational basis {|0 , |1 } (which is an eigenbasis of σ z in H I = ασ z ⊗ B), we have ρ m e +iα k|σ z |k Bt }.
Let us substitute j|σ z |j = (−1) j and rewrite Since all the matrices are diagonal, they commute and we can collect the terms by qubits:

68
Let us denote (−1) j − (−1) k = 2ǫ jk . The trace can be easily computed to be Thus the final expression for the system density matrix elements is Notice that ǫ 00 = ǫ 11 = 0, hence the diagonal matrix elements do not depend on time as before: For the off-diagonal matrix elements ǫ 01 = 1, ǫ 10 = −1, and the evolution is described by where In terms of the Bloch vector components, this can be written in the form of Eq. (170), where

Short Times
Consider the evolution for short times where αt ≪ 1. Then where (see Appendix A at the end of this chapter) Note that for the above approximation to be valid, we need 2(αt) 2 Q 2 ≪ 1. The total phase of f (t) in Eq. (177) is where we have used Eq. (174). Thus, the off-diagonal elements of the system density matrix become Finally, the dynamics of the Bloch vector components are: This represents the well known behavior [113] of the evolution of an open quantum system in the Zeno regime. In this regime coherence does not decay exponentially but is initially flat, as is the case here due to the vanishing time derivative of ρ 01 S (t) at t = 0. As we will see in Section 4.3, the dynamics in the Born approximation (which is also the second order time-convolutionless approximation) exactly matches the last result.

Large N
When N ≫ 1 and the values of g n are random, then the different terms in the product of Eq. (177) are smaller than 1 most of the time and have recurrences at different times. Therefore, we expect the function f (t) to be close to zero in magnitude for most of the time and full recurrences, if they exist, to be extremely rare. When g n are equal and so are Ω n , then partial recurrences occur periodically, independently of N .
Full recurrences occur with a period which grows at least as fast as N . This can be argued from Eq. (170) by imposing the condition that the arguments of all the cosines and sines are simultaneously equal to an integer multiple of 2π. When J(Ω) has a narrow high peak, e.g., one g n is much larger than the others, then the corresponding terms in the products in Eq. (177) oscillate faster than the rate at which the whole product decays. This is effectively a modulation of the decay.

Discontinuous spectral density co-domain
As can be seen from Eq. (177), the coupling constants g n determine the oscillation periods of the product terms, while the temperature factors β n determine their modulation depths. If the codomain of spectral density is not continuous, i.e. it can be split into non-overlapping intervals G j , j = 1, ..., J, then Eq. (177) can be represented in the following form: where In this case, if G j are separated by large enough gaps, the evolution rates of different P j (t) can be significantly different. This is particularly noticeable if one P j (t) undergoes partial recurrences while another P j ′ (t) slowly decays.
For example, one can envision a situation with two intervals such that one term shows frequent partial recurrences that slowly decay with time, while the other term decays faster, but at times larger than the recurrence time. The overall evolution then consists in a small number of fast partial recurrences. In an extreme case, when one g n is much larger then the others, this results in an infinite harmonic modulation of the decay with depth dependent on β n , i.e., on temperature.

Alternating signs
If the bath has the property that every bath qubit m has a pair −m with the same frequency Ω −m = Ω m , but opposite coupling constant g −m = −g m , the exact solution can be simplified. First, β −m = β m , and θ = 0. Next, Eq. (177) becomes This function is real, thus Eq. (178) becomes Eq. (179) remains unchanged, and since Q 2 depends on g 2 n , but not g n , it and all second order approximations also remain unchanged. In the special case |g m | = g, the exact solution exhibits full recurrences with period T = π/αg.

Approximation methods
In this section we discuss the performance of different approximation methods developed in the open quantum systems literature [8,30]. The corresponding master equations for the system density matrix can be derived explicitly and since the model considered here is exactly solvable, we can compare the approximations to the exact dynamics. We use the Bloch vector representation and since the z component has no dynamics, a fact which is reflected in all the master equations, we omit it from our comparisons.

Born and Born-Markov approximations
Both the Born and Born-Markov approximations are second order in the coupling strength α.

Born approximation
The Born approximation is equivalent to a truncation of the Nakajima-Zwanzig projection operator method at the second order, which is discussed in detail in Section 4.3.2. The Born approximation is given by the following integro-differential master Since in our case the interaction Hamiltonian is time-independent, the integral becomes easy to solve. We obtaiṅ where Q 2 is the second order bath correlation function in Eq. (180). Writing ρ S (t) in terms of Bloch vectors as (I + v · σ)/2 [Eq. (156)], we obtain the following integro- These equations can be solved by taking the Laplace transform of the variables.
The equations become where V x,y (s) is the Laplace transform of v x,y (t). This gives which can be readily solved by taking the inverse Laplace transform. Doing so, we obtain the solution of the Born master equation for our model: Note that this solution is symmetric under the interchange v x ↔ v y , but the exact dynamics in Eq. (170) does not have this symmetry. The exact dynamics respects the symmetry: v x → v y and v y → −v x , which is a symmetry of the Hamiltonian. This means that higher order corrections are required to break the symmetry v x ↔ v y in order to approximate the exact solution more closely.
One often makes the substitution v x,y (t) for v x,y (s) in Eq. (189) since the integrodifferential equation obtained in other models may not be as easily solvable. This approximation, which is valid for short times, yieldṡ i.e., we recover Eq. (183). This is the same solution obtained in the second order approximation using the time-convolutionless (TCL) projection method discussed in Section 4.3.2.

Born-Markov approximation
In order to obtain the Born-Markov approximation, we use the following quantities [30][Ch.3]: where and P E i is the projector onto the eigenspace corresponding to this eigenvalue. In our case H S is diagonal in the eigenbasis of σ z , and only ω = 0 is relevant. This leads Hence the Lamb shift Hamiltonian H L = 0, and the Lindblad form of the Born-Markov approximation isρ A different approach to the derivation of a Markovian semigroup master equation was proposed in [104]. In this approach, a Lindblad equation is derived from the Kraus operator-sum representation by a coarse-graining procedure defined in terms of a phenomenological coarse-graining time scale τ . The general form of the equation is: where the operators A 0 = I and A α , α = 1, ..., M form an arbitrary fixed operator basis in which the Kraus operators (164) can be expanded as The quantities χ α,β (t) and Q(t) are defined through and For our problem we find with C(t) and S(t) defined in Eq. (171). In order for this approximation to be justified, it is required that the coarse-graining time scale τ be much larger than any characteristic time scale of the bath [104]. However, in our case the bath correlation time is infinite which, once again, shows the inapplicability of the Markovian approximation.
This is further supported by the performance of the optimal solution that one can achieve by varying τ , which is discussed in Section 4.4. There we numerically examine the average trace-distance between the solution to Eq. (202) and the exact solution as a function of τ . The average is taken over a time T , which is greater than the decay time of the exact solution. We determine an optimal τ for which the average trace distance is minimum and then determine the approximate solution. The solution of Eq. (202) for a particular τ in terms of the Bloch vector components is . The average trace distance as a function of τ is given by, where ρ CG represents the coarse-grained solution and where |X| = √ X † X. The results are presented in Section 4.4. Next we consider the Nakajima-Zwanzig (NZ) and the time-convolutionless (TCL) master equations for higher order approximations.

NZ and TCL master equations
Using projection operators one can obtain approximate non-Markovian master equations to higher orders in αt. A projection is defined as follows, and serves to focus on the "relevant dynamics" (of the system) by removing the bath (a recent generalization is discussed in Ref. [28]). The choice of ρ B is somewhat arbitrary and can be taken to be ρ B (0) which significantly simplifies the calculations. Using the notation introduced in [29], define for any superoperator S. Thus S n denote the moments of the superoperator. Note that for the Liouvillian superoperator, L = 0 by virtue of the fact that Tr B {Bρ B (0)} = 0 (see [30]). Since we assume that the initial state is a product state, both the NZ and TCL equations are homogeneous equations. The NZ master equation is an integrodifferential equation with a memory kernel N (t, s) and is given bẏ The TCL master equation is a time-local equation given bẏ When these equations are expanded in αt and solved we obtain the higher order corrections. When the interaction Hamiltonian is time independent (as in our case), the above equations simplify to and 79 for the NZ and TCL equations, respectively, where the time-ordered integral operator I n (t, s) is defined as The definitions of the partial cumulants L pc and the ordered cumulants L oc are given in Refs. [142,134,80]. For our model we have and Explicit expressions for these quantities are given in Appendix at the end of this chapter. Substituting these into the NZ and TCL equations (211) and (212), we obtain what we refer to below as the NZn and TCLn master equations, with n = 2, 3, 4. These approximate master equations are, respectively, second, third and fourth order in the coupling constant α, and they can be solved analytically. The second order solution 80 of the NZ equation (NZ2) is exactly the Born approximation and the solution is given in Eq. (192). The third order NZ master equation is given bẏ and the fourth order iṡ These equations are equivalent to, respectively, 6th and 8th order differential equations (with constant coefficients) and are difficult to solve analytically. The results we present in the next section were therefore obtained numerically.
The situation is simpler in the TCL approach. The second order TCL equation is given byρ whose solution is as given in Eq. (194) in terms of Bloch vector components. For and for TCL4 we finḋ These equation can be solved analytically, and the solutions to the third and fourth order TCL equations are given by where It is interesting to note that the second order expansions of the TCL and NZ master equations exhibit a v x ↔ v y symmetry between the components of the Bloch vector, and only the third order correction breaks this symmetry. Notice that the coefficient of α 3 does not vanish in this model unlike in the one considered in [29] because both L 3 pc = 0 and L 3 oc = 0 and hence the third order (and other odd order) approximations exist.

Post-Markovian (PM) master equation
In this section we study the performance of the post-Markovian master equation recently proposed in [139]: This equation was constructed via an interpolation between the exact dynamics and the dynamics in the Markovian limit. The operator D is the dissipator in the Lindblad 82 equation (353), and k(t) is a phenomenological memory kernel which must be found by fitting to data or guessed on physical grounds. As was discussed earlier, the Markovian approximation fails for our model, nevertheless, one can use the form of the dissipator we obtained in Eq. (353) It is interesting to examine to what extent Eq. (222) can approximate the exact dynamics. As a measure of the performance of the post-Markovian equation, we will take the trace-distance between the exact solution ρ exact (t) and the solution to the post-Markovian equation ρ 1 (t). The general solution of Eq. (222) can be found by expressing ρ(t) in the damping basis [31] and applying a Laplace transform [139]. The solution is where } and the eigenvalues are {0, −2, −2, 0}. Therefore, we can immediately write the formal solution in terms of the Bloch vector components: We see that v x (t) has no dependence on v y (0), and neither does v y (t) on v x (0), in contrast to the exact solution. The difference comes from the fact that the dissipator D does not couple v x (t) and v y (t). This reveals an inherent limitation of the post-Markovian master equation: it inherits the symmetries of the Markovian dissipator D, which may differ from those of the generator of the exact dynamics. In order to rigorously determine the optimal performance, we use the trace distance between the exact solution and a solution to the post-Markovian equation: Obviously this quantity reaches its minimum for ξ(t) = C(t), ∀t independently of the initial conditions. The kernel for which the optimal performance of the post-Markovian master equation is achieved, can thus be formally expressed, using Eq. (226), as: It should be noted that the condition for complete positivity of the map generated by , amounts here to |ξ(t)| = |C(t)| ≤ 1, which holds for all t. Thus the minimum achievable trace-distance between the two solutions is given by The optimal fit is plotted in Section 4.4.
Finding a simple analytical expression for the optimal kernel Eq. (228) seems difficult due to the complicated form of C(t). One way to approach this problem is to expand C(t) in powers of αt and consider terms which give a valid approximation for small times αt ≪ 1. For example, Eq. (179) yields the lowest non-trivial order as: Note that this solution violates the complete positivity condition for times larger than The corresponding kernel is: Alternatively we could try finding a kernel that matches some of the approximate solutions discussed so far. For example, it turns out that the kernel leads to an exact match of the NZ2 solution. Finding a kernel which gives a good description of the evolution of an open system is an important but in general, difficult question which remains open for further investigation. We note that this question was also taken up in the context of the PM in the recent study [109], where the PM was applied to an exactly solvable model describing a qubit undergoing spontaneous emission and stimulated absorption. No attempt was made to optimize the memory kernel and hence the agreement with the exact solution was not as impressive as might be possible with optimization.

Comparison of the analytical solution and the different approximation techniques
In the results shown below, all figures express the evolution in terms of the dimensionless parameter αt (plotted on a logarithmic scale). We choose the initial condition v x (0) = v y (0) = 1/ √ 2 and plot only v x (t) since the structure of the equations for v x (t) and v y (t) is similar. In order to compare the different methods of approximation, we consider various choices of parameter values in our model. Among these choices we consider both low and high temperature cases. We note that in a spin bath model it is assumed that the environment degrees of freedom are localized and this is usually the case at low temperatures. At higher temperatures one may need to consider delocalized environment degrees of freedom in order to account for such environment modes such as phonons, magnons etc. A class of models known as oscillator bath models (e.g., Ref. [167]) consider such effects. In this study, we restrict attention to the spin bath model described here for both low and high temperatures.

Exact solution
We first assume that the frequencies of the qubits in the bath are equal (Ω n = 1, ∀n), and so are the coupling constants (g n = 1, ∀n). In this regime, we consider large and We also consider the case where the frequencies Ω n and the coupling constants g n can take different values. We generated uniformly distributed random values in the interval [−1, 1] for both Ω n and g n . In Figs. (3) and (4) we plot the ensemble average of the solution over 50 random ensembles. The main difference from the solution with equal Ω n and g n is that the partial recurrences decrease in size, especially as N increases. We attribute this damping partially to the fact that we look at the ensemble average, which amounts to averaging out the positive and negative oscillations that arise for different values of the parameters. The main reason, however, is that for a generic ensemble of random Ω n and g n the positive and negative oscillations in the sums (171) tend to average out. This is particularly true for large N , as reflected in We also looked at the case where one of the coupling constants, say g i , has a much larger magnitude than the other ones (which were made equal). The behavior was similar to that for a bath consisting of only a single spin.

NZ
In this subsection, we compare the solutions of different orders of the NZ master equation for Ω n = g n = 1. and β = 10, while NZ2 has a periodic behavior. A clear conclusion from these plots is that the NZ approximation is truly a short-time one: it becomes completely unreliable for times longer than αt ≪ 1. This can occur when the bath density matrices of some of the bath spins are maximally mixed or in the limit of a very high bath temperature. Clearly, when the Kraus map is non-invertible, the TCL approach becomes invalid since it relies on the assumption that the information about the initial state is contained in the current state. This fact has also been observed for the spin-boson model with a damped Jaynes-Cummings

TCL
Hamiltonian [30]. At the point where the Kraus map becomes non-invertible, the TCL solution deviates from the exact solution (see Fig. 9). We verified that both v x and v y vanish at this point.

NZ, TCL, and PM
In this subsection, we compare the exact solution to TCL4, NZ4 and the solution of the optimal PM master equation.   As discussed above, this approximation (which also coincides with TCL2) is valid when 2Q 2 (αt) 2 ≪ 1. As temperature decreases, so does the magnitude of Q 2 , which leads to a better approximation at fixed αt. Since NZ2 gives the lowest-order correction, this improvement is reflected in NZ4 as well.
In Figs. 14 and 15 we plot the averaged solutions over 50 ensembles of random values for Ω n and g n in the interval [−1, 1]. We see that on average TCL4, NZ4 and the optimal PM solution behave similarly to the case when Ω n = g n = 1. Due to the damping of the recurrences, especially when N = 100, the TCL4 and the PM solutions match the exact solution closely for much longer times than in the deterministic case.
Again, the PM solution is capable of qualitatively matching the behavior of the exact solution at long times.

Coarse-graining approximation
Finally, we examine the coarse-graining approximation discussed in Section 4.3.1. We choose the time over which the average trace distance is calculated to be the time where the exact solution dies down. In Fig. 16 we plot the coarse-grained solution for the value of τ for which the trace distance to the exact solution is minimum. As can be seen, the coarse-graining approximation does not help since the Markovian assumption is not valid for this model. In deriving the coarse-graining approximation [104] one makes the assumption that the coarse-graining time scale is greater than any

Summary and conclusions
We studied the performance of various methods for approximating the evolution of an Ising model of an open quantum system for a qubit system coupled to a bath bath consisting of N qubits. The high symmetry of the model allowed us to derive the exact dynamics of the system as well as find analytical solutions for the different master equations. We saw that the Markovian approximation fails for this model due to the time independence of the bath correlation functions. This is also reflected in the fact that the coarse-graining method [104] does not approximate the exact solution well. We discussed the performance of these solutions for various parameter regimes.
Unlike other spin bath models discussed in literature (e.g., Ref. axis of the Bloch sphere, a fact which is reflected in the exact solution. We showed that up to fourth order TCL performs better than NZ at medium and high temperatures.
For low temperatures we demonstrated an enhancement in the performance of NZ and showed that NZ and TCL perform equally well. We showed that the TCL approach breaks down for certain parameter choices and related this to the non-invertibility of the Kraus map describing the system dynamics. We also studied the performance of the post-Markovian master equation obtained in [139] with an optimal memory kernel.
We discussed possible ways of approximating the optimal kernel for short times and derived the kernel which leads to an exact fit to the NZ2 solution. It turns out that PM master equation performs as well as the TCL2 for a large number of spins and outperforms all orders of NZ and TCL considered here at long times, as it captures the recurrences of the exact solution.

Appendix A: Bath correlation functions
Here we show how to calculate the bath correlation functions used in our simulations.
The k th order bath correlation function is defined as where B and ρ B were given in Eqs. (162) and (158), respectively. This yields: or where Z = l exp(−βE l ) and the expressions for E l andẼ l were given in Eqs. (167) and (169), respectively.
The above formulas are useful when the energy levels E l andẼ l are highly degenerate, which is the case for example when g n ≡ g and Ω n ≡ Ω for all n. For a general choice of these parameters, it is computationally more efficient to consider θ in the form (174) and the initial bath density matrix in the form (172). For example, the second order bath correlation function is .
Higher order correlation functions are computed analogously. Second order: Third order: Fourth order:

Continuous quantum error correction
In general, error probabilities increase with time. No matter how complicated a code or how many levels of concatenation are involved, the probability of uncorrectable errors is never truly zero, and if the system is exposed to noise for a sufficiently long time, the weight of uncorrectable errors can accumulate. To combat this, error correction must be applied repeatedly and sufficiently often. If one assumes that the time for an errorcorrecting operation is small compared to other relevant time scales of the system, error-correcting operations can be considered instantaneous. Then the scenario of repeated error correction leads to a discrete evolution which often may be difficult to describe. To study the evolution of a system in the limit of frequently applied instantaneous error correction, Paz and Zurek proposed to describe error correction as a continuous quantum jump process [125]. In this model, the infinitesimal errorcorrecting transformation that the density matrix of the encoded system undergoes during a time step dt is where Φ(ρ) is the completely positive trace-preserving (CPTP) map describing a full error-correcting operation, and κ is the error-correction rate. The full error-correcting operation Φ(ρ) consists of a syndrome detection, followed (if necessary) by a unitary correction operation conditioned on the syndrome.
Consider, for example, the three-qubit bit-flip code whose purpose is to protect an unknown qubit state from bit-flip (Pauli X) errors. The code space is spanned by |0 = |000 and |1 = |111 , and the stabilizer generators are ZZI and IZZ (see Section 8.3). Here by X = σ x , Y = σ y , Z = σ z and I we denote the usual Pauli operators and the identity, respectively, and a string of three operators represents the tensor product of operators on each of the three qubits. The standard error-correction procedure involves a measurement of the stabilizer generators, which projects the state onto one of the subspaces spanned by |000 and |111 , |100 and |011 , |010 and |101 , or |001 and |110 ; the outcome of these measurements is the error syndrome. Assuming that the probability for two-or three-qubit errors is negligible, then with high probability the result of this measurement is either the original state with no errors, or with a single X error on the first, the second, or the third qubit. Depending on the outcome, one then applies an X gate to the erroneous qubit and transforms the state back to the original one. The CPTP map Φ(ρ) for this code can be written explicitly as The quantum-jump process (239) can be viewed as a smoothed version of the discrete scenario of repeated error correction, in which instantaneous full error-correcting operations are applied at random times with rate κ. It can also be looked upon as arising from a continuous sequence of infinitesimal CPTP maps of the type (239). In practice, such a weak map is never truly infinitesimal, but rather has the form where ǫ ≪ 1 is a small but finite parameter, and the weak operation takes a small but nonzero time τ c . For times t much greater than τ c (τ c ≪ t), the weak error-correcting map (241) is well approximated by the infinitesimal form (239), where the rate of error correction is A weak map of the form (241) could be implemented, for example, by a weak coupling between the system and an ancilla via an appropriate Hamiltonian, followed by discarding the ancilla. A closely related scenario, where the ancilla is continuously cooled in order to reset it to its initial state, was studied in [136].
Another way of implementing the weak map (241) is via weak measurements followed by weak unitaries dependent on the outcome. In the appendix at the end of this chapter, we give an example of such an implementation for the case of the bitflip code-when Φ(ρ) is given by (240). We also present a scheme in terms of weak measurements for codes that correct arbitrary single-qubit errors. In the latter case, the resulting weak map is not the same as (241), but also yields the strong errorcorrecting map Φ(ρ) when exponentiated. We point out that the weak measurements

Markovian decoherence
So far, continuous quantum error correction has been studied only for Markovian error models. As we discussed in the previous chapter, the Markovian approximation describes situations where the bath-correlation times are much shorter than any characteristic time scale of the system [30]. In this limit, the dynamics can be described by a semi-group master equation in the Lindblad form [106]: Here H is the system Hamiltonian and the {L j } are suitably normalized Lindblad operators describing different error channels with decoherence rates λ j . For example, the Liouvillian where X j denotes a local bit-flip operator acting on the j-th qubit, describes independent Markovian bit-flip errors.
For a system undergoing Markovian decoherence and error correction of the type (239), the evolution is given by the equation where Γ(ρ) = Φ(ρ) − ρ. In [125], Paz and Zurek showed that if the set of errors {L j } are correctable by the code, in the limit of infinite error-correction rate (strong errorcorrecting operations applied continuously often) the state of the system freezes and is protected from errors at all times. The effect of freezing can be understood by noticing that the transformation arising from decoherence during a short time step ∆t, is i.e., the weight of correctable errors emerging during this time interval is proportional to ∆t, whereas uncorrectable errors (e.g., multi-qubit bit flips in the case of the threequbit bit-flip code) are of order O(∆t 2 ). Thus, if errors are constantly corrected, in the limit ∆t → 0 uncorrectable errors cannot accumulate, and the evolution stops.

The Zeno effect. Error correction versus error prevention
The effect of "freezing" in continuous error correction strongly resembles the quantum Zeno effect [111], in which frequent measurements slow down the evolution of a system, freezing the state in the limit where they are applied continuously. The Zeno effect arises when the system and its environment are initially decoupled and they undergo a Hamiltonian-driven evolution, which leads to a quadratic change with time of the state during the initial moments [113] (the so called Zeno regime). Let the initial state of the system plus the bath be ρ SB (0) = |0 0| S ⊗ ρ B (0). For small times, the fidelity of the system's density matrix with the initial state can be approximated as In terms of the Hamiltonian H SB acting on the entire system, the coefficient C is According to Eq. (247), if after a short time step ∆t the system is measured in an orthogonal basis which includes the initial state |0 , the probability to find the system in a state other than the initial state is of order O(∆t 2 ). Thus if the state is continuously measured (∆t → 0), this prevents the system from evolving.
It has been proposed to utilize the quantum Zeno effect in schemes for error prevention [180,15,158], in which an unknown encoded state is prevented from errors simply by frequent measurements which keep it inside the code space. The approach is similar to error correction in that the errors for which the code is designed send a codeword to a space orthogonal to the code space. The difference is that different errors need not be distinguishable, since the procedure does not involve correction of errors, but their prevention. In [158] it was shown that with this approach it is possible to use codes of smaller redundancy than those needed for error correction and a four-qubit encoding of a qubit was proposed, which is capable of preventing arbitrary independent errors arising from Hamiltonian interactions. The possibility of this approach implicitly assumes the existence of a Zeno regime, and fails if we assume Markovian decoherence for all times. This is because the probability of errors emerging during a time step dt in a Markovian model is proportional to dt (rather than dt 2 ), and hence errors will accumulate with time if not corrected.
From the above observations we see that error correction is capable of achieving results in noise regimes where error prevention fails. Of course, this advantage is at the expense of a more complicated procedure-in addition to the measurements used in error prevention, error correction involves unitary correction operations, and in general requires codes with higher redundancy. At the same time, we see that in the Zeno regime it is possible to reduce decoherence using weaker resources than those needed in the case of Markovian noise. This suggests that in this regime error correction may exhibit higher performance than it does for Markovian decoherence.

Non-Markovian decoherence
Markovian decoherence is an approximation valid for times much larger than the memory of the environment. As we saw in the previous chapter, however, in many situations of practical significance the memory of the environment cannot be neglected and the evolution is highly non-Markovian. Furthermore, no evolution is strictly Markovian, and for a system initially decoupled from its environment a Zeno regime is always present, short though it may be [113]. If the time resolution of error-correcting operations is high enough so that they "see" the Zeno regime, this could give rise to different behavior. Nevertheless, it follows from a result in [141] that an error-correction procedure which is capable of correcting a certain class of completely positive (CP) maps, can also correct any linear noise map whose operator elements can be expressed as linear combinations of the operator elements in a correctable CP map. This implies, in particular, that an error-correction procedure that can correct arbitrary single-qubit CP maps can correct arbitrary single-qubit linear maps. In this context, we note that the effects of system-environment correlations in non-Markovian error models have also been studied from the perspective of fault tolerance, and it has been shown that the threshold theorem can be extended to various types of non-Markovian noise [156,12,3].
Another important difference from the Markovian case is that error correction and the effective noise on the reduced density matrix of the system cannot be treated as independent processes. One could derive an equation for the effective evolution of the system alone subject to interaction with the environment, like the Nakajima-Zwanzig [112,181] or the time-convolutionless (TCL) [143,142] master equations, but the generator of transformations at a given moment in general will depend (implicitly or explicitly) on the entire history up to this moment. Therefore, adding error correction can nontrivially affect the effective error model. This means that in studying the performance of continuous error correction one either has to derive an equation for the effective evolution of the encoded system, taking into account error correction from the very beginning, or one has to look at the evolution of the entire system-including the bath-where the error generator and the generator of error correction can be considered independent. In the latter case, for sufficiently small τ c , the evolution of the entire system including the bath can be described by where ρ is the density matrix of the system plus bath, H is the total Hamiltonian, and the error-correction generator Γ acts locally on the encoded system. In this study, we take this approach for a sufficiently simple bath model which allows us to find a solution for the evolution of the entire system.

Plan of this chapter
The rest of the chapter is organized as follows. To develop understanding of the workings of continuous error correction, in Section 5.2 we look at a simple example: an error-correction code consisting of only one qubit which aims at protecting a known state. We discuss the difference in performance for Markovian and non-Markovian decoherence, and argue the implications it has for the case of multi-qubit codes. In Section 5.3, we study the three-qubit bit-flip code. We first review the performance of continuous error correction in the case of Markovian bit-flip decoherence, which was first studied in [125]. We then consider a non-Markovian model, where each qubit in the code is coupled to an independent bath qubit. This model is a simple version of the one studied in the previous chapter, and it allows us to solve for its evolution analytically. In the limit of large error-correction rates, the effective evolution approaches the evolution of a single qubit without error correction, but the coupling strength is now decreased by a factor which scales quadratically with the error-correction rate.
This is opposed to the case of Markovian decoherence, where the same factor scales linearly with the rate of error-correction. In Section 5.4, we show that the quadratic enhancement in the performance over the case of Markovian noise can be attributed to the presence of a Zeno regime and argue that for general stabilizer codes and independent errors, the performance of continuous error correction would exhibit the same qualitative characteristics. In Section 5.5, we conclude. In the Appendix (Section 5.6), we present an implementation of the quantum-jump error correcting model that uses weak measurements and weak unitary operations.

The single-qubit code
and If the state lies on the z-axis of the Bloch sphere, it will never leave it, since both the noise generator (250) and the error-correction generator (251) keep it on the axis. We will take the qubit to be initially in the desired state |0 , and therefore at any later moment it will have the form has the interpretation of a fidelity with the trivial code space spanned by |0 . For an infinitesimal time step dt, the effect of the noise is to decrease α(t) by the amount λ(2α(t) − 1)dt and that of the correcting operation is to increase it by κ(1 − α(t))dt. The net evolution is then described by The solution is where and r = κ/λ is the ratio between the rate of error correction and the rate of decoherence. We see that the fidelity decays, but it is confined above its asymptotic value α M * , which can be made arbitrarily close to 1 for a sufficiently large r.
Now let us consider a non-Markovian error model. We choose the simple scenario where the system is coupled to a single bath qubit via the Hamiltonian where γ is the coupling strength. This is the Ising Hamiltonian (147)  We will assume that the bath qubit is initially in the maximally mixed state, which can be thought of as an equilibrium state at high temperature. From Eq. (249) one can verify that if the system is initially in the state |0 , the state of the system plus the bath at any moment will have the form In the tensor product, the first operator belongs to the Hilbert space of the system and the second to the Hilbert space of the bath. We have α(t) ∈ [0, 1], and |β(t)| ≤ The reduced density matrix of the system has the same form as the one for the Markovian case. The traceless term proportional to β(t) can be thought of as a "hidden" part, which nevertheless plays an important role in the errorcreation process, since errors can be thought of as being transferred to the "visible" part from the "hidden" part (and vice versa). This can be seen from the fact that during an infinitesimal time step dt, the Hamiltonian changes the parameters α and β as follows: The effect of an infinitesimal error-correcting operation is Note that the hidden part is also being acted upon. Putting it all together, we get the system of equations The solution for the fidelity α(t) is We see that as time increases, the fidelity stabilizes at the value where R = κ/γ is the ratio between the error-correction rate and the coupling strength.
In Fig. 1 we have plotted the fidelity as a function of the dimensionless parameter γt for three different values of R. For error-correction rates comparable to the coupling strength (R = 1), the fidelity undergoes a few partial recurrences before it stabilizes close to α NM * . For larger R = 2, however, the oscillations are already heavily damped and for R = 5 the fidelity seems confined above α NM * . As R increases, the evolution becomes closer to a decay like the one in the Markovian case.
A remarkable difference, however, is that the asymptotic weight outside the code  For multi-qubit codes, error correction cannot preserve a high fidelity with the initial codeword for all times, because there will be multi-qubit errors that can lead to errors within the code space itself. But it is natural to expect that the code-space fidelity can be kept above a certain value, since the effect of the error-correcting map (239) is to oppose its decrease. If similarly to the single-qubit code there is a quadratic difference in the code-space fidelity for the cases of Markovian and non-Markovian decoherence, this could lead to a different performance of the error-correction scheme with respect to the rate of accumulation of uncorrectable errors inside the code space. This is because multi-qubit errors that can lead to transformations entirely within the code space during a time step dt are of order O(dt 2 ). This means that if the state is kept constantly inside the code space (as in the limit of an infinite errorcorrection rate), uncorrectable errors will never develop. But if there is a finite nonzero portion of correctable errors, by the error mechanism it will give rise to errors not distinguishable or misinterpreted by the code. Therefore, the weight outside the code space can be thought of as responsible for the accumulation of uncorrectable errors, and consequently a difference in its magnitude may lead to a difference in the overall performance. In the following sections we will see that this is indeed the case.

5.3
The three-qubit bit-flip code 5

.3.1 A Markovian error model
Even though the three-qubit bit-flip code can correct only bit-flip errors, it captures most of the important characteristics of nontrivial stabilizer codes. Before we look at a non-Markovian model, we will review the Markovian case which was studied in [125].
Let the system decohere through identical independent bit-flip channels, i.e., L(ρ) is of the form (244) with λ 1 = λ 2 = λ 3 = λ. Then one can verify that the density matrix at any moment can be written as where are equally-weighted mixtures of single-qubit, two-qubit and three-qubit errors on the original state.

118
The effect of decoherence for a single time step dt is equivalent to the following transformation of the coefficients in Eq. (262): If the system is initially inside the code space, combining Eq. (264) with the effect of the weak error-correcting map ρ → (1 − κdt)ρ + κdtΦ(ρ), where Φ(ρ) is given in Eq. (240), yields the following system of first-order linear differential equations for the evolution of the system subject to decoherence plus error correction: The exact solution has been found in [125]. Here we just note that for the initial conditions a(0) = 1, b(0) = c(0) = d(0) = 0, the exact solution for the weight outside the code space is where r = κ/λ. We see that similarly to what we obtained for the trivial code in the previous section, the weight outside the code space quickly decays to its asymptotic value 3 4+r which scales as 1/r. But note that here the asymptotic value is roughly three times greater than that for the single-qubit model. This corresponds to the fact that there are three single-qubit channels. More precisely, it can be verified that if for a given κ the uncorrected weight by the single-qubit scheme is small, then the uncorrected weight by a multi-qubit code using the same κ and the same kind of decoherence for each qubit scales approximately linearly with the number of qubits.
Similarly, the ratio r required to preserve a given overlap with the code space scales linearly with the number of qubits in the code.
The most important difference from the single-qubit model is that in this model there are uncorrectable errors that cause a decay of the state's fidelity inside the code space. Due to the finiteness of the resources employed by our scheme, there always remains a nonzero portion of the state outside the code space, which gives rise to uncorrectable three-qubit errors. To understand how the state decays inside the code space, we ignore the terms of the order of the weight outside the code space in the exact solution. We obtain: Comparing this solution to the expression for the fidelity of a single decaying qubit without error correction-which can be seen from Eq. (253) for κ = 0-we see that the encoded qubit decays roughly as if subject to bit-flip decoherence with rate 6λ/r. Therefore, for large r this error-correction scheme can reduce the rate of decoherence approximately r/6 times. In the limit r → ∞, it leads to perfect protection of the state for all times.

A non-Markovian error model
We consider a model where each qubit independently undergoes the same kind of non-Markovian decoherence as the one we studied for the single-qubit code. Here the system we look at consists of six qubits-three for the codeword and three for the environment. We assume that all system qubits are coupled to their corresponding environment qubits with the same coupling strength, i.e., the Hamiltonian is where the operators X S act on the system qubits and X B act on the corresponding bath qubits. The subscripts label the particular qubit on which they act. Obviously, the types of effective single-qubit errors on the density matrix of the system that can result from this Hamiltonian at any time, whether they are CP or not, will have operator elements which are linear combinations of I and X S , i.e., they are correctable by the procedure according to [141]. Considering the forms of the Hamiltonian (372) and the error-correcting map (240), one can see that the density matrix of the entire system at any moment is a linear combination of terms of the following type: Here the first term in the tensor product refers to the Hilbert space of the system, and the following three refer to the Hilbert spaces of the bath qubits that couple to the first, second and third qubits from the code, respectively. The powers l, m, n, p, q, r take values 0 and 1 in all possible combinations, and X 1 = X, X 0 = X 2 = I. Note that ̺ lmn,pqr should not be mistaken for the components of the density matrix in the computational basis. Collecting these together, we can write the density matrix in the form ρ(t) = l,m,n,p,q,r where the coefficients C lmn,pqr (t) are real. The coefficient C 000,000 is less than or equal to the codeword fidelity (with equality when ρ(0) = |0 0 | or ρ(0) = |1 1 |). Since the scheme is intended to protect an unknown codeword, we are interested in its worst-case performance; we will therefore use C 000,000 as a lower bound on the codeword fidelity.
Using the symmetry with respect to permutations of the different system-bath pairs of qubits and the Hermiticity of the density matrix, we can reduce the description of the evolution to a system of equations for only 13 of the 64 coefficients. (In fact, 12 coefficients are sufficient if we invoke the normalization condition Trρ = 1, but we have found it more convenient to work with 13.) The equations are linear, and we write them as a single 13-dimensional vector equation: where R = κ/γ. Each nonzero component in this matrix represents an allowed transition process for the quantum states; these transitions can be driven either by the decoherence process or the continuous error-correction process. We plot these allowed transitions in Fig. 2.
We can use the symmetries of the process to recover the 64 coefficients of the full state. Each of the 13 coefficients represents a set of coefficients having the same number of 1s on the left and the same number of 1s on the right, as well as the same number of places which have 1 on both sides. All such coefficients are equal at all times. For example, the coefficient C 110,011 is equal to all coefficients with two 1s on the left, two In determining the transfer rate from one coefficient to another in Fig. 2 Table I.
Obviously all eigenvalues except the first one and the last two describe fast decays with rates ∼ κ. They correspond to terms in the solution which will vanish quickly after the beginning of the evolution. The eigenvalue 0 corresponds to the asymptotic (t → ∞) solution, since all other terms will eventually decay. The last two eigenvalues are those that play the main role in the evolution on a time scale t ≫ 1 κ . We see that on such a time scale, the solution will contain an oscillation with an angular frequency approximately equal to (24/R 2 )γ which is damped by a decay factor with a rate of approximately (144/R 3 )γ. In Fig. 3 we have plotted the codeword fidelity C 000,000 (t) as a function of the dimensionless parameter γt for R = 100. The graph indeed represents this type of behavior, except for very short times after the beginning (γt ∼ 0.1), where one can see a fast but small in magnitude decay (Fig. 4). The maximum magnitude of this quickly decaying term obviously decreases with R, since in the limit of R → ∞ the fidelity should remain constantly equal to 1. From the form of the eigenvalues one can see that as R increases, the frequency of the main oscillation decreases as 1/R 2 while the rate of decay decreases faster, as 1/R 3 .
Thus in the limit R → ∞, the evolution approaches an oscillation with an angular frequency (24/R 2 )γ. (We formulate this statement more rigorously below.) This is the same type of evolution as that of a single qubit interacting with its environment, but the coupling constant is effectively reduced by a factor of R 2 /12.
While the coupling constant serves to characterize the decoherence process in this particular case, this is not valid in general. To handle the more general situation, we propose to use the instantaneous rate of decrease of the codeword fidelity F cw as a measure of the effect of decoherence: (In the present case, F cw = C 000,000 .) This quantity does not coincide with the decoherence rate in the Markovian case (which can be defined naturally from the Lindblad equation), but it is a good estimate of the rate of loss of fidelity and can be used for any decoherence model. From now on we will refer to it simply as an error rate, but we note that there are other possible definitions of instantaneous error rate suitable for non-Markovian decoherence, which in general may depend on the kind of errors they describe. Since the goal of error correction is to preserve the codeword fidelity, the quantity (273) is a useful indicator for the performance of a given scheme. Note that Λ(F cw ) is a function of the codeword fidelity and therefore it makes sense to use it for a comparison between different cases only for identical values of F cw . For our example, the fact that the coupling constant is effectively reduced approximately R 2 /12 times implies that the error rate for a given value of F cw is also reduced R 2 /12 times. Similarly, the reduction of λ by the factor r/6 in the Markovian case implies a reduction of Λ by the same factor. We see that the effective reduction of the error rate increases quadratically with κ 2 in the non-Markovian case, whereas it increases only linearly with κ in the Markovian case. Now let us rigorously derive the approximate solution to this model of non-Markovian decoherence with continuous error correction. Assuming that γ ≪ κ (or equivalently, R ≫ 1), the superoperator driving the evolution of the system during a time step δt can be written as We have denoted the Liouvillian by L = L γ + L κ , where L κ ρ = κΓ(ρ), and L γ ρ =

−i[H, ρ].
Let γδt ≪ 1 ≪ κδt. We will derive an approximate differential equation for the evolution of ρ(t) by looking at the terms of order δt in the change of ρ according to Eq. (274). When κ = 0, we have dρ/dt = L γ ρ, so the effect of L γ on the state of the system can be seen from Eq. (272) with κ taken equal to 0. By the action of exp(L κ t), is to multiply each of these new terms by e −κ(t ′′ −t ′ ) . After the action of the second L γ , the action of the third exponent on the relevant resultant terms will be again to multiply them by a factor e −κ(t ′′′ −t ′′ ) . Thus the second and the third exponents yield a net factor of e −κ(t ′′′ −t ′ ) . After the second and the third L γ , the relevant terms that we get are ̺ 000,000 and ̺ 100,100 , ̺ 010,010 , ̺ 001,001 , each with a corresponding factor. Finally, the last exponent acts as the identity on ̺ 000,000 and transforms each of the terms ̺ 100,100 , ̺ 010,010 , ̺ 001,001 into (1 − e −κ(δt−t ′′′ ) )̺ 000,000 . Counting the number of different terms that arise at each step, and taking into account the factors that accompany them, we obtain: )C 111,000 + · · · ≈ C 000,000 + C 111,000 24 Using that C 000,000 + C 111,111 ≈ 1, in a similar way one obtains C 111,000 → C 111,000 − (2C 000,000 − 1) 12 For times much larger than δt, we can write the approximate differential equations which yield for the fidelity C 000,000 (t) = 1 + e −144γt/R 3 cos(24γt/R 2 ) 2 .
We see that in addition to the effective error process which is of the same type as that of a single qubit, there is an extra Markovian bit-flip process with rate 72γ/R 3 .
This Markovian behavior is due to the Markovian character of our error-correcting procedure which, at this level of approximation, is responsible for the direct transfer of weight between ̺ 000,000 and ̺ 111,111 , and between ̺ 111,000 and ̺ 000,111 . The exponential factor explicitly reveals the range of applicability of the solution (278): with precision O(1/R), it is valid only for times γt of up to order R 2 . For times of the order of R 3 , the decay becomes significant and cannot be neglected. The exponential factor may also play an important role for short times of up to order R, where its contribution is bigger than that of the cosine. But in the latter regime the difference between the cosine and the exponent is of order O(1/R 2 ), which is negligible for the precision that we consider.
In general, the effective evolution that one obtains in the limit of high errorcorrection rate does not have to approach a form identical to that of a single decohering qubit. The reason we obtain such behavior here is that for this particular model the lowest order of uncorrectable errors that transform the state within the code space is 3, and three-qubit errors have the form of an encoded X operation. Furthermore, the symmetry of the problem ensured an identical evolution of the three qubits in the code. For general stabilizer codes, the errors that a single qubit can undergo are not limited to bit flips only. Therefore, different combinations of single-qubit errors may lead to different types of lowest-order uncorrectable errors inside the code space, none of which in principle has to represent an encoded version of the single-qubit operations that compose it. In addition, if the noise is different for the different qubits, there is no unique single-qubit error model to compare to. Nevertheless, we will show that with regard to the effective decrease in the error-correction rate, general stabilizer codes will exhibit the same qualitative performance.

Relation to the Zeno regime
The effective continuous evolution (277)  Let us consider again the single-qubit code from Section 5.2, but this time let the error model be any Hamiltonian-driven process. We assume that the qubit is initially in the state |0 , i.e., the state of the system including the bath has the form ρ(0) = |0 0| ⊗ ρ B (0). For times smaller than the Zeno time δt Z , the evolution of the fidelity without error correction can be described by Eq. (247). Equation (247) naturally defines the Zeno regime in terms of α itself: For a single time step ∆t ≪ δt Z , the change in the fidelity is On the other hand, the effect of error correction during a time step ∆t is i.e., it tends to oppose the effect of decoherence. If both processes happen simultaneously, the effect of decoherence will still be of the form (282) In agreement with what we obtained in Section 5.2, the equilibrium fidelity differs from 1 by a quantity proportional to 1/κ 2 . This quantity is generally quasi-stationary and

132
But what happens in the case of non-trivial codes? As we saw, there the state decays inside the code space and therefore can be highly correlated with the environment. Can we talk about a Zeno regime then? It turns out that the answer is positive.
Assuming that each qubit undergoes an independent error process, then up to first order in ∆t the Hamiltonian cannot map terms in the code space to other terms with- implies that for small enough 1 − α, the change in the code-space fidelity is of the type (282), which is Zeno-like behavior. Then using only the properties of the Zeno behavior as we did above, we can conclude that the weight outside the code space will be kept at a quasi-stationary value of order 1/κ 2 . Since uncorrectable errors enter the code space through the action of the error-correction procedure, which misinterprets some multi-qubit errors in the error space, the effective error rate will be limited by a factor proportional to the weight in the error space. That is, this will lead to an effective decrease of the error rate at least by a factor proportional to 1/κ 2 .
The accumulation of uncorrectable errors in the Markovian case is similar, except that in this case there is a direct transfer of errors between the code space and the visible part of the error space. In both cases, the error rate is effectively reduced by a factor which is roughly proportional to the inverse of the weight in the error space, and therefore the difference in the performance comes from the difference in this weight.
The quasi-stationary equilibrium value of the code-space fidelity establishes a quasistationary flow between the code space and the error space. One can think that this flow effectively takes non-erroneous weight from the code space, transports it through the error space where it accumulates uncorrectable errors, and brings it back into the code space. Thus by minimizing the weight outside the code space, error correction creates a "bottleneck" which reduces the rate at which uncorrectable errors accumulate.
Finally, a brief remark about the resources needed for quadratic reduction of the error rate. As pointed out above, two conditions are involved: one concerns the rate of error correction; the other concerns the time resolution of the weak error-correcting operations. Both of these quantities must be sufficiently large. There is, however, an interplay between the two, which involves the strength of the interaction required to implement the weak error-correcting map (241). Let us imagine that the weak map is implemented by making the system interact weakly with an ancilla in a given state, after which the ancilla is discarded. The error-correction procedure consists of a sequence of such interactions, and can be thought of as a cooling process which takes away the entropy accumulated in the system as a result of correctable errors. If the time for which a single ancilla interacts with the system is τ c , one can verify that the parameter ǫ in Eq. (241) would be proportional to g 2 τ 2 c , where g is the coupling strength between the system and the ancilla. From Eq. (242) we then obtain that The two parameters that can be controlled are the interaction time and the interaction strength, and they determine the error-correction rate. Thus if g is kept constant, a decrease in the interaction time τ c leads to a proportional decrease in κ, which may be undesirable. In order to achieve a good working regime, one may need to adjust both τ c and g. But it has to be pointed out that in some situations decreasing τ c alone can prove advantageous, if it leads to a time resolution revealing the non-Markovian character of an error model which was previously described as Markovian.
The quadratic enhancement of the performance as a function of κ may compensate the decrease in κ, thus leading to a seemingly paradoxical result: better performance with a lower error-correction rate.

Summary and outlook
In this chapter we studied the performance of a particular continuous quantum error- While here we have only considered codes for the correction of single-qubit errors, our results can be extended to other types of codes and errors as well. As long as the error process only produces errors correctable by the code to lowest order, an argument analogous to the one given here shows that a Zeno regime will exist, which leads to an enhancement in the error-correction performance. Unfortunately, it is very difficult to describe the evolution of a system with a continuous correction protocol, based on a general error-correction code and subject to general non-Markovian interactions with the environment. This is especially true if one must include the evolution of a complicated environment in the description, as would be necessary in general. A more practical step in this direction might be to find an effective description for the evolution of the reduced density matrix of the system subject to decoherence plus error correction, using projection techniques like the Nakajima-Zwanzig or the TCL master equations. Since one is usually interested in the evolution during initial times before the codeword fidelity decreases significantly, a perturbation approach could be useful.
This is a subject for further research.

Appendix: Implementation of the quantum-jump errorcorrecting process via weak measurements and weak unitary operations
Here we show how the weak CPTP map (241) for the bit-flip code can be implemented using weak measurements and weak unitary operations. We also present a similar 136 scheme for codes that correct arbitrary-single qubit errors, which yields a weak map different from (241) but one that also results in the strong error-correcting map Φ(ρ) when exponentiated. To introduce our construction, we start again with the singlequbit code with stabilizer Z .

The single-qubit model
Consider the completely positive map corresponding to the strong error-correcting operation for the single-qubit code: Observe that this transformation can also be written as where |± = (|0 ± |1 )/ √ 2 and is the Hadamard gate. Therefore the same error-correcting operation can be implemented as a measurement in the |± basis (measurement of the operator X), followed by a unitary conditioned on the outcome: if the outcome is '+', we apply ZR; if the outcome is '-', we apply XR. This choice of unitaries is not unique-for example, we could apply just R instead of ZR after outcome '+'. But this particular choice has a convenient geometric interpretation-the unitary ZR corresponds to a rotation around the Y-axis by an angle π/2: ZR = e i π 2 Y 2 , and XR corresponds to a rotation around the same axis by an angle −π/2: ZR = e −i π 2 Y 2 .

137
A weak version of the above error-correcting operation can be constructed by taking the corresponding weak measurement of the operator X, followed by a weak rotation around the Y-axis, whose direction is conditioned on the outcome: Here ǫ and ǫ ′ are small parameters. From the symmetry of this map it can be seen that if the map is applied to a state which lies on the Z-axis, the resultant state will still lie on the Z-axis. Whether the state will move towards |0 0| or towards |1 1|, depends on the relation between ǫ and ǫ ′ . Since our goal is to protect the state from drifting away from |0 0| due to bit-flip decoherence, we will assume that the state lies on the Z-axis in the northern hemisphere (although the transformation we will obtain works for any kind of decoherence where the state need nor remain on the Z-axis). We would like, if possible, to choose the relation between the parameters ǫ and ǫ ′ in such a way that the effect of this map on any state on the Z-axis to be to move the state towards |0 0|.
In order to calculate the effect of this map on a given state, it is convenient to write the state in the |± basis. For a state on the Z-axis, ρ = α|0 0| + (1 − α)|1 1|, we For the action of our map on the state (290) we obtain: Thus we can think that upon this transformation the parameter α transforms to α ′ , If it is possible to choose the relation between ǫ and ǫ ′ in such a way that α ′ ≥ α for every 0 ≤ α ≤ 1, then clearly the state must remain invariant when α = 1. Imposing this requirement, we obtain or equivalently Substituting back in (292), we can express We see that the coefficient α (which is the fidelity of our state with |0 0|) indeed increases after every application of our weak completely positive map (Fig.1). The amount by which it increases for fixed ǫ ′ depends on α and becomes smaller as α approaches 1.
Since we will be taking the limit ǫ → 0, we can write Eq. (294) as If we define the relation between the time step τ c and ǫ as in Eq. (242), for the effect of the CPTP map (289) on an arbitrary state of the form ρ = α|0 0| + β|0 1| + This is exactly the map (241) for Φ(ρ) given by Eq. (286).

The bit-flip code
While in the toy model from the previous section we had to protect a given state from errors, here we have to protect the whole subspace spanned by |0 and |1 . This makes a geometric visualization of the problem significantly more difficult than in the previous case, which is why we will take a different approach.
In the single-qubit model we saw how to protect a qubit in state |0 from bit-flip errors. Similarly we could protect a qubit in state |1 ; the only difference is that the weak unitaries following the two outcomes of the weak measurement of X have to be exchanged. For the three-qubit bit-flip code, every block of the code lies in the subspace spanned by the codewords |000 and |111 , i.e., each qubit is in state |0 when the other two qubits are in state |00 , or in state |1 when the other qubits are in state |11 . This correlation is what makes it possible for the code to correct singlequbit bit-flip errors without ever acquiring information about the actual state of the system. We propose to utilize this correlation in a three-qubit scheme which protects each qubit by applying to it the corresponding single-qubit scheme for either |0 or |1 depending on the value of the other two qubits. This, of course, has to be done without acquiring information about the encoded state.
Just as in the single-qubit case, the scheme consist of weak measurements followed by weak unitaries conditioned on the outcomes of the measurements. For error correction on the first qubit, we propose the weak measurement with measurement operators where I±ǫX 2 are the same weak measurement operators that we used in (289), acting on the first qubit. This measurement can be thought of as a weak measurement of the operator X ⊗ (|00 00| + |11 11|). In order to understand its effect better, consider the expansion of the density matrix of our system in the computational basis of the three qubits in a given block of the code. Assuming that the state begins inside the code space and that the system decoheres through single-qubit bit-flip channels, the density matrix at any time can be written as a linear combination of the following |110 001|, |110 110|. For those terms in the expansion for which the second and third qubits are in the subspace spanned by |00 and |11 , the effect of this measurement will be the same as the effect of a weak single-qubit measurement of X on the first qubit. Those terms in which the second and third qubits are in the subspace spanned by |01 and |10 will not be affected by the measurement. This is because the threequbit bit-flip code cannot distinguish multi-qubit errors from single-qubit errors; the subspaces corresponding to two-and three-qubit errors are the same as the subspaces corresponding to single-qubit or no errors. This is why, if the second and third qubits have different values, the error-correction scheme will assume that an error has occurred on one of these two qubits and will not apply any correction on the first qubit.
The unitary operation conditioned on the outcome of the measurement is This is a weak unitary driven by the Hamiltonian ±Y ⊗(|00 00|−|11 11|). Again, it is designed in such a way that those components of the density matrix which correspond to an error on the second or third qubits will undergo no transformation, while the terms for which the second and third qubits have the same value (these are the same terms that have undergone non-trivial transformation during the measurement) will undergo a rotation of the first qubit analogous to that from the single-qubit model.
One can verify that the only terms that undergo non-trivial transformation after the completely positive map ρ → U 1 where |φ is any state orthogonal to the subspace spanned by |100 and |011 . We see that the effect of this operation on the terms that correspond to bit flip on the 142 first qubit is to correct these terms by the same amount as in the single-qubit errorcorrection scheme. All other terms remain unchanged. If we write the state of the system at a given moment as where ρ(0) is the initial state, then the effect of the above completely positive map is: We apply the same correction (ρ → U i

General single-error-correcting stabilizer codes
We now proceed to generalizing this scheme to error-correcting codes that correct arbitrary single-qubit errors. A stabilizer code which is able to correct arbitrary singlequbit errors, has the property that a single-qubit X, Y or Z error on a state inside the code space, sends that state to a subspace orthogonal to the code space [67]. One can verify that this implies that any two orthogonal codewords can be written as where |ψ i j , i, j = 0, 1 form an orthonormal set. Here we have expanded the codewords in the computational basis (the eigenbasis of Z) of the first qubit, but the same can be done with respect to any qubit in the code. Note that an X, Y , or Z error on one of the other qubits sends each of the vectors |ψ i j to a subspace orthogonal to the subspace spanned by |ψ i j , i, j = 0, 1. This can be shown to follow from the fact that different single-qubit errors send the code space to different orthogonal subspaces. An exception is the case of degenerate codes where the error in question has the same effect on a codeword as an error on the first qubit. In such a case, however, we can assume that the error has occurred on the first qubit. The weak operation for correcting bit flips on a given qubit (say the first one) is therefore constructed similarly to that for the bit-flip code. We first apply the weak measurement where I n−1 is the identity on the space of all qubits in the code except the first one.
This can be thought of as a weak measurement of the operator X(|ψ 0 0 ψ 0 0 |+|ψ 0 1 ψ 0 1 |+ |ψ 1 0 ψ 1 0 | + |ψ 1 1 ψ 1 1 |). The measurement is followed by the unitary conditioned on the outcome. The Hamiltonian driving this unitary is ±Y (|ψ 0 0 ψ 0 0 | + It is easy to verify that the effect of the corresponding completely positive map is analogous to that for the bit-flip code. The action of each of the operators U 1 + M 1 + and U 1 − M 1 − can be summarized as follows: This implies that the effect of the map σ → U 1 a bit-flip error on the first qubit of a codeword ρ is: Just like in the bit-flip code, the error-correcting procedure for the case where each qubit decoheres through an independent bit-flip channel consists of simultaneous corrections of all qubits (i = 1, 2, ..., n) by continuous application of the maps σ → From (304) it can be seen that the codewords have analogous forms when expanded in the eigenbasis of another Pauli operator (X or Y ) acting on a given qubit: Here and are eigenbases of X and Y respectively, and and To see this, consider all possible single-qubit errors on a density matrix ρ ∈ C. The most general form of a single-qubit error on the i th qubit is a single application of the weak map causes the transformation Using that at any moment a + b + c + d = 1 and taking the limit τ c → 0, from (329) we obtain that the evolution of a is described by The solution is i.e., in the limit of t → ∞ we obtain a(t) → 1 (and therefore b, c, d → 0). We don't need to look at the evolution of the traceless terms in ρ i because our map is completely positive and therefore the transformed ρ i is also a density matrix, which implies that if a = 1, b = 0, c = 0, d = 0, all traceless terms have to vanish. This completes the proof that in the limit of infinitely many applications, our weak error-correcting map is able to correct arbitrary single-qubit errors.

Preliminaries
Operator quantum error correction (OQEC) [94,95,24] is a unified approach to error correction which uses the most general encoding for the protection of informationencoding in subsystems [88,163] (see also [26]). This approach contains as special cases the standard quantum error-correction method [144,150,22,89] as well as the methods of decoherence-free subspaces [55,176,105,103] and subsystems [90,51,83,174]. In the OQEC formalism, noise is represented by a completely positive trace-preserving (CPTP) linear map or a noise channel, and correctability is defined with respect to such channels. In practice, however, noise is a continuous process and if it can be represented by a CPTP map, that map is generally a function of time. Correctability is therefore a time-dependent property. Furthermore, the evolution of an open system is completely positive if the system and the environment are initially uncorrelated, and necessary and sufficient conditions for CPTP dynamics are not known. As pointed out in the previous chapter, for more general cases one might need a notion of correctability that can capture non-CP transformations [141].
Whether completely positive or not, the noise map is a result of the action of the generator driving the evolution and possibly of the initial state of the system and the environment. Therefore, our goal will be to understand the conditions for correctability in terms of the generator that drives the evolution. We will consider conditions on the system-environment Hamiltonian, or in the case of Markovian evolution-on the Lindbladian.
Conditions on the generator of evolution have been derived for decoherence-free subsystems (DFSs) [140], which are a special type of operator codes. DFSs are fixed subsystems of the system's Hilbert space, inside which all states evolve unitarily. One generalization of this concept are the so called unitarily correctable subsystems [95].
These are subsystems, all states inside of which can be corrected via a unitary operation up to an arbitrary transformation inside the gauge subsystem. Unlike DFSs, the unitary evolution followed by states in a unitarily correctable code are not restricted to the initial subsystem. An even more general concept is that of unitarily recoverable subsystems [94,95], for which states can be recovered by a unitary transformation up to an expansion of the gauge subsystem. It was shown that any correctable subsystem is in fact a unitarily recoverable subsystem [96]. This reflects the so called subsystem principle [88,163], according to which protected information is always contained in a subsystem of the system's Hilbert space. The connection between DFSs and unitarily recoverable subsystems suggests that similar conditions on the generators of evolution to those for DFSs can be derived in the case of general correctable subsystems. This is the subject of the present study.
The chapter is organized as follows. In Section 6.2 we review the definitions of correctable subsystems and unitarily recoverable subsystems. In Section 6.3, we discuss the necessary and sufficient conditions for such subsystems to exist in the case of CPTP maps. In Section 6.4, we derive conditions for the case of Markovian decoherence. The conditions for general correctability in this case are essentially the same as those for unitary correctability except that the dimension of the gauge subsystem is allowed to suddenly increase. For the case when the evolution is non-correctable, we conjecture a procedure for tracking the subsystem which contains the optimal amount of undissipated information and discuss its possible implications for the problem of optimal error correction. In Section 6.5, we derive conditions on the system-environment Hamiltonian. In this case, the conditions for unitary correctability concern only the effect of the Hamiltonian on the system, whereas the conditions for general correctability concern the entire system-environment Hamiltonian. In the latter case, the state of the noisy subsystem plus environment belongs to a particular subspace which plays an important role in the conditions. We extend the conditions to the case where the environment is initialized inside a particular subspace. In Section 6.6, we conclude.

Correctable subsystems
For simplicity, we consider the case where information is stored in only one subsystem.
Then there is a corresponding decomposition of the Hilbert space of the system, where the subsystem H A is used for encoding of the protected information. The subsystem H B is referred to as the gauge subsystem, and K denotes the rest of the Hilbert space. In the formulation of OQEC [94,95], the noise process is a completely where the Kraus operators The subsystem H A in Eq. (335) is called noiseless with respect to the noise process for all σ ∈ B(H S ) such that σ = P AB (σ) , where P AB (·) = P AB (·)P AB (339) with P AB being the projector of H S onto H A ⊗ H B , Similarly, a correctable subsystem is one for which there exists a correcting CPTP map R : B(H S ) → B(H S ), such that the subsystem is noiseless with respect to the map R • E: for all σ ∈ B(H S ) such that σ = P AB (σ) .
When the correcting map R is unitary, R = U, the subsystem is called unitarily correctable: for all σ ∈ B(H S ) such that σ = P AB (σ) .
A similar but more general notion is that of a unitarily recoverable subsystem, for shown that the reverse is also true-if H A is correctable, it is unitarily recoverable.
This equivalence will provide the basis for our derivation of correctability conditions for continuous decoherence.
Before we proceed with our discussion, we point out that condition () can be equivalently written as [94,95] U

Completely positive linear maps
Let H S and H E denote the Hilbert spaces of a system and its environment, and let H = H S ⊗H E be the total Hilbert space. As we pointed out earlier, a common example of a CP map is the transformation that the state of a system undergoes if the system is initially decoupled from its environment, ρ(0) = ρ S (0) ⊗ ρ E (0), and both the system and environment evolve according to the Schrödinger equation: Equation (344) gives rise to the unitary transformation with where T denotes time ordering. Under the assumption of an initially-decoupled state of the system and the environment, the transformation of the state of the system is described by the time-dependent CPTP map with Kraus operators where {|µ } is a basis in which the initial environment density matrix is diagonal, ρ B (0) = µ λ µ |µ µ|. We already saw one example of such a map in Chapter 4 where we studied the evolution of a qubit coupled to a spin bath (Eq. (164)).
The Kraus representation (336) applies to any CP linear map which need not necessarily arise from evolution of the type (344). This is why in the following theorem we derive conditions for discrete CP maps. For correctability under continuous decoherence, the same conditions must apply at any moment of time, i.e., one can think that the quantities M i , U , C i , as well as the subsystem H B ′ in the theorem are implicitly time-dependent.
Proof: The sufficiency of condition (348) is obvious-using that ρ ⊗ τ in Eq. (343) where for now we assume that g i and |φ ′ i may depend on |ψ . In other words, But if we impose (350) on a linear superposition |ψ = a|ψ 1 + b|ψ 2 , (a, b = 0), we Since Eq. (350) has to be satisfied for all |ψ ∈ H A and all |φ ∈ H B , we obtain Applying U † from the left yields condition (348).
We remark that condition (348) is equivalent to the conditions obtained in Ref. [95].

Markovian dynamics
The most general continuous completely positive time-local evolution of the state of a quantum system is described by a semi-group master equation in the form (243) but with time dependent coefficients, (For a discussion of the situations in which such time-dependent Markovian evolution can arise, see, e.g., Ref. [101].) Here H(t) is a system Hamiltonian, L j (t) are Lindblad operators, and L(t) is the Liouvillian superoperator corresponding to this dynamics.
(The decoherence rates λ j that appear in Eq. (243), here have been absorbed in L j (t).) The general evolution of a state is given by We will first derive necessary and sufficient conditions for unitarily correctable subsystems under the dynamics (353), and then will extend them to the case of unitarily recoverable subsystems.
In the case of continuous dynamics, the error map E and the error-correcting map U in Eq. (342) are generally time dependent. If we set t = 0 as the initial time at which the system is prepared, the error map resulting from the dynamics (353) is Let the U(t) = U (t)(·)U † (t) be the unitary error-correcting map in Eq. (342). We can define the rotating frame corresponding to U † (t) as the transformation of each operator as In this frame, the Lindblad equation (353) can be written as where H ′ (t) is defined through i.e., The CPTP map resulting from the dynamics (357) is and and for all t, where P K denotes the projector on K.
Notice that since H A is unitarily correctable by U (t), at time t the initially encoded information can be thought of as contained in the subsystem H A (t) defined through i.e., this subsystem is obtained from H A in Eq. We now extend the above conditions to the case of unitarily recoverable subsystems.
As we pointed out earlier, the difference between a unitarily correctable and a unitarily recoverable subsystem is that in the latter the dimension of the gauge subsystem may increase. Since the dimension of the gauge subsystem is an integer, this increase can happen only in a jump-like fashion at particular moments. Between these moments, the evolution is unitarily correctable. Therefore, we can state the following with respect to which the evolution during this interval is unitarily correctable.
Remark: An increase of the gauge subsystem at time t i happens if the operator C j (t) in Eq. (361) obtains non-zero components that map states from From that moment on, t i ≤ t ≤ t i+1 , Eq. (361) must hold for the new decomposition The unitary U (t) is determined from Eq. (362) and Eq. (363) as described earlier.
The conditions derived in this section provide insights into the mechanism of information preservation under Markovian dynamics, and thus could have implications for the problem of error correction when perfect correctability is not possible [138,86,133,173,58,92]. For example, it is possible that the unitary operation constructed according to Eq. (358) with the appropriate modification for the case of increasing gauge subsystem, may be useful for error-correction also when the conditions of the theorems are only approximately satisfied. Notice that the generator driving the effective evolution of the subspace U † (t)H A ⊗ H B whose projector we denote by can be written as where is an effective Hamiltonian, is a dissipator, and in order to optimally track the retrievable information. We leave this as a problem for future investigation.

Conditions on the system-environment Hamiltonian
We now derive conditions for correctability of a subsystem when the dynamics of the system and the environment is described by the Schrödinger equation (344). While the CP-map conditions can account for such dynamics when the states of the system and the environment are initially disentangled, they depend on the initial state of the environment. Below, we will first derive conditions on the system-environment Hamiltonian that hold for any state of the environment, and then extend them to the case when the environment is initialized inside a particular subspace.
We point out that the equivalence between unitary recoverable subsystems and correctable subsystems has been proven for CPTP maps. Here, we could have a non-CP evolution since the initial state of the system and the environment may be entangled.
Nevertheless, since correctability must hold for the case when the initial state of the system and the environment is separable, the conditions we obtain are necessary. They are obviously also sufficient since unitary recoverability implies correctability.
Let us write the system-environment Hamiltonian as where H S (t) and H E (t) are the system and the environment Hamiltonians respectively, and is the interaction Hamiltonian.
From the point of view of the Hilbert space of the system plus environment, the decomposition (335) reads

Conditions independent of the state of the environment
We will consider again conditions for unitary correctability first, and then conditions for general correctability. In the rotating frame (356), the Schrödinger equation (344) becomes Since in this picture a unitarily-correctable subsystem is noiseless, we can state the i.e., This simply means that the evolution of the subspace that contains the encoded information is driven by the system Hamiltonian.
The conditions again can be separated into two parts.
(Here P BE (t) denotes the projector on H BE (t).) Note that as an operator on the entire Hilbert space, this unitary has the form W 0 (t) Let us define the frame where Then the evolution driven by a Hamiltonian G(t), in this frame will be driven by where H B ′ was defined in (380), such that Proof: Assume that the information encoded in H A is unitarily recoverable by U (t). Consider the evolution in the frame defined through the unitary operation At the same time, we can choose H ′′ (t) so that D BE (t) = 0. This corresponds to where H BE (t) was defined in the discussion before Theorem 5. To ensure D BE (t) = 0, we can choose where P ABE  It is tempting to perform an argument similar to the one we presented for the Markovian case about the possible relation of the specified recovery unitary operation U (t) and the optimal error-correcting map in the case of approximate error correction.
If the encoded information is not perfectly preserved, we can construct the unitary operation U (t) as explained in the comment after Theorem 5 by optimally approximating Eq. (385) and Eq. (387). However, in this case the evolution is not irreversible and the information that leaks out of the system may return back to it. Thus we cannot argue that the unitary map specified in this manner would optimally track the remaining encoded information.

Conditions depending on the initial state of the environment
We can easily extend Theorem 5 to the case when the initial state of the environment belongs to a particular subspace H E 0 ∈ H E . The only modification is that instead of P ABE in Eq. (384), we must have P ABE 0 , where P ABE 0 is the projector on H A ⊗ H B ⊗ H E 0 , and on the right-hand side must have The following two theorems follow by arguments analogous to those for Theorem   such that Notice that the conditions of Theorem 6 and Theorem 7 do not depend on the particular initial state of the environment but only on the subspace to which it belongs.
This can be understood by noticing that different environment states inside the same subspace give rise to Kraus operators (347) which are linear combinations of each other.
The discretization of errors in operator quantum error correction [94,95] implies that all such maps will be correctable.
The conditions for correctable dynamics dependent on the state of the environment could be useful if we are able to prepare the state of the environment in the necessary subspace. The environment, however, is generally outside of the experimenter's control.
Nevertheless, it is conceivable that the experimenter may have some control over the environment (for example, by varying its temperature), which for certain Hamiltonians could bring the environment state close to a subspace for which the evolution of the system is correctable. It is important to point out that according to the result we derive in the next chapter, the error due to imperfect initialization of the bath will not increase under the evolution.

Summary and outlook
We have derived conditions for correctability of subsystems under continuous decoherence. We first presented conditions for the case when the evolution can be described by a CPTP linear map. These conditions are equivalent to those known for operator codes [94,95] except that we consider them for time-dependent noise processes. We then derived condition for the case of Markovian decoherence and general Hamiltonian evolution of the system and the environment. We derived conditions for both unitary correctability and general correctability, using the fact that correctable subsystems are unitarily recoverable [96].
The conditions for correctability in both Markovian and Hamiltonian evolution can be understood as consisting of two parts-the first is necessary and sufficient for lack of noise inside the instantaneous subsystem that contains the information, and the second is necessary and sufficient for states not to leave the subsystem as it evolves with time. In this sense, the new conditions can be thought of a generalizations of the conditions for noiseless subsystems to the case where the subsystem is time-dependent.
In the Hamiltonian case, the conditions for unitary correctability concern only the action of the Hamiltonian on the system, whereas the conditions for general correctability concern the entire system-bath Hamiltonian. The reason for this is that the state of the gauge subsystem plus the environment generally belongs to a particular subspace, which does not factor into sectors belonging separately to the system and the environment. We also derived conditions in the Hamiltonian case that depend on the initial state of the environment. These conditions could be useful, in principle, since errors due to imperfect initialization of the environment do not increase under the evolution. Furthermore, these conditions could provide a better understanding of correctability under CPTP maps, since a CPTP map that results from Hamiltonian evolution depends on both the Hamiltonian and the initial state of the environment.
An interesting generalization of this work would be to derive similar condition for the case of the Nakajima-Zwanzig or the TCL master equations.
We discussed possible implications of the conditions we derived for the problem of optimal recovery in the case of imperfectly preserved information. We hope that the results obtained in this study will provide insight into the mechanisms of information flow under decoherence that could be useful in the area of approximate error correction as well.
Chapter 7: Robustness of operator quantum error correction against initialization errors The conditions we derived in the previous chapter, as well as the standard OQEC conditions for discrete errors, depend on the assumption that states are perfectly initialized inside the subspace factored by the correctable subsystem. In practice, however, perfect initialization of the state may not be easy to achieve. Hence, it is important to understand to what extent the preparation requirement can be relaxed. In this chapter, we examine the performance of OQEC in the case of imperfect encoding.

Preliminaries
As can be seen from the definitions (338) and (341), the concept of noiseless subsystem is a cornerstone in the theory of OQEC; it serves as a basis for the definition of correctable subsystem and error correction in general. As shown in Ref. [140], in order to ensure perfect noiselessness of a subsystem in the case of imperfect initialization, the noise process has to satisfy more restrictive conditions than those required in the case of perfect initialization. It was believed that these conditions are necessary if a noiseless (or more generally decoherence-free) subsystem is to be robust against arbitrarily large initialization errors. The fundamental relation between a noiseless subsystem and a correctable subsystem implies that in the case of imperfect initialization, more restrictive conditions would be needed for OQEC codes as well.
In this chapter we show that with respect to the ability of a code to protect from errors, more restrictive conditions are not necessary. For this purpose, we define a measure of the fidelity between the encoded information in two states for the case of subsystem encoding. We first give an intuitive motivation for the definition, and then study the properties of the measure. We then show that the effective noise that can arise inside the code due to imperfect initialization under the standard conditions, is such that it can only increase the fidelity of the encoded information with the information encoded in a perfectly prepared state. This robustness against initialization errors is shown to hold also when the state is subject to encoded operations. If, however, one allows imperfect initialization, ρ(0) = P AB (ρ(0)), this need not be the case. Consider the "initialization-free" analogue of the definition (338):

Kraus operators
Obviously Eq. () implies Eq. (338), but the reverse is not true. As shown in [140], the definition () imposes more restrictive conditions on the channel E than those imposed by (338). To see this, consider the form of the Kraus operators M i of E ((336)) in the block basis corresponding to the decomposition (335). From a result derived in [140] it follows that the subsystem H A is noiseless in the sense of Eq. (338), if and only if the Kraus operators have the form where the upper left block corresponds to the subspace H A ⊗ H B , and the lower right block corresponds to K. The completeness relation (337) implies the following conditions on the operators C B i , D i , and G i : In the same block basis, a perfectly initialized state ρ and its image under the map (392) have the form Using the linearity and cyclic invariance of the trace together with Eq. (393), we obtain i.e., the reduced operator on H A remains invariant.
On the other hand, an imperfectly initialized stateρ and its image have the form Hereρ 2 and/orρ 3 are non-vanishing, and In this case, using the linearity and cyclic invariance of the trace together with Eq. (393) and Eq. (394), we obtain i.e., the reduced operator on H A is not preserved. It is easy to see that the reduced operator would be preserved for every imperfectly initialized state if and only if we impose the additional condition Since the weight of any state already present in the mixture can only increase by this process, this should not worsen the faithfulness with which information is encoded inρ. In order to make this argument rigorous, however, we need a measure that quantifies the faithfulness of the encoding.

Motivating the definition
If we consider two states with density operators τ and υ, a good measure of the faithfulness with which one state represents the other is given by the fidelity between the states: This quantity can be thought of as a square root of a generalized "transition probability" between the two states τ and υ as defined by Uhlmann [157]. Another interpretation due to Fuchs [59] gives an operational meaning of the fidelity as the minimal overlap between the probability distributions generated by all possible generalized measurements on the states: Here, minimum is taken over all positive operators {M i } that form a positive operator valued measure (POVM) [93], In our case, we need a quantity that compares the encoded information in two states. Clearly, the standard fidelity between the states will not do since it measures the similarity between the states on the entire Hilbert space. The encoded information, however, concerns only the reduced operators on subsystem H A . In view of this, we propose the following Definition 1: Let τ and υ be two density operators on a Hilbert space H S with decomposition (335). The fidelity between the information encoded in subsystem H A in the two states is given by: where maximum is taken over all density operators τ ′ and υ ′ that have the same reduced operators on H A as τ and υ: The intuition behind this definition is that by maximizing over all states that have the same reduced operators on H A as the states being compared, we ensure that the measure does not penalize for differences between the states that are not due specifically to differences between the reduced operators.

Properties of the measure
Property 1 (Symmetry): Since the fidelity is symmetric with respect to its inputs, it is obvious from Eq. (406) that F A is also symmetric: Although intuitive, the definition (406) does not allow for a simple calculation of F A . We now derive an equivalent form for F A , which is simple and easy to compute.
Let P K (·) = P K (·)P K denote the superoperator projector on B(K), and let denote the normalized reduced operator of ρ on H A .

Theorem 1: The definition (406) is equivalent to
where Proof: Let τ * and υ * be two states for which the maximum on the right-hand side of Eq. (406) is attained. From the monotonicity of the standard fidelity under CPTP maps [16] it follows that where Π(·) = P AB (·) + P K (·). But the states Π(τ * ) and Π(υ * ) satisfy i.e., they are among those states over which the maximum in Eq. (406) is taken. Therefore, Using Eq. (404) and the fact that in the block basis corresponding to the decomposition (335) the states Π(τ * ) and Π(υ * ) have block-diagonal forms, it is easy to see that whereF is a function that has the same expression as the fidelity (404) 182 Since τ * and σ * should maximize the right-hand side of Eq. (415), and the only restriction on P K (τ * ) and P K (υ * ) is Tr{P K (τ * )} = Tr{P K (τ )}, Tr{P K (υ * )} = Tr{P K (υ)}, we must have i.e., Thus we obtainF The term (416) also must be maximized.
This completes the proof.
We next provide an operational interpretation of the measure F A . For this we need the following Lemma: The function f A (τ, υ) defined in Eq. (410) equals the minimum overlap between the statistical distributions generated by all local measurements on subsystem Note that since the operators M i do not form a complete POVM on the entire Hilbert space, the probability distributions p τ (i) = Tr{M i τ } and p υ (i) = Tr{M i υ} generated by such measurements generally do not sum up to 1. This reflects the fact that a measurement on subsystem H A requires a projection onto the subspace , it is realized through post-selection.
Proof: Using that we can write Eq. (422) in the form From Eq. (405), we see that (424) is equivalent to (410).
Theorem 2: F A (τ, υ) equals the minimum overlap between the statistical distributions generated by all possible measurements of the Proof: The proof follows from Eq. (409) and Eq. (422).
Note that the measure F A compares the information stored in subsystem H A , which is the information extractable through local measurements on H A . The last result reflects the intuition that extracting information encoded in H A involves a measurement that projects on the subspaces H A ⊗ H B or K.

Property 2 (Normalization): From the definition (406) it is obvious that
From Eq. (409) we can now see that as one would expect from a measure that compares only the encoded information in Proposition: Using that the maximum in Eq. (406) is attained for states of the form Π(τ * ) and Π(υ * ) (Eq. (414)) where τ * and υ * satisfy Eq. (419) and Eq. (421), without loss of generality we can assume that for all τ and υ, where with |0 B and |0 K being some fixed states in H B and K, respectively.
This means that the property of strong concavity of the fidelity [115] (and all weaker concavity properties that follow from it) as well as the concavity of the square of the fidelity [157], are automatically satisfied by the measure F A .

Definition 2:
Similarly to the concept of angle between two states [115] which can be defined from the standard fidelity, we can define an angle between the encoded information in two states: Property 4 (Triangle inequality): From Eqs. (428)-(430) it follows that just as the angle between states satisfies the triangle inequality, so does the angle between the encoded information: for

Robustness of OQEC with respect to initialization errors
Let us now consider the fidelity between the encoded information in an ideally prepared state (396) and in a state which is not perfectly initialized (398): = Tr Tr B ρ 1 Tr Bρ1 Tr B ρ 1 ≡F (Tr B ρ 1 , Tr Bρ1 ).
After the noise process E with Kraus operators (392), the imperfectly encoded state transforms to E(ρ). Its fidelity with the perfectly encoded state becomes where we have used the expressions for Tr B ρ ′ 1 and Tr Bρ ′ 1 obtained in Eq. (397) and Eq. (402). As we pointed out earlier, the operator Then from the concavity of the square of the fidelity [157], it follows thať (Here, the transition from the first to the second line is obtained by pulling out the normalization factors of the operators inF so that the latter can be expressed in terms of the fidelity F . The transition form the second to the third line is by using the concavity of the square of the fidelity. The last line is obtained by expressing the quantities again in terms ofF ). Therefore, we can state the following We see that even if the "initialization-free" constraint (403) is not satisfied, no further decrease in the fidelity occurs as a result of the process. The effective noise that arises due to violation of that constraint, can only decrease the initialization error.
The above result can be generalized to include the possibility for information processing on the subsystem. Imagine that we want to perform a computational task which ideally corresponds to applying the CPTP map C A on the encoded state. In general, the subsystem H A may consist of many subsystems encoding separate information units (e.g., qubits), and the computational process may involve many applications of error correction. The noise process itself generally acts continuously during the computation. Let us assume that all operations following the initialization are performed fault-tolerantly [145,2,85,91,67] so that the overall transformation C on a perfectly initialized state succeeds with an arbitrarily high probability (for a model of fault-tolerant quantum computation on subsystems, see, e.g., Ref. [11]). This means that the effect of C on the reduced operator of a perfectly initialized state is up to an arbitrarily small error.
Again, the preparation error is not amplified by the process. The problem of how to deal with preparation errors has been discussed in the context of fault-tolerant computation on standard error-correction codes, e.g., in Ref. [131]. The situation for general OQEC is similar-if the initial state is known, the error can be eliminated by repeating the encoding. If the state to be encoded is unknown, the preparation error generally cannot be corrected. Nevertheless, encoding would still be worthwhile as long as the initialization error is smaller than the error which would result from leaving the state unprotected.

Summary and outlook
In summary, we have shown that a noiseless subsystem is robust against initialization errors without the need for modification of the noiseless subsystem conditions. Similarly, we have argued that general OQEC codes are robust with respect to imperfect preparation in their standard form. This property is compatible with fault-tolerant methods of computation, which is essential for reliable quantum information processing. In order to rigorously prove our result, we introduced a measure of the fidelity F A (τ, υ) between the encoded information in two states. The measure is defined as 190 the maximum of the fidelity between all possible states which have the same reduced operators on the subsystem code as the states being compared. We derived a simple form of the measure and discussed many of its properties. We also gave an operational interpretation of the quantity.
Since the concept of encoded information is central to quantum information science, the fidelity measure introduced in this study may find various applications. It provides a natural means for extending key concepts such as the fidelity of a quantum channel [89] or the entanglement fidelity [137] to the case of subsystem codes.
Chapter 8: A fault-tolerant scheme for holonomic quantum computation

Preliminaries
There are two main sources of errors in quantum computers-environment-induced decoherence and imperfect control. According to the theory of fault tolerance [145,53,2,85,91,68,67,131], if the errors of each type are sufficiently uncorrelated and their rates are bellow a certain threshold, it is possible to implement reliably an arbitrarily long computational task with an efficient overhead of resources. Quantum error correction thus provides a universal software strategy to combat noise in quantum computers.
In addition to the software approach, there have also been proposals to deal with the effects of noise by hardware methods that provide robustness through their inherent properties. One such method is holonomic quantum computation (HQC) [177,123]-an adiabatic, all-geometric method of computation which uses non-Abelian generalizations [168] of the Berry phase [25]. It has been shown that due to its geometric nature, this approach is robust against various types of errors in the control parameters driving the evolution [43,49,148,60,179], and thus provides a degree of built-in resilience at the hardware level.
In Ref. [172] HQC was combined with the method of decoherence-free subspaces (DFSs) [55,176,105,103], which was the first step towards systematic error protection in conjunction with the holonomic approach. DFSs provide passive protection against certain types of correlated noise; however, they cannot protect against independent errors. The standard tool to deal with the latter is active error correction [144,150]. Active error correction is also the basis of quantum fault tolerance, which is necessary for the scalability of any method of computation. Even if the system is perfectly isolated from its environment, when the size of the circuit increases, errors due to imperfect operations would accumulate detrimentally unless they are corrected.
Therefore, scalability of HQC requires combining the holonomic approach with active error correction.
In this chapter, we presented a scheme which combines HQC with the techniques for fault-tolerant computation on stabilizer codes. This demonstrates that HQC is a scalable method of computation [121]. The scheme uses Hamiltonians which are elements of the stabilizer, or in the case of subsystem codes-elements of the gauge group. Gates are implemented by slowly varying the Hamiltonians along suitable paths in parameter space, such that the resulting geometric transformation in each eigenspace of the Hamiltonian is transversal. On certain codes such as the 9-qubit Shor code [144] or its subsystem generalizations [13,14], universal computation according to our scheme can be implemented with Hamiltonians of weight 2 and 3.
The parameters λ µ are local coordinates on M (1 ≤ µ ≤ dimM) and {|nα; λ } dn α=1 is an orthonormal basis of the n th eigenspace of the Hamiltonian at the point λ.
When the path λ(t) forms a loop γ(t), γ(0) = γ(T ) = λ 0 , the unitary matrix is called the holonomy associated with the loop. In the case when the n th energy level is non-degenerate (d n = 1), the corresponding holonomy reduces to the Berry phase In Refs. [177,123] it was shown that if the dimension of the control manifold is sufficiently large, quantum holonomies can be used as a means of universal quantum computation. In the proposed approach, logical states are encoded in the degenerate eigenspace of a Hamiltonian and gates are implemented by adiabatically varying the Hamiltonian along suitable loops in the parameter manifold (for a construction of a universal set of gates, see also Ref. [116]).

Stabilizer codes and fault tolerant computation
A large class of quantum error-correcting codes can be described by the so called stabilizer formalism [66,37,38]. A stabilizer S is an Abelian subgroup of the Pauli group G n on n qubits, which does not contain the element −I [115]. operator stabilizer code encoding k qubits into n with r gauge qubits, has n − r − k stabilizer generators, while the gauge group has 2r generators [130]. According to the error-correction condition for stabilizer codes [115,130] A quantum information processing scheme is called fault-tolerant if a single error occurring during the implementation of any given operation introduces at most one error per block of the code [68]. This property has to apply for unitary gates as well as measurements, including those that constitute the error-correcting operations themselves. Fault-tolerant schemes for computation on stabilizer codes generally depend on the code being used-some codes, like the Bacon-Shor subsystem codes [13,14] for example, are better suited for fault-tolerant computation than others [11].
In spite of these differences, however, it has been shown that fault-tolerant information processing is possible on any stabilizer code [68,67]. The general procedure can be and Knill's [87] methods). Since single-qubit unitaries together with the C-NOT gate form a universal set of gates, fault-tolerant computation can be realized entirely in terms of single-qubit operations and C-NOT operations between qubits from different blocks, assuming that the "cat" state can be prepared reliably. Hence, our goal will be to construct holonomic realizations of these operations as well as of the preparation of the "cat" state. It is not evident that by doing so we will obtain a fault-tolerant construction, because the geometric approach requires that we use degenerate Hamiltonians which generally couple qubits within the same block. Nevertheless, we will see that it is possible to design the scheme so that single-qubit errors do not propagate.

The scheme
Consider an [[n, 1, r, 3]] stabilizer code. This is a code that encodes 1 qubit into n, has r gauge qubits, and can correct arbitrary single-qubit errors. In order to perform a holonomic operation on this code, we need a nontrivial starting Hamiltonian which leaves the code space invariant. It is easy to verify that the only Hamiltonians that satisfy this property are linear combinations of the elements of the stabilizer, or in the case of subsystem codes-elements of the gauge group. Note that the stabilizer and the gauge group transform during the course of the computation under the operations being applied. At any stage when we complete an encoded operation, they return to their initial forms. During the implementation of a standard encoded gate, the Pauli group G n on a given codeword may spread over other codewords, but it can be verified that this spreading can be limited to at most 4 other codewords counting the "cat" state. This is because the encoded C-NOT gate can be implemented fault-tolerantly on any stabilizer code by a transversal operation on 4 encoded qubits [67], and any encoded Clifford gate can be realized using only the encoded C-NOT provided that we are able to do fault-tolerant measurements (the encoded Clifford group is generated by the encoded Hadamard, Phase and C-NOT gates). Encoded gates outside of the Clifford group, such as the encoded π/8 or Toffoli gates, can be implemented faulttolerantly using encoded C-NOT gates conditioned on the qubits in a "cat" state, so they may require transversal operations on a total of 5 blocks. More precisely, the faulttolerant implementation of the Toffoli gate requires the preparation of a special state of three encoded qubits [145], which involves a sequence of conditional encoded Phase operations and conditional encoded C-NOT operations with conditioning on the qubits in a "cat" state [67]. But the encoded Phase gate has a universal implementation using an encoded C-NOT between the qubit and an ancilla, so the conditional Phase gate may require applying a conditional encoded C-NOT. The procedure for implementing an encoded π/8 gate involves applying an encoded SX gate conditioned on the qubits in a "cat" state [27], where is the Phase gate, but the encoded S gate generally also involves an encoded C-NOT on the qubit and an ancilla, so it may also require the interaction of 4 blocks. For CSS codes, however, the spreading of the Pauli group of one block during the implementation of a basic encoded operation can be limited to a total of 3 blocks, since the encoded C-NOT gate has a transversal implementation [67].
It also has to be pointed out that fault-tolerant encoded Clifford operations can be implemented using only Clifford gates on the physical qubits [67]. These operations transform the stabilizer and the gauge group into subgroups of the Pauli group, and their elements remain in the form of tensor products of Pauli matrices. The faulttolerant implementation of encoded gates outside of the Clifford group, however, involves operations that take these groups outside of the Pauli group. We will, therefore, consider separately two cases-encoded operations in the Clifford group, and encoded operations outside of the Clifford group.

Encoded operations in the Clifford group
In Ref. [67] it was shown that every encoded operation in the Clifford group can be implemented fault-tolerantly using Clifford gates on physical qubits. The Clifford group is generated by the Hadamard, Phase and C-NOT gates, but in addition to these gates, we will also demonstrate the holonomic implementation of the X and Z gates which are standard for quantum computation. We will restrict our attention to implementing single-qubit unitaries on the first qubit in a block, as well as C-NOT operations between the first qubits in two blocks. The operations on the rest of the qubits can be obtained analogously.

Single-qubit unitary operations
In order to implement a single-qubit holonomic operation on the first qubit in a block, we will choose as a starting Hamiltonian an element of the stabilizer (with a minus sign) or an element of the gauge group that acts non-trivially on that qubit. Since we are considering codes that can correct arbitrary single-qubit errors, one can always find an element of the initial stabilizer or the initial gauge group that has a factor σ 0 = I, σ 1 = X, σ 2 = Y or σ 3 = Z acting on the first qubit, i.e., where G is a tensor product of Pauli matrices and the identity on the rest n − 1 qubits.
It can be verified that under Clifford gates the stabilizer and the gauge group transform in such a way that this is always the case except that the factor G may spread on qubits in other blocks. From now on, we will use "hat" to denote operators on all these qubits and "tilde" to denote operators on all the qubits excluding the first one.
Without loss of generality we will assume that the chosen stabilizer or gauge-group element for that qubit has the form As initial Hamiltonian, we will take the operator Thus, if G is an element of the stabilizer, the code space will belong to the ground space of H(0). Our goal is to find paths in parameter space such that when the Hamiltonian is varied adiabatically along these paths, it gives rise to single-qubit transformations on the first qubit in each of its eigenspaces.
Proposition: If the initial Hamiltonian (451) is varied adiabatically so that only the factor acting on the first qubit changes, where the geometric transformation resulting in each of the eigenspaces of this Hamiltonian will be equivalent to a local unitary on the first qubit, U (t) ≈ U (t) ⊗ I.
Proof. Observe that (452) can be written as where are orthogonal complementary projectors. The evolution driven by H(t) is therefore where Let |φ 0 (t) and |φ 1 (t) be the instantaneous ground and excited states of H(t) with eigenvalues E 0,1 (t) = ∓E(t) (E(t) > 0). Using Eq. (444) for the expressions (457), we obtain that in the adiabatic limit where ω(t) = t 0 dτ E(τ ) and The projectors on the instantaneous ground and excited eigenspaces of H(t) are and i.e, up to an overall dynamical phase its effect on each of the eigenspaces is the same as that of the unitary where We next show how by suitably choosing H(t) we can implement all necessary single-qubit gates. We will identify a set of points in parameter space, such that by interpolating between these points we can draw various paths resulting in the desired transformations. We remark that if a path does not form a loop, the resulting geometric transformation (465) is an open-path holonomy [98].
Consider the single-qubit unitary operator where θ is a real parameter (note that V θ− = V † θ+ ). Define the following single-qubit Hamiltonian: 202 Let H(t) in Eq. (452) be a Hamiltonian which interpolates between H(0) = Z and H(T ) = H θ± (up to a factor) as follows: where f (0), g(T ) > 0, f (T ) = g(0) = 0. To simplify our notations, we will drop the indices f and g of the Hamiltonian, since the exact form of these functions is not important for our analysis as long as they are sufficiently smooth (see discussion below). This Hamiltonian has eigenvalues ± f (t) 2 + g(t) 2 and its energy gap is nonzero unless the entire Hamiltonian vanishes. We will show that in the adiabatic limit, the Hamiltonian (452) with H(t) = H θ± (t) gives rise to the geometric transformation To prove this, observe that where W θ is the Hermitian unitary The unitaries U θ±0,1 , given by Eq. (457) for H(t) = H θ± (t), are then related by 203 Let us define the eigenstates of H θ± (t) at time T as |φ θ±0 (T ) = V θ± |0 and |φ θ±1 (T ) = V θ± |1 . Expression (459) can then be written as where α θ±0 and α θ±1 are geometric phases. Without explicitly calculating the geometric phases, from Eq. (474) and Eq. (473) we obtain Therefore, up to a global phase, Eq. (465) yields We will use this result, to construct a set of standard gates by sequences of operations of the form V θ± , which can be generated by interpolations of the type (468) run forward or backward. For single-qubit gates in the Clifford group, we will only need three values of the parameter θ: 0, π/2 and π/4. For completeness, however, we will also demonstrate how to implement the π/8 gate, which together with the Hadamard gate is sufficient to generate any single-qubit unitary transformation [27]. For this we will need θ = π/8. Note that 204 so for these values of θ we have H 0± = ±X, H π/2± = ±Y , H π/4± = ±( 1 Consider the adiabatic interpolations between the following Hamiltonians: According to the above result, the first interpolation yields the transformation V π/2+ .
The second interpolation can be regarded as the inverse of and − H(t) yield the same geometric transformations. Thus the second interpolation results in V † π/2− = V π/2+ . The net result is therefore V π/2+ V π/2+ = iX. We see that up to a global phase the above sequence results in a geometric implementation of the X gate.
Similarly, one can verify that the Z gate can be realized via the loop The Phase gate can be realized by applying followed by the X gate.
The Hadamard gate can be realized by first applying Z, followed by Finally, the π/8 gate can be implemented by first applying XZ, followed by

A note on the adiabatic condition
Before we show how to implement the C-NOT gate, let us comment on the conditions under which the adiabatic approximation assumed in the above operations is satisfied.
Because of the form (456) of the overall unitary, the adiabatic approximation depends on the extent to which each of the unitaries (457) approximate the expressions (458).
The latter depends only on the properties of the single-qubit Hamiltonian H(t), for which the adiabatic condition [110] reads where and ∆ = min is the minimum energy gap of H(t).
Along the segments of the parameter paths we described, the Hamiltonian is of the form (468) and its derivative is This derivative is well defined as long as df (t) dt and dg(t) dt are well defined. The curves we described, however, may not be differentiable at the points connecting two segments. In order for the Hamiltonians (468) that interpolate between these points to be differentiable, the functions f (t) and g(t) have to satisfy df (T ) dt = 0 and dg(0) dt = 0.
This means that the change of the Hamiltonian slows down to zero at the end of each segment (except for a possible change in its strength), and increases again from zero 206 along the next segment. We point out that when the Hamiltonian stops changing, we can turn it off completely by decreasing its strength. This can be done arbitrarily fast and it would not affect a state which belongs to an eigenspace of the Hamiltonian.
Similarly, we can turn on another Hamiltonian for the implementation of a different operation.
The above condition guarantees that the adiabatic approximation is satisfied with . It is known, however, that under certain conditions on the Hamiltonian, we can obtain better results [70]. Let us write the Schrödinger equation as where ǫ > 0 is small. IfH(t) is smooth and all its derivatives vanish at the end points t = 0 and t = T , the error would scale super-polynomially with ǫ, i.e., it will decrease with ǫ faster than O(ǫ N ) for any N . (Notice that ε ∆ 2 ∝ ǫ, i.e., the error according to the standard adiabatic approximation is of order O(ǫ 2 ).) In our case, the smoothness condition translates directly to the functions f (t) and g(t). For any choice of these functions which satisfies the standard adiabatic condition, we can ensure that the stronger condition is satisfied by the reparameterization f (t) → f (y(t)), g(t) → g(y(t)) where y(t) is a smooth function of t which satisfies y(0) = 0, y(T ) = T , and has vanishing derivatives at t = 0 and t = T . Then by slowing down the change of the Hamiltonian by a constant factor ǫ, which amounts to an increase of the total time T by a factor 1/ǫ, we can decrease the error super-polynomially in ǫ. We will use this result to obtain a low-error interpolation in Section 8.5 where we estimate the time needed to implement a holonomic gate with certain precision.
holonomic computation is possible in principle. In Section 8.6 we show that depending on the code one can find more natural implementations of these operations.
If we want to apply a second conditional Clifford operation Q on the first qubits in the block, we can do this via the Hamiltonian where H Q (t)⊗G 2,...,n is now the Hamiltonian we would use to implement the operation Q, had we implemented the operation O before that. Here again, the factor β(t) guarantees that there is no splitting of the energy levels of the Hamiltonian. Subsequent operations are applied analogously. Using this general method, we can implement holonomically any transversal Clifford operation conditioned on the "cat" state.

Using the "cat" state
In addition to transversal operations, a complete fault-tolerant scheme requires the ability to prepare, verify and use a special ancillary state such as the "cat" state (|00...0 + |11...1 )/ √ 2 proposed by Shor [145]. This can also be done in the spirit of our holonomic scheme. Since the "cat" state is known and its construction is nonfault-tolerant, we can prepare it by simply treating each initially prepared qubit as a simple code (with G in Eq. (450) being trivial), and updating the stabilizer of the code via the applied geometric transformation as the operation progresses. The stabilizer of the prepared "cat" state is generated by Z i Z j , i < j. Transversal unitary operations between the "cat" state and other codewords are applied as described in the previous section.
We also have to be able to measure the parity of the state, which requires the ability to apply successively C-NOT operations from two different qubits in the "cat" state to one and the same ancillary qubit initially prepared in the state |0 . We can regard the qubit in state |0 as a simple code with stabilizer Z , and we can apply the first C-NOT as described before. Even though after this operation the state of the target qubit is unknown, the second C-NOT gate can be applied via the same interaction, since the transformation in each eigenspace of the Hamiltonian is the same and at the end when we measure the qubit we project on one of the eigenspaces.

Fault-tolerance of the scheme
We showed how we can generate any transversal operation on the code space holonom- It is remarkable that even though the Hamiltonian couples qubits within the same block, single-qubit errors do not propagate. This is because the coupling between the qubits amounts to a change in the relative phase between the ground and excited spaces, but the latter is irrelevant since it is either equivalent to a gauge transformation, or when we apply a correcting operation we project on one of the eigenspaces. In the case of the C-NOT gate, an error can propagate between the control and the target qubits, but it never results in two errors within the same codeword.

Effects on the accuracy threshold for environment noise
Since the method we presented conforms completely to a given fault-tolerant scheme, it would not affect the error threshold per operation for that scheme. Some of its features, however, would affect the threshold for environment noise.
First, observe that when applying the Hamiltonian (452), we cannot at the same time apply operations on the other qubits on which the factor G acts non-trivially.
Thus, some operations at the lowest level of concatenation that would otherwise be implemented simultaneously might have to be implemented serially. The effect of this is equivalent to slowing down the circuit by a constant factor. (Note that we could also vary the factor G simultaneously with H(t), but in order to obtain the same precision as that we would achieve by a serial implementation, we would have to slow down the change of the Hamiltonian by the same factor.) The slowdown factor resulting from this loss of parallelism is usually small since this problem occurs only at the lowest level of concatenation. For example, for the Bacon-Shor code, we can implement operation on up to 6 out of the 9 qubits in a block simultaneously. As we show in Section 8.6, when implementing an encoded single-qubit gate, we can address any two qubits in a row or column using our method by taking G in Eq. (452) to be a single-qubit operator Z or X on the third qubit in the same row or column. The Hamiltonians used for applying operations on the two qubits commute with each other at all times and do not interfere. A similar thing holds for implementation of the encoded C-NOT and the operations involving the "cat" state. Thus for the Bacon-Shor code we have a slowdown due to parallelism by a factor of 1.5.
A more significant slowdown results from the fact that the evolution is adiabatic.
In order to obtain a rough estimate of the slowdown due specifically to the adiabatic requirement, we will compare the time T h needed for the implementation of a holonomic gate with precision 1 − δ to the time T d needed for a dynamical realization of the same gate with the same strength of the Hamiltonian. We will consider a realization of the X gate via the interpolation where τ (0) = 0, τ (T h ) = T h . Thus the energy gap of the Hamiltonian is always at maximum. The optimal dynamical implementation of the same gate is via the Hamiltonian −X for time T d = π 2 .
As we argued in Section 8.4, the accuracy with which the adiabatic approximation holds for the Hamiltonian (499) is the same as that for the Hamiltonian We now present estimates for two different choices of the function τ (t). The first one is τ (t) = t.
In this case the Schrödinger equation can be easily solved in the instantaneous eigenbasis of the Hamiltonian (500). For the probability that the initial ground state remains a ground state at the end of the evolution, we obtain where 215 Expanding in powers of ε and averaging the square of the cosine whose period is much smaller than T h , we obtain the condition Assuming, for example, that δ ≈ 10 −4 (approximately the threshold for the 9-qubit Bacon-Shor [11]), we obtain that the time of evolution for the holonomic case must be about 70 times longer than that in the dynamical case.
It is known, however, that if H(t) is smooth and its derivatives vanish at t = 0 and t = T h , the adiabatic error decreases super-polynomially with T h [70]. To achieve this, we will choose For this interpolation, by a numerical solution we obtain that when T h /T d ≈ 17 the error is already of the order of 10 −6 , which is well below the threshold values obtained for the Bacon-Shor codes [11]. This is a remarkable improvement in comparison to the previous interpolation which shows that the smoothness of the Hamiltonian plays an important role in the performance of the scheme.
An additional slowdown in comparison to a perfect dynamical scheme may result from the fact that the constructions for some of the standard gates we presented involve long sequences of loops. With more efficient parameter paths, however, it should be possible to reduce this slowdown to minimum. An approach for finding loops presented in Ref. [116] may be useful in this respect.
In comparison to a dynamical implementation, the allowed rate of environment noise for the holonomic case would decrease by a factor similar to the slowdown factor.
In practice, however, dynamical gates are not perfect and the holonomic approach may be advantageous if it gives rise to a better operational precision.
We finally point out that an error in the factor H(t) in the Hamiltonian (452) would result in an error on the first qubit according to Eq. (465). Such an error clearly has to be below the accuracy threshold. More dangerous errors, however, are also possible. For example, if the degeneracy of the Hamiltonian is broken, this can result in an unwanted dynamical transformation affecting all qubits on which the Hamiltonian acts non-trivially. Such multi-qubit errors have to be of higher order in the threshold, which imposes more severe restrictions on the Hamiltonian.

Fault-tolerant holonomic computation with low-weight Hamiltonians
The weight of the Hamiltonians needed for the scheme we described depend on the weight of the stabilizer or gauge-group elements. Remarkably, certain codes possess stabilizer or gauge-group elements of low weight covering all qubits in the code, which allows us to perform holonomic computation using low-weight Hamiltonians. Here we will consider as an example a subsystem generalization of the 9-qubit Shor code [144]-the Bacon-Shor code [13,14]-which has particularly favorable properties for fault-tolerant computation [10,11]. In the 9-qubit Bacon-Shor code, the gauge group is generated by the weight-two operators Z k,j Z k,j+1 and X j,k X j+1,k , where the subscripts label the qubits by row and column when they are arranged in a 3 × 3 square lattice.
Since the Bacon-Shor code is a CSS code, the C-NOT gate has a direct transversal implementation. We now show that the C-NOT gate ca be realized using at most weight-three Hamiltonians.
If we want to apply a C-NOT gate between two qubits each of which is, say, in the first row and column of its block, we can use as a starting Hamiltonian −Z t 1,1 ⊗ Z t 1,2 , where the superscript t signifies that these are operators in the target block. We can then apply the C-NOT gate as described in the previous section. After the operation, however, this gauge-group element will transform to −Z t 1,1 ⊗Z c 1,1 ⊗Z t 1,2 . If we now want to implement a C-NOT gate between the qubits with index {1, 2} using as a starting Hamiltonian the operator −Z t 1,1 ⊗ Z c 1,1 ⊗ Z t 1,2 according to the same procedure, we will have to use a four-qubit Hamiltonian. Of course, at this point we can use the starting Hamiltonian −Z t 1,2 ⊗ Z t 1,3 , but if we had also applied a C-NOT between the qubits labeled {1, 3}, this operator would not be available-it would have transformed to −Z t 1,2 ⊗ Z t 1,3 ⊗ Z c 1,3 .
What we can do instead, is to use as a starting Hamiltonian the operator −Z t 1,1 ⊗ Z t 1,2 ⊗ Z c 1,2 which is obtained from the gauge-group element Z t 1,1 ⊗ Z c 1,1 ⊗ Z t 1,2 ⊗ Z c 1,2 after the application of the C-NOT between the qubits with index {1, 1}. Since the C-NOT gate is its own inverse, we can regard the factor Z t 1,1 as G in Eq. (488) and use this starting Hamiltonian to apply our procedure backwards. Thus we can implement any transversal C-NOT gate using at most weight-three Hamiltonians.
Since the encoded X, Y and Z operations have a bitwise implementation, we can always apply them according to our procedure using Hamiltonians of weight 2. For the Bacon-Shor code, the encoded Hadamard gate can be applied via bitwise Hadamard transformations followed by a rotation of the grid to a 90 degree angle [11]. The encoded Phase gate can be implemented using the encoded C-NOT and an ancilla.
We point out that the preparation and measurement of the "cat" state can also be done using Hamiltonians of weight 2. To prepare the "cat" state, we prepare first all qubits in the state |f 0 + = (|0 +|1 )/ √ 2, which can be done by measuring each of them in the |0 , |1 basis (this ability is assumed for any type of computation) and applying the transformation −Z → −X or Z → −X depending on the outcome. To complete the preparation of the "cat" state, apply a two-qubit transformation between the first qubit and each of the other qubits (j > 1) via the transformation Single-qubit transformations on qubits from the "cat" state can be applied according to the method described in the previous section using at most weight-two Hamiltonians.
To measure the parity of the state, we need to apply successively C-NOT operations from two different qubits in the "cat" state to the same ancillary qubit initially prepared in the state |0 . As described in the previous section, this can also be done according to our method and requires Hamiltonians of weight 2.
For universal computation with the Bacon-Shor code, we also need to be able to apply one encoded transformation outside of the Clifford group. As we mentioned earlier, in order to implement the Toffoli gate or the π/8 gate, it is sufficient to be able to implement a C-NOT gate conditioned on a "cat" state. For the Bacon-Shor code, the C-NOT gate has a transversal implementation, so the conditioned C-NOT gate can be realized by a series of transversal Toffoli operations between the "cat" state and the two encoded states. We now show that the latter can be implemented using at most three-qubit Hamiltonians.
Ref. [115] provides a circuit for implementing the Toffoli gate as a sequence of oneand two-qubit gates. We will use the same circuit, except that we flip the control and target qubits in every C-NOT gate using the identity where R i denotes a Hadamard gate on the qubit labeled by i and C i,j denotes a C-NOT gate between qubits i and j with i being the control and j being the target.
Let Toffoli i,j,k denote the Toffoli gate on qubits i, j and k with i and j being the two control qubits and k being the target qubit, and let S i and T i denote the Phase and π/8 gates on qubit i, respectively. Then the Toffoli gate on three qubits (the first one of which we will assume to belong to the "cat" state), can be written as: To show that each of the above gates can be implemented holonomically using Hamiltonians of weight at most 3, we will need an implementation of the C-NOT gate which is suitable for the case when we have a stabilizer or gauge-group element of the form where the factor X acts on the target qubit and G acts trivially on the control qubit.
By a similar argument to the one in Section 8.4, one can verify that in this case the C-NOT gate can be implemented as follows: apply the operation S † on the control qubit (we describe how to do this for our particular case below) together with the transformation −X ⊗ G → −Z ⊗ G → X ⊗ G (510) on the target qubit, followed by the transformation Since the second and the third qubits belong to blocks encoded with the Bacon-Shor code, there are weight-two elements of the initial gauge group of the form Z ⊗ Z covering all qubits. The stabilizer generators on the "cat" state are also of this type.
Following the transformation of these operators according to the sequence of operations (508), one can see that before every C-NOT gate in this sequence, there is an element of the form (509) with G = Z which can be used to implement the C-NOT gate as described provided that we can implement the gate S † on the control qubit. We also point out that all single-qubit operations on qubit 1 in this sequence can be implemented according to the procedure describes in Section 8.4, since at every step we have a weight-two stabilizer element on that qubit with a suitable form. Therefore, all we need to show is how to implement the necessary single-qubit operations on qubits 2 and 3. Due to the complicated transformation of the gauge-group elements during the sequence of operations (508), we will introduce a method of applying a single-qubit operation with a starting Hamiltonian that acts trivially on the qubit.
For implementing single-qubit operations on qubits 2 and 3 we will use as a starting Hamiltonian the operator where the first factor (I i ) acts on the qubit on which we want to apply the operation (2 or 3), and X 1 ⊗ Z is the transformed (after the Hadamard gate R 1 ) stabilizer element of the "cat" state that acts non-trivially on qubit 1 (the factor Z acts on some other qubit in the "cat" state).
To implement a single-qubit gate on qubit 3 for example, we first apply the inter- This results in a two-qubit geometric transformation U 1,3 on qubits 1 and 3. We do not have to calculate this transformation exactly since we will undo it later, but the fact that each eigenspace undergoes the same two-qubit geometric transformation can be verified similarly to the C-NOT gate we described in Section 8.4.
At this point, the Hamiltonian is of the form (451) with respect to qubit 3, and we can apply any single-qubit unitary gate V 3 according to the method described in Section 8.4. This transforms the Hamiltonian to −V 3 Z 3 V † 3 ⊗ X 1 ⊗ Z. We can now "undo" the transformation U 1,3 by the interpolation The latter transformation is the inverse of Eq. (513) up to the single-qubit unitary transformation V 3 , i.e., it results in the transformation V 3 U † 1,3 V † 3 . Thus the net result is which is the desired single-qubit unitary transformation on qubit 3. We point out that during this transformation, a single-qubit error can propagate between qubits 1 and 3, but this is not a problem since we are implementing a transversal Toffoli operation and such an error would not result in more that one error per block of the code.
We showed that for the BS code our scheme can be implemented with at most 3-local Hamiltonians. This is optimal for the construction we presented, since there are no non-trivial codes with stabilizer or gauge-group elements of weight smaller than 2 covering all qubits. One could argue that since the only Hamiltonians that leave the code space invariant are superpositions of elements of the stabilizer or the gauge group, one cannot do better than this. However, it may be possible to approximate the necessary Hamiltonians with sufficient precision using 2-local interactions. A possible direction to consider in this respect is the technique introduced in Ref. [84] for approximating three-local Hamiltonians by two-local ones. This is left as a problem for future investigation.

Conclusion
We described a scheme for fault-tolerant holonomic computation on stabilizer codes, which demonstrates that HQC is a scalable method of computation. The scheme opens the possibility of combining the software protection of error correction with the inherent robustness of HQC against control imperfections. Our construction uses Hamiltonians that are elements of the stabilizer or the gauge group for the code. The Hamiltonians needed for implementing two-qubit gates are at least 3-local. We have shown that computation with at most 3-local Hamiltonians is possible with the Bacon-Shor code.
It is interesting to point out that the adiabatic regime in which our scheme operates is consistent with the model of Markovian decoherence. In Ref. [9] it was argued that the standard dynamical paradigm of fault tolerance is based on assumptions that are in conflict with the rigorous derivation of the Markovian limit. Although the threshold theorem has been extended to non-Markovian models [156,12,3], the Markovian assumption is an accurate approximation for a wide range of physical scenarios [42]. It also allows for a much simpler description of the evolution in comparison to non-Markovian models, as we saw in Chapter 5. In Ref. [9] it was shown that the weak-coupling-limit derivation of the Markovian approximation is consistent with computational methods that employ slow transformations, such as adiabatic quantum computation [57] or HQC. A theory of fault-tolerance for the adiabatic model of computation at present is not known, although significant steps in this direction have been undertaken [79,102]. Our hybrid HQC-QEC scheme provides a solution for the case of HQC. We point out, however, that it is an open problem whether the Markovian approximation makes sense for a fixed value of the adiabatic slowness parameter when the circuit increases in size.
Applying the present strategy to actual physical systems might require modifying our abstract construction in accordance with the available interactions, possibly using superpositions of stabilizer or gauge-group elements rather than single elements as the basic Hamiltonians. Given that simple QEC codes and two-qubit geometric transformations have been realized using NMR [47,78] and ion-trap [50,100] techniques, these systems seem particularly suitable for hybrid HQC-QEC implementations.
We hope that the techniques presented in this study might prove useful in other areas as well. It is possible that some combination of transversal adiabatic transformations and active correction could provide a solution to the problem of fault tolerance in the adiabatic model of computation.
8.8 Appendix: Calculating the holonomy for the Z gate

Linear interpolation
We first demonstrate how to calculate the ground-space holonomy for the Z gate for the case of linear interpolation along each segment of the path, i.e., when f (t) and g(t) in Eq. (468) are In order to calculate the holonomy (447) corresponding to our construction of the Z gate, we need to define a single-valued orthonormal basis of the ground space of the Hamiltonian along the loop described by Eq. (479). Since the Hamiltonian has the form (454) at all times, it is convenient to choose the basis of the form j = 0, 1; k = 1, ..., 2 n−2 , where |χ 0 (λ(t)) and |χ 1 (λ(t)) are ground and excited states of H(t), and | ψ 0k and | ψ 1k are fixed orthonormal bases of the subspaces that support the projectors P 0 and P 1 defined in Eq. (455), respectively. The eigenstates |χ 0 (λ(t)) and |χ 1 (λ(t)) are defined up to an overall phase, but we have to chose the phase such that the states are single-valued along the loop.
Observe that because of this choice of basis, the matrix elements (446) become i.e., the matrix A µ is diagonal. (Since we are looking only at the ground space, we are not writing the index of the energy level). We can therefore drop the path-ordering operator. The resulting unitary matrix U γ jk,j ′ k ′ acting on the subspace spanned by {|jk; λ(0) } is also diagonal and its diagonal elements are These are precisely the Berry phases for the loops described by the states |χ j (λ ).
Since the loop in parameter space consist of four line segments, we can write the last expression as where γ i , i = 1, 2, 3, 4 are the segments indexed in the order corresponding to Eq. (479).
If we parameterize each line segment by the dimensionless time 0 ≤ s ≤ 1, we get U γ jk,jk = exp where the superscript i in |χ i j (s) indicates the segment. In the |0 , |1 basis, we will write these states as The decomposition into weak measurements presents a practical prescription for the implementation of any generalized measurement using weak measurements and feedback control. It also presents a powerful mathematical tool for the study of measurement processes. In practice, there may exist limitations on the type of weak measurements an experimenter can implement, and hence it would be interesting to look at the inverse problem-given a set of weak measurements, what are the generalized measurements that one can generate with them. It might be convenient to recast this problem in terms of the system-ancilla interactions that are available for the implementation of such measurements. The decomposition may prove useful in other problems involving feedback control as well. One of its interesting features is that the evolution that corresponds to it is confined on a specific manifold (the simplex). In that sense, the procedure avoids dissipation into areas from which the state could drift away from the desired outcomes. This property could be helpful in designing optimal feedback-control protocols.
The decomposition into weak measurements furthermore suggests that it may be possible to find a unified description of measurement protocols. The operations applied at a given time during the measurement procedure for generating generalized 232 measurements, drive the evolution of a stochastic process on the simplex, i.e., they can be represented by a stochastic matrix on the coordinate space. This suggests that there may exist a general coordinate space, which includes all such simplexes, on which the most general notion of a measurement protocol can be represented by a stochastic process. The basic object in such a description would not be a quantum state but a classical probability distribution on a space whose coordinates correspond to quantum states. Since stochastic processes are well understood, such a unified description could be useful for studies of measurement-driven schemes.
We also used the decomposition into weak measurements for deriving necessary and sufficient conditions for entanglement monotones. These conditions may be useful for proving monotonicity of conjectured monotones, finding new classes of entanglement measures, or finding measures with particularly nice properties such as additivity.
Another interesting possibility suggested by the existence of necessary and sufficient differential conditions for monotonicity under all types of CPTP transformations, is that it may be possible to think of all quantum operations as generated by infinitesimal operations. It is known that CPTP maps cannot be generated by weak CPTP maps, however, the differential form of the convexity condition can be thought of as a condition for monotonicity under infinitesimal loss of information. Therefore, if we adopt the approach in which the basic objects are ensembles of states and loss of classical information is a basic operation on these objects, it may be possible to arrive at a unified description of the most general form of quantum operations where every operation can be continuously connected to the identity. As we mentioned in Chapter 5, the TCL or NZ projection techniques might be useful also for the effective description of the reduced dynamics of a system subject to non-Markovian decoherence and continuous error correction. Here too, it would be interesting to consider expanding the solution directly. We presented a generalized notion of a Zeno regime applicable for the problem of error correction and identified the bottle-neck mechanism through which the performance of the error-correction scheme depends on this regime. As the Zeno regime plays a central role in the workings of another error-correction approach-dynamical decoupling (DD) [164,56]-it might be useful to apply the insights developed here in the design of hybrid EC-DD schemes.
Another direction for future research is expanding our scheme for continuous error correction based on weak measurement and weak unitary operations to include more sophisticated feedback. Since making full use of the available information about the state can only help, we expect that this approach would lead to schemes with better performance.
One of the problems suggested by our study of the conditions for exact correctability under continuous decoherence, was whether a similar approach to the one we used could be useful in studies concerning approximate error correction. A question we raised is whether the Markovian decoherence process during an infinitesimal time step can be separated into completely correctable and non-correctable parts. If this is the case, it could allow us to formulate conditions for optimal correctability by tracking the evolution of the maximal information that remains during the process. As we argued, for non-Markovian decoherence such an approach cannot be optimal since the information may flow out to the environment and later return back, but it could nevertheless be helpful for finding locally optimal solutions. Even if the answer to this question is negative, the differential approach in studying information loss certainly seems promising. One interesting extension of this work would be to derive conditions for correctability in the context of the TCL or NZ master equations.
Another promising tool introduced in this thesis is the measure of fidelity for encoded information that we used to prove the robustness of operator error correction against imperfect encoding. As we pointed out, this measure provides a natural means of extending concepts such as the fidelity of a quantum channel and the entanglement fidelity to the case of subsystem codes. As subsystem encoding provides the most general method of encoding, this measure could also be useful in studies concerning optimal quantum error correction. Its simple form makes it suitable for computation which is important in this respect.
Finally, our scheme for fault-tolerant holonomic computation has also opened a number of interesting questions. We have shown that for universal computation this scheme requires three-local Hamiltonians. It may be possible, however, to use perturbative techniques to approximate three-local Hamiltonians using two-local ones in a manner similar to the one introduced in Ref. [84]. Another direction for future research is suggested by the fact that the gap of the adiabatic Hamiltonian provides a natural protection against those types of errors that lead to excitations. It is interesting whether it is possible to design more efficient error correction schemes that make use of this property. Another question is whether the holonomic approach could provide a solution to the problem of the inconsistency between the standard faulttolerance assumptions and the rigorous derivation of the Markovian limit. Giving a definitive answer to this question requires a rigorous analysis of the accumulation of non-Markovian errors due to deviation from perfect adiabaticity.

235
Our scheme for the Bacon-Shor code uses an approach to holonomic computation in which the Hamiltonian acts trivially on the subsystem code and non-trivially on the gauge subsystem. It would be interesting to formulate this approach as a general method for holonomic computation on subsystems. The techniques introduced in this study may also prove useful for the problem of fault tolerance in the adiabatic model of computation. It is possible that some combination of transversal adiabatic operations and active error correction could provide a solution for this case too.