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Dynamical decoupling sequences for multi-qubit dephasing suppression and long-time quantum memory

We consider a class of multi-qubit dephasing models that combine classical noise sources and linear coupling to a bosonic environment, and are controlled by arbitrary sequences of dynamical decoupling pulses. Building on a general transfer filter-function framework for open-loop control, we provide an exact representation of the controlled dynamics for arbitrary stationary non-Gaussian classical and quantum noise statistics, with analytical expressions emerging when all dephasing sources are Gaussian. This exact characterization is used to establish two main results. First, we construct multi-qubit sequences that ensure maximum high-order error suppression in both the time and frequency domain and that can be exponentially more efficient than existing ones in terms of total pulse number. Next, we show how long-time multi-qubit storage may be achieved by meeting appropriate conditions for the emergence of a fidelity plateau under sequence repetition, thereby generalizing recent results for single-qubit memory under Gaussian dephasing. In both scenarios, the key step is to endow multi-qubit sequences with a suitable displacement anti-symmetry property, which is of independent interest for applications ranging from environment-assisted entanglement generation to multi-qubit noise spectroscopy protocols.


Introduction
Characterizing and counteracting decoherence from noise environments which may exhibit both temporal and spatial correlations is a central challenge for realizing high-fidelity quantum information processing (QIP) and fault-tolerant quantum computation [1,2]. Of particular interest are purely dephasing environments, which provide an accurate physical description whenever relaxation processes associated with energy exchange occur over a characteristic time scale that is substantially longer than that associated with dephasing dynamics. Provided that the noise arises predominantly from low-frequency components, and that external control is available over time scales that are short compared to the resulting temporal correlations, open-loop techniques based on dynamical decoupling (DD) and dynamically error-corrected gates [1,[3][4][5][6][7][8] provide a powerful tool for boosting operational fidelities-potentially eliminating 'coherent' (highly 'non-Markovian') errors that dominate worst-case error estimates in rigorous threshold analyses [9,10].
Our focus in this work is DD for a class of purely dephasing models, that describe stationary but not necessarily Gaussian correlated noise from combined classical and quantum sources on a multi-qubit system. More precisely, as detailed in section 2.1, we consider arbitrary single-and two-qubit classical time-dependent phase noise, along with quantum noise from a dephasing 'free field' Hamiltonian-namely, a linear diagonal spin-boson model where non-Gaussian statistics may arise solely from the bath initial state [11][12][13][14]. Our motivation is twofold: on the one hand, while general-purpose multi-qubit DD sequences based on concatenation and nesting are known, which may in principle ensure the desired error suppression up to arbitrarily high order [5,15,16] (section 2.2), it may be possible to design less resource-intensive DD protocols by tailoring construction to dephasing models with particular characteristics; on the other hand, while an approach to practical long-time high-fidelity quantum memory has been recently proposed based on the idea of engineering a 'coherent plateau' [17] by sequence repetition, the existing analysis is only applicable to a single qubit exposed to Gaussian phase noise.
We tackle the above issues under the simplifying assumptions that the required DD pulses may be instantaneously and perfectly effected on selected (subsets of) qubits-subject, however, to a realistic constraint on their minimum separation, or 'switching time' [17,18]. We leverage a general filter-transfer function formalism for characterizing open-loop error-suppression capabilities in both the time and the frequency domain using fundamental filter functions (FFs) as building blocks [19]. Our first result, and starting point for subsequent analysis, is an exact analytical characterization for the multi-qubit controlled dynamics in terms of a suitably defined time-ordered cumulant expansion, given in section 2.3. While it is known that an exact solution exists for the free evolution of a system subject to Gaussian bosonic phase noise [20][21][22], and is in fact equivalent to the one resulting from a second-order Magnus perturbative treatment [12], our analysis extends this equivalence to DD-controlled evolution. In the process, we recover a number of partial results that have appeared in the literature, limited to two qubits under special symmetry assumptions [23][24][25][26][27]. For generic non-Gaussian dephasing noise, our representation shows how arbitrary high-order cumulants may still be expressed in terms of only two FFs associated with the control modulation of single-and two-qubit terms.
Sections 3 and 4 contain our core results on two control tasks of increasing complexity: (i) suppressing errors as effectively as possible, so that quantum information is preserved over a short-time regime of interest (e.g., a single gating period); (ii) ensuring that quantum information is preserved with high fidelity for long storage times, with on-demand access. Provided that selective control over individual qubits is available, we show that new multi-qubit DD sequences may be constructed, so that a displacement anti-symmetry property is obeyed by the control switching functions, both in the simplest case of N=2 qubits (section 3.2) and, by appropriately orchestrating the sign pattern for every qubit pair, for general N (section 3.3). Such sequences ensure the same order of error suppression as the best known nested or concatenated protocols, while also maximizing their 'filtering order' in the frequency domain [19]. Further to that, for a fixed system size N, DD sequences incorporating displacement anti-symmetry are exponentially more efficient in terms of total pulse number, as long as any direct coupling between qubits is time-independent. The possibility of achieving multi-qubit longtime storage by meeting appropriate conditions for the emergence of a fidelity plateau under sequence repetition is first established for the important case of non-Gaussian classical stationary phase noise in section 4.1. Generalization of this result to dephasing scenarios that also include bosonic noise sources again relies crucially on incorporating displacement anti-symmetry in the DD sequences used for repetition (section 4.2).
Provided that the relevant conditions for the emergence of a fidelity plateau can be satisfied, we argue in section 5.1 that the combined use of multi-qubit DD sequences that do not (or, respectively, do) obey displacement anti-symmetry provides a systematic method for controlled generation and storage of multipartite entanglement mediated by a common quantum environment, extending known schemes for bipartite entanglement generation [28,29]. Section 5.2 briefly discusses aspects related to the practical significance and main anticipated limitations of our findings in the light of realistic considerations, whereas a summary and outlook to future research conclude in section 6. By way of concrete illustration of our general FF formalism, we provide in appendix A additional technical detail on the simplest yet practically important case of a DDcontrolled two-qubit system under Gaussian dephasing from combined classical and quantum bosonic sources. Lastly, we include in appendix B a discussion of the impact of timing errors on the ideal fidelity plateau, based on a simple error model that breaks the required underlying displacement anti-symmetry in a two-qubit system.

Multi-qubit controlled dephasing dynamics 2.1. Gaussian versus non-Gaussian dephasing models
We consider a class of purely-dephasing noise models on N qubits in which no energy exchange takes place between the system and an environment ('bath') modeled by a continuum of classical and/or quantum modes -see section 5.2 for further discussion on their physical relevance. , implying the existence of a preferred (energy or computational) basis [1]. Without loss of generality, we take this to be the z basis, and use ℓ Z to denote the Pauli operator s z acting on qubit ℓ. A general multi-qubit dephasing Hamiltonian that includes up to two-body noisy qubit interactions and linear system-bath coupling may then be written as where we allow for dephasing due to both classical and quantum noise sources, as represented by the fluctuating system Hamiltonian H S (t) and the system-bath interaction Hamiltonian H SB , for suitable (Hermitian) operators ℓ B acting on B. Specifically, ( ) are classical stochastic processes describing random fluctuations of the local energy splittings and two-local (Ising) coupling strengths, ℓ d and ℓ ℓ¢ d , . A formally similar treatment of classical and quantum noise sources is possible upon moving to the interaction picture associated with the bath Hamiltonian H B . The total Hamiltonian becomes ( )˜( )˜( )˜( ) and where, for notational convenience, we simply use ( ) ℓ B t to denote the interaction-picture representation of ℓ B . For the purpose of achieving decoherence suppression and arbitrary state preservation, the dynamics generated by˜( ) H t corresponds to unwanted ('error') evolution, thus˜( ) ( ) º H t H t e in the framework of dynamical error suppression [8,19]. In a multi-qubit setting, the noise acting on different (subsets of) qubits may exhibit different kinds of temporal as well as spatial correlations. In particular, two limiting situations may be envisioned for the coupling of the quantum bath to different qubits: 0 , ℓ ℓ " ¹ ¢ for at least some ¢ t t , .
• A private (or independent) bath, in which case [ ( ) While intermediate situations are clearly possible, an additional distinction is relevant for the common-bath scenario depending on whether qubit-permutation symmetry is present. Specifically, collective dephasing corresponds to ( ) ( ) ℓ = B t B t , ℓ " , as extensively studied in the context of decoherence-free subspaces [1,30].
Noise processes and their correlations can be further characterized by their spectral properties in the frequency domain.
Spectral properties of classical phase noise-Classical noise processes may be compactly characterized by their statistical moments [31], obtained via the ensemble average over noise realizations, henceforth denoted by · á ñ c .
Given the fluctuating Hamiltonian H S (t) in equation (2), we are most generally interested in cross-correlations of the form , labels qubit pairs, and we include joint moments of a single noise source, ( ) , as a particular case. A simpler description is obtained by using cumulants, which we denote and define via general moment-cumulant relations [31]. If { } A i denotes a set of random variables, then For example: 3 1 2  3  3 1 2  1  3  1  1  1  2  2  3 1  1  2  2  3 2  1  1  2  1 2  1  3 and so on. The above definition of cumulant, which respects the order of the arguments, is equivalent to the traditional one when the random variables commute, but solves the ambiguity that arises when ordering becomes important, as it does for non-commuting operator-variables in the quantum case. The expressions in equation (4) can be inverted as usual in probability theory and statistics, allowing one to write the cumulants in terms of the moments [31].
A description of the noise process in frequency space may be given by considering the Fourier transform of cumulants. Specifically, the Fourier transform of a kth order (cross-) cumulant defines the kth order polyspectrum [14,32], to denote a vector of length 5 k. The noise process is stationary if arbitrary cumulants, , depend solely on time differences, say, t º - 1, rather that absolute time values. Thus, the stationarity assumption translates into Of particular interest are stationary zero-mean classical Gaussian noise processes, whose statistical properties are completely characterized by the second-order cumulants. That is , are statistically independent, one may further simplify to ( ) . Spectral properties of quantum phase noise-The spectral properties of a quantum dephasing environment may likewise be characterized in terms of statistical moments and cumulants. The starting point is to define statistical averages with respect to the initial state of the bath, say, r B , by using (· ) r Tr B B instead of · á ñ c . While in the most general case initial system-bath correlations may be present (hence , throughout the present analysis we shall work under the standard assumption of an initial factorized state of the form or in the frequency Fourier transform of the corresponding cumulant is the Fourier transform of ( ) ℓ B t . Operator cumulants are defined recursively via the momentcumulant relations given earlier, equation (4), upon replacing · á ñ c with · á ñ q , e.g.
. . In particular, a stationary zero-mean quantum Gaussian noise process may be defined by requiring that, for all t, ( ) ℓ á ñ = B t 0, q and, in analogy to equation (7), that arbitrary cumulants vanish for > k 2. The corresponding k=2 quantum power spectrum then obeys the relation q , In what follows, we focus on dephasing noise originating from bosonic sources, representing for instance phonon or photon modes. In this context, a paradigmatic example is the linear spin-boson model [3, 11, 13, 20-22, 33, 34], defined by the following bath Hamiltonian and interaction operators in equations (1) and (3) (in units where  = 1): a a a a , 0 , k k k k , and ℓ g k describes the coupling strength between qubit ℓ and bath mode k. If the initial bath state r B is thermal, describes a stationary noise process which also obeys freefield Gaussian statistics [13]. The above quantum power spectrum is explicitly given by 5 Noise cross-power spectra may be also defined in terms of the Fourier transform of the moments (and related high-order time-correlation functions) [31]. Following standard practice in classical signal processing theory [32] and our previous analysis [19], we employ cumulants in what follows. In addition, we will assume sufficient regularity for all the relevant Fourier transforms to be mathematically well-defined.  [11], where  k is the momentum associated with the kth mode and ℓ  r the position of the ℓth qubit, one gets where we have further expressed the exponent in terms of the mode frequency and the transit time [22] via the . Under these assumptions is the spectral density function of the oscillator bath in the continuum limit 6 .
Since, by definition, 2 is the second-order cumulant of the Fourier-transformed bath operators, we may write upon separating operator products into anti-commutators and commutators. If the latter are, respectively, denoted by [ ]  , , direct calculation yields Thus, ) have symmetric (anti-symmetric) character in the sense that a feature that will be useful in understanding their contribution to different dynamical aspects. While the free dynamics under multi-qubit classical plus bosonic dephasing will be recovered as a special case of the DD-controlled dynamics, two remarks are in order. First, our approach may be applied to a wider class of quantum noise scenarios, for instance environments consisting of multiple interacting types of bosons, as long as the relevant Hamiltonian can be mapped to equation (9) via an appropriate Bogoliubov transformation [35]. Second, for all the noise sources considered, a finite norm of the error dynamics to be filtered out will be assumed-or, in more physical terms, a finite correlation time ('non-Markovian' behavior in the sense of [4,13]) relative to which fast control regimes may be accessed. This is a crucial requirement for DD methods to be effective [1,4,18,19,36], which will translate into assuming that all relevant noise spectra decay sufficiently rapidly at high frequencies.

Control protocols
To facilitate presentation of the essential new results, we assume access to perfect instantaneous π rotations, and postpone discussion of realistic effects to section 5.2. This ideal 'bang-bang' DD setting has been extensively investigated in both single-qubit [3,5,6,33,34] and multi-qubit systems [4,15,16,37] (see [1] for a review). For the sake of completeness and consistency, we summarize the basic ingredients in this section, building in particular on [16,38]. In the process, we introduce a new composition rule which unifies existing high-order sequence constructions and may be of independent interest within DD theory.
For the dephasing noise models under consideration, control implemented via instantaneous rotations along a fixed axis (say, x), suffices. Let a DD sequence, labeled by an integer index s, be specified in terms of a control operation ( ) s º X x , and a number n(s) of control intervals { }  denotes free evolution over ( ) t j s , the controlled propagator induced by such a sequence reads where operators are applied from right to left and a final pulse is included at time t=T depending on the parity of s in order to ensure a cyclic control propagator. Following [19,38], we say that the DD sequence achieves cancellation order (CO) α, with respect to the control operation X, if the norm of unwanted (non-commuting) 6 Note that, unlike classical noise, quantum noise is spectrally asymmetric in general, with the ratio respectively. We can build a new sequence over time T by using the following composition rule: We are now ready to define the specific high-order sequences we will be using.
Single-qubit DD sequences-The two relevant sequences are single-axis CDD [5] and Uhrig DD (UDD) [6]. The former may be defined recursively via , so that, for example and so on. On the other hand, the αth order UDD protocol is defined by the pulse timings Multi-qubit DD sequences-Both single-qubit CDD and UDD sequences admit extensions to multiple qubits via the composition rule given in equation (17). In particular, by letting , one can define the following N-qubit sequences:  growing exponentially with N in either case. Depending on the sequences used as building blocks, all known high-order DD sequences may be recovered. Specifically, we shall consider the following αth order Nqubit sequences: (i) Multi-qubit CDD [5]: 1, , 1 1, , 1 1, , 1 times (ii) Nested UDD (NUDD) [15]: To date, NUDD is the most efficient known sequence, in terms of the required number of pulses capable of ensuring CO α for general multi-qubit dephasing noise as in equations (2) and (3), at least for sufficiently hard noise spectral cutoffs [39].
Once a multi-qubit DD sequence is chosen, the effect of the control is most simply described in the interaction picture associated to the control propagator U c (t), which in the present setting simply leads to time- 7 Note that the sequences given in (ii) and (iii) are two extremes of the possible range of sequences that may be built using the composition rule of equation (17) and CDD as building blocks. The composition rule may be naturally extended to multi-axis DD sequences, recovering for instance concatenated UDD or quadratic DD [1].
where the control switching function for the ℓth qubit has the form n P denotes the total number of pulses applied on qubit ℓ and Θ is, as usual, the step function. We say that the control is (qubit) non-selective (or global) if ( ) ( ) ℓ = y t y t ℓ " , and that it is selective otherwise. Note that nested multi-qubit sequences necessarily require qubit-selective control. A simple illustrative example is depicted in figure 1.

Exact representation of the controlled dynamics and fundamental FFs
In order to determine the reduced dynamics of the controlled multi-qubit system, the first step is to obtain an expression for the unitary 'error propagator' resulting from equation (1) under DD, evaluated in the combined interaction picture associated to the control and bath evolution. Thanks to the fact that the bosonic algebra guarantees simple commutation relationships between noise operators, namely, [ ( ) for all ¹ ¢ t t , the Magnus expansion [1] truncates exactly at the second order. Accordingly, the desired propagator, which we denote henceforth˜( ) U T e , may be compactly expressed as follows (up to an irrelevant global phase factor):˜( where equation (21) explicitly defines the relevant effective Hamiltonian H eff , and we have introduced new random variables as well as time-averaged quantities by letting It is instructive to examine the propagator in the frequency domain, by exploiting the fundamental FF formalism introduced in [19]. Substituting the explicit forms of the Magnus terms in equations (23) and (24) in the   where the relevant firstand second-order generalized FFs, , may in turn be expressed in terms of fundamental FFs: and the corresponding firstand second-order fundamental FFs are given by We recall that this general filtering formalism allows one to determine the CO of a control protocol as well as its filtering order (FO)-in particular, around  w = 0, as relevant to DD. If , , j 1 , respectively. In general, it is crucial to analyze the suppression capabilities of a protocol in both the frequency and time domain to fully characterize the control performance [19]. As defined in equation (29), the FO contains information about how 'flat' the filter implemented by ( ) is around  w = 0, hence how well it attenuates low-frequency noise. The CO, on the other hand, quantifies the way in which uncorrected error contributions are expected to scale with the total sequence time, irrespective of frequency. In this sense, the CO can be thought of as providing information about the overall magnitude of the filtered signal. Notice that dimensional analysis establishes a relation between the CO and FO of any generalized FF. Since ( ) Minimization of the left hand-side also implies that the overall CO and FO of a protocol obey the simple relationship 8  a F . In the light of these considerations, the high-order sequences described in the previous section have interesting filtering properties: because only FFs of order a = 1, 2 are relevant to the analysis, one may verify that  CDD and NUDD sequences have CO equal to α and FO { } a a Î -1, .
The effective Hamiltonian H eff defined in equation (21) (or equation (25) in Fourier space) comprises different physical contributions. One-body terms, proportional to ℓ Z , are present for qubits coupled to arbitrary classical or private quantum baths. Two-body terms, proportional to ℓ ℓ¢ Z Z and capable of inducing quantum correlations between different qubits, may have two distinct origins within our model: • They may result from the 'direct' Ising coupling in the free Hamiltonian, corresponding to the term ( ) ℓ z ¢ T in equation (21) and filtered by ( ) • They may result from the 'induced' coupling mediated by the quantum bath via its non-commuting character, corresponding to the term ( ) in equation (21) and filtered by ( ) in equation (25). 8 We stress that for noise models more general than dephasing as considered here, Φ and α need not be tightly related, and one can only Even in the absence of direct coupling ( ), the bath-mediated interaction poses a main obstacle to designing less resource-intensive DD sequences, as we will see in the following section. Notwithstanding, it is important to iterate that, thanks to the bosonic algebra, the frequency-domain commutator 1 2 B depends only on one frequency variable, and thus only ( ) will be relevant to our analysis.
Depending on the initial state of the system and/or bath, the cancellation or filtering order of a DD sequence may be higher [19] than they are at the propagator level. In order to proceed to the exact solution for the reduced system dynamics under the initial factorization assumption where the sum ranges over all possible binary strings  º a a a N 1 and The interaction-picture reduced density matrix at time T may then be expressed as where the complex factor allows for both non-trivial decay and phase evolution of each coherence element. Specifically, direct calculation yields: u v and ℓ a is the ℓth entry of the string a. Formally, irrespective of whether we consider the classical ensemble average or the quantum statistical average with respect to r B , the following cumulant expansion holds for a random variable or operator Q: where · á ñ stands for either · á ñ q or · á ñ c . If Q has stationary zero-mean Gaussian statistics, the only non-vanishing Note that all the quantities in the exponents that define , which involve sums over single (classical or quantum) noise sources, inherit the properties of the corresponding noise processes: e.g., if ( ) ℓ z t is zero-mean Gaussian, the same holds for in equation (33). Furthermore, in view of the bosonic algebra, (equation (13)), the two contributions (equation (34)) and (equation (35)) are independent of r B and thus constants with respect to · á ñ q . This allows us to obtain exact analytical expressions for the controlled dynamics when all noise sources are Gaussian.
For non-Gaussian dephasing models in which the cumulant series does not truncate to the second order (including nonlinear spin-boson models [40]), one needs to consider, in principle, all the infinite multiple-point correlations which, in the Fourier domain, implies dealing with an infinite hierarchy of FFs. Remarkably, however, for the family of classical plus linear bosonic noise models studied in this paper, arbitrary high-order cumulants can still be written in a compact way in terms of only two generalized FFs, regardless of stationarity: and ℓ ℓ¢ -S , B, are functions of two independent frequency variables. Similarly the kth order polyspectra depend on the full set of k frequency variables. Together with the exact expressions in equations (33)- (35), the above still provide a closed-form representation of the reduced multi-qubit dynamics, which may be used to infer general results or serve as a basis to build approximations via truncation of the cumulant expansion.

Dynamical decoupling versus multi-qubit dephasing noise: short-time memory
Having introduced the necessary tools, we are now poised to address the first of the two control problems we set out to explore, namely, efficient short-term memory using DD. Different DD protocols will be contrasted, and in each case we will characterize their decoherence suppression capabilities and extract their CO and FO directly from the relevant generalized FFs. While certain control strategies will only allow a fixed, small CO, we will see how, as sufficient structure is added to the control in terms of selectivity and symmetry, arbitrarily high CO is possible in principle. Beside re-establishing CDD and NUDD sequences as capable of arbitrary CO against general dephasing noise [1,5,6,15,16], a main goal is to construct more efficient DD sequences tailored to dephasing noise with particular features. We will present a new family of multi-qubit sequences that, by satisfying a particular displacement anti-symmetry condition, can achieve arbitrary CO using exponentially less resources, so long as no direct Ising coupling is present. The symmetry that these new sequences possess will also prove fundamental in the context of building long-term multi-qubit memories (see section 4).

Non-selective multi-qubit control sequences
The most naıve, yet often readily available, strategy to decouple multiple qubits from their environment is to rely on non-selective control. In the formalism of section 2.2, this means applying global control operations of the form Since, however, the same pulse is applied synchronously to each qubit, ( ) This affects the ability of non-selective DD to suppress time-independent noise: while ( ) i.e., the noise induced by any constant two-qubit direct coupling in equation (2)) cannot be suppressed by non-selective DD. This is expected, as global pulses commute with the direct (system-only) coupling term and thus cannot affect it.
Interestingly, the bath-induced two-qubit coupling and the phase evolution ensuing from the second-order Magnus term can nevertheless be suppressed, albeit not according to an arbitrarily high power-law behavior. If either a a CDD or a UDD non-selective protocol is applied, the FF suppressing the induced coupling term, , is found to scale as ( ( ) )   w m T 1 3 , similar to free evolution (note that, using equation (28), is a real function). Thus, non-selective DD cannot suppress the induced coupling term any better than free evolution, namely, only up to a fixed order independent of α. While at first it would then seem that increasing α would not improve error suppression, as claimed in some two-qubit analysis [23,26], numerical exploration clearly demonstrates that as α increases, the absolute value does decrease, i.e., better suppression is achieved (see figure 2) .
This behavior may be understood in terms of a formal analogy between DD and a Zeno effect resulting from the increasingly more frequent pulses within [ ] T 0, , as suggested in [3] for a single qubit. Consider implementing M repetitions of a given non-selective DD sequence ( ) While the CO is fixed, the number of pulses grows as M 2 . In the simplest case of ( Thus, while increasing M does not change the power law of ω or T, a suppression effect still takes place, stemming from the increasing number of pulses within the storage time.
Assuming that no significant direct Ising couplings are present, simple non-selective control sequences may still offer adequate short-time suppression in practical scenarios, depending upon specific system and noise features. Given their inability to access high CO, however, the benefits of such sequences become increasingly limited as technological limitations prevent access to sufficiently short minimum switching times, making qubit-selective sequences essential to achieve high-fidelity control.

Selective two-qubit control sequences
When qubit-selective control is available, multi-qubit DD protocols with arbitrarily high CO may be devised using the composition rule in equation (17). Both for clarity and relevance to near-term implementations, we first consider two-qubit systems, though we will use notation that may be easily adapted to the multi-qubit case (section 3.3). Let the DD sequences being composed have CO a 1 , a 2 , and require ( ) a n 1 , ( ) a n 2 pulses. The composition guarantees a high CO, { } a a a = min , 1 2 , hence all the generalized FFs are ( )  a+ T 1 . We seek strategies that achieve the same CO, but that also ensure high FO and require less than ( ) ( ) a a = n n n P tot 1 2 pulses.

Vanishing direct qubit-qubit coupling
Assume that no direct inter-qubit coupling is present, i.e., ( ) for all t (hence, no contribution to decay due to¯( ) ℓ ℓ h ¢ T , in equation (33)). Similar to the non-selective protocols ( ) ℓ ℓ a ¢ U X X considered above, one can see that simply executing the single-qubit sequences ( ) independently on the two qubits achieves the desired power-law dependence for the local (one-body) terms but not for the induced (two-body) coupling terms: that is, 1 in general. However, adding more structure to the control design can overcome the issue, yielding arbitrarily high FO and CO for all the generalized FFs relevant to the problem, as we show next.
• Mirror anti-symmetry-While, as remarked, the commutator spectrum 2 that enters the error integrals for two-body terms, equation (25), is in general complex albeit anti-symmetric in the sense of equation (15), a simplified solution is possible if the noise is known to obey additional symmetry properties. Suppose that and consider any DD protocol that satisfies the following mirror anti-symmetry condition(see also top panel in figure 3 for a pictorial illustration): with mirror symmetry, corresponding to a reflection about = t T 2, being correspondingly described by a + sign in the right hand-side of equation (38). Then it is easy to verify that . In all cases the CO and FO are the same as for free evolution (lower left panel), but the value of the FF decreases as the number of pulses within a fixed time T grows. This is verified by studying M repetitions of CDD 1 Thus, mirror anti-symmetry guarantees that no contribution to the propagator in equation (25) arises from the above integral. A simple recipe to generate a control protocol with the required anti-symmetry is to execute independent CDD or UDD sequences with CO ℓ a and ℓ a ¢ obeying the 'odd-parity condition' that ℓ ℓ a a + ¢ is odd. If so, ( ) ℓ y t has mirror symmetry, ( pulses for mirror anti-symmetric DD. The symmetry condition imposed on the noise spectra by equation (37) corresponds to requiring that ℓ ℓ *  Î ¢ g g k k . In view of equation (11), this in turn implies real coupling strengths, [ ] ℓ = g Im 0 k for all ℓ, with collective dephasing as a special case. Interestingly, it had already been shown [26] that enhanced DD protocols obeying the above-mentioned odd-parity condition guarantee high CO for two qubits exposed to collective spin-boson dephasing from a Gaussian (thermal) bath state. Beside shedding light on the underlying symmetry enabling this result, our analysis at the propagator level shows something stronger: as long as equation (37) is obeyed, mirror anti-symmetric sequences achieve arbitrary CO irrespective of the initial state r B , hence including arbitrary non-Gaussian bath states.
• Displacement anti-symmetry-Since the environmental couplings may not be precisely characterized in realistic scenarios, or need not comply with equation (37), assuming such a symmetry is too stringent in general. We thus proceed to design model-robust efficient high-order sequences without any further symmetry assumptions beyond those stemming from the bosonic algebra. Demanding a suitable symmetry on the control switching functions will still be crucial to achieve this goal. Building on the explicit form of the second-order fundamental FF relevant to the problem, in equation (28), we propose the use of a displacement anti-symmetry, defined as follows(see also bottom panels in figure 3): with displacement symmetry being instead associated to the + sign in the right hand-side of the above equation. Direct calculation shows that the second-order fundamental FF obeys  (38)) entails a reflection of the pattern with respect to T 2, whereas displacement symmetry (equation (39)    where ¢ =t t T 2 j j for j=1, 2. Thus, the anti-symmetry requirement implies that with each fundamental FF appearing in an integral over (w 1 , ) w 2 and being multiplied by 2 . Accordingly, and the first term in each of the above expressions never contributes. It follows that the relevant second-order generalized FF factorizes into a product of first-order fundamental FFs: Let the two basic single-qubit DD sequences correspond to CO pulses. Notice that it is straightforward to build a control sequence possessing the required displacement anti-symmetry: given a DD sequence over an interval of duration T 2, say ( ) . While the relevant firstand second-order generalized FFs give us information on the worst-case filtering and cancellation capabilities that a DD protocol can ensure under minimal knowledge about the noise process, higher effective CO and FO can be achieved if additional assumptions hold [19]. Recall, in particular, the pure spin-boson stationary Gaussian noise model on two qubits discussed in the appendix. As remarked there, ( ) ( ) ℓ w G T , Z 1 never appears alone in the reduced dynamics and thus, assuming e.g. that qubit 1 corresponds to an anti-symmetric switching function, we have the effective generalized FFs Thus, the extra symmetry associated with the zero-mean Gaussian nature of the initial bath state has effectively increased the CO and FO of the sequence. The above predicted power-law behaviors are demonstrated in figure 4.
• Sequence comparison-In order to gain quantitative insight, it is useful to compare the more efficient displacement-anti-symmetric DD sequences we have just built with other DD sequences applicable to our noise model. The term which is most relevant to such a comparison is the induced coupling term, since guaranteeing a high CO for this term is the whole point of introducing anti-symmetry or using the composition rule. Specifically, let us focus on the noise-induced phase evolution , inspired by phenomenological treatments of nuclear-spininduced dephasing in semiconductor quantum-dot qubits [17,41]. For fixed total time T and minimum switching time τ, we construct sequences with the highest possible CO α within these constraints, using CDD as building blocks and incorporating mirror anti-symmetry, displacement anti-symmetry, or using nesting-see table 1 for a summary. is key to effectively suppressing phase evolution, with similar conclusions holding for any spectrum ( ) w that is heavily weighted around w = 0. Because displacement-anti-symmetric DD achieves ( is purely imaginary whenever displacement anti-symmetry is imposed, whereas , , Multi-qubit CDD, = T T 2 the highest CO and FO for the induced coupling term, it outperforms all other strategies. Interestingly, two suppression mechanisms are evident: the first, stemming from the FO and CO, is manifest in the relative amplitudes of the oscillating functions, especially at low frequency; the second is the oscillatory character of the integrand itself, with positive and negative contributions partially canceling each other. The absolute values of ( ) w I T , quoted in figure 5 further show the importance of the FO: amongst all sequences, displacement anti-symmetry-enhanced DD yields substantially better suppression. While one might naively expect such a difference in performance to originate solely from the difference in CO, it has already been shown that sequences with the same CO can have very different performance because their FO differs [19]. Here, we observe a related behavior. Since . The sequence with the highest value of ( ) F 2 will thus also have the largest ( ) a 2 . Accordingly, a higher FO does not only imply better protection around w = 0, but indeed a higher protection in the sense of the power law in T. Also notice that when the FO/CO of a sequence are below those of free evolution, DD may become a liability, effectively increasing the unwanted noise effect; e.g., this is the case for both the nested CDD sequence and for the mirror anti-symmetric sequences for m = T 8 s.
for the corresponding sequence. Specifically, we compare (see also table 1):

Non-vanishing direct coupling
In the presence of a direct Ising coupling in the system Hamiltonian, i.e., ( ) , DD sequences applied independently to each qubit will not suppress such terms to arbitrary order. Using the displacement anti-symmetry-enhanced sequences described earlier and recalling equation (25) , we find that, in general Thus, displacement anti-symmetry alone guarantees a high FO and CO only in the presence of a static Ising coupling. In such a case, , . If a time-dependent noisy coupling is present, it is necessary to resort to nested two-qubit DD sequences built via composition, i.e., . These sequences, which include NUDD and multi-qubit CDD, are the only known protocols capable of achieving arbitrary CO α. However, their FO need not be maximum: direct calculation of ( ) , combined with dimensional analysis, shows that indeed only achieves a F = -1 in general. As before, by further imposing a displacement anti-symmetry on NUDD, the FO can be maximized to a F = .

Selective multi-qubit control sequences 3.3.1. General construction
The above results may be extended beyond the N=2 qubit scenario. Doing so requires generalizing displacement anti-symmetry to multiple qubits. Since satisfying the requirement of pairwise displacement antisymmetry, equation (39), for all pairs simultaneously is clearly impossible, we seek a control structure capable of satisfying an analogue to displacement anti-symmetry at different timescales. We start by subdividing the total evolution time T in -2 N 1 segments of length t 0 , i.e., t = -T 2 N 1 0 . One can then demand that for every ℓand t t = 2 s s We can generalize to multiple qubits via an appropriate matrix In this way, for every pair ℓ ℓ¢ , there exists an s such that, when t 1 and t 2 are in an interval . We then proceed recursively: given  . The resulting matrix ( ) Q N gives us a way to build-in the symmetries and anti-symmetries of the switching functions ( ) ℓ y t at different timescales; e.g., in the two qubit case described earlier

Given a base DD sequence on an interval
, the displacement anti-symmetryenhanced version is then given by in agreement with our construction in equation (41). For N qubits, the generalized-displacement antisymmetry-enhanced sequence It remains to show that the above construction achieves in general the desired arbitrary-order noise suppression, similar to the two-qubit case. Our strategy is to first show the result for a fixed pair of qubits for which generalized displacement anti-symmetry at scale t s holds, in the sense of equation (40), and then generalize the argument by noticing that, by construction, equation (45) is satisfied for at least one s for every ℓ ℓ¢ , pair. Given pair ℓ ℓ , one then has , while the first summation can be simplified by invoking displacement anti-symmetry:  , as relevant to stationary noise. Accordingly, in the absence of a timedependent direct Ising coupling, generalized displacement-antisymmetry-enhanced DD achieves CO and FO equal to { } ℓ a a = min , as desired.

Example and resource scaling
We illustrate the above general construction in the simplest non-trivial multi-qubit setting, N=3, in which case we have Explicitly, the following set of symmetries are enforced by ( ) P 3 : at time scales t 0 , for  for m=1. Starting from ( ) y t 3 , for instance, and with t = T 4 0 , these conditions require which may be simultaneously obeyed by the pattern , we may then build an enhanced DD sequence using equation (46) and the (independent) base sequence . Explicitly: As in section 2.2, let ( ) ( ) ℓ  a a n n denote the number of pulses in a single-qubit DD sequence achieving CO ℓ a , with for multi-qubit  CDD or NUDD. For a sequence that incorporates generalized displacement anti-symmetry, the required number of pulses grows instead as . While this is still exponential in N, it may translate into a significant resource reduction: which represents an exponential reduction in the required pulse number whenever a > 1.
Beside affording a smaller total pulse number, the presence of displacement anti-symmetry also implies less stringent requirements on the minimum switching time of the corresponding protocols. For instance, the minimum switching time for a multi-qubit NUDD sequence achieving minimum CO α scales as 1 that is, our sequences also have an exponentially larger minimum switching time, making them potentially more amenable to experimental implementation in multi-qubit systems.
As in the two-qubit case, the above advantages in terms of pulse number and timing constraints are meaningful only so long as an arbitrarily high CO/FO is achievable, which in our noise model is possible only when no fluctuating direct coupling is present. Even if the latter is non-zero, however, it is beneficial to incorporate displacement anti-symmetry in NUDD or achieve both CO and FO equal to α for the most general version of the noise model under consideration. It is also worth noting that, out of the multi-qubit sequences described in section 2.2, the multi-qubit ( ) already has a form of displacement anti- symmetry built in. This is easily seen by noticing that may be thought of as the result of imposing anti-symmetry on a sequence ( ) using an 'enlarged' ( ) P N matrix, with N×N entries, given by˜( 1 pulses. While the latter is exponentially larger than than for pulses), multi-qubit CDD is capable of suppressing an arbitrary (direct and induced) coupling term. As we shall see next, displacement anti-symmetry is also the key to establishing our results for long-term storage of multiple qubits, hence our results will apply in particular to ( ) sequences.

Dynamical decoupling versus multi-qubit dephasing noise: long-time storage
If the promised benefits of QIP are to be fully realized, techniques that enable high-fidelity preservation of arbitrary quantum states for extended time periods (i.e., periods longer than those over which DD protocols have been traditionally designed to be effective) must be available. For example, in a quantum computer it may be necessary to store information at error rates well below threshold in idle qubits, while additional processing tasks are carried out in parallel on other qubits. Furthermore, in order minimize negative effects on total processing time, this stored information should, ideally, be accessible 'on-demand'. For these reasons, we now turn attention to the problem of high-fidelity long-time storage of information, under realistic timing and accesslatency constraints. That is, we are not solely interested in protecting an arbitrary multi-qubit state for a fixed (potentially arbitrarily long) storage time T, but also in being able to retrieve it on demand, with guaranteed high fidelity and sufficiently small access latency. This problem was studied for a single-qubit exposed to Gaussian dephasing noise in [17], where a systematic approach to meet the above requirements was proposed, based on periodic repetition of a high-order DD 'base' -resulting in total storage time = T MT p and access latency capped at  T T p . Provided that the relevant noise spectrum decays to zero sufficiently rapidly at high frequency (ideally, with a 'hard cutoff' of the form ( ) ( ) w w w w µ Q -S s c ), the key observation is that the long-time limiting value of the qubit coherence need not be zero; rather, coherence (hence fidelity) may be made to 'plateau' at a controllable limiting value by appropriately choosing the base DD sequence, so that the following conditions are obeyed: as a kcomponent vector with entries L r , direct calculation yields: In order for the above kth-order cumulant to have a finite value, one must then avoid the ( )  M divergences at the peaks located at every In practice, meeting this requirement is impossible, since no DD sequence is known, such that its FFs have a high-order zero for all { } q L r , as required. However, building on the single-qubit case [17], it is unrealistic to expect that a plateau regime may emerge at all, unless the noise is sufficiently 'well-behaved' at high frequencies.
Let us work in the simplest (idealized) scenario, where a hard frequency cutoff exists for each frequency variable, such that the power spectra may be Taylor-expanded around the   w = = q 0 as follows: where w c L , r is a high-frequency cutoff parameter and s L r characterizes the power-law behavior as w  0 L r . Similarly, we assume that the low-frequency behavior of the relevant FFs reads Lr r r r Lr in terms of the CO a L r corresponding to each frequency variable. By demanding that, in analogy to equation (47), , for all L r , we may then ensure that there are no high-frequency noise components at any of the resonant frequencies, so that only the divergence at { } = q 0 L r contributes to the kth order cumulant. The strategy is then to decompose the integral in equations (48) and (49) into a sum of integrals over hypercubes of side p T 2 p centered around  = q 0 and, as in [17], to make sure that the dependence upon each frequency variable is sufficiently well-behaved close to zero for the corresponding integration to be convergent. That is, we require that each frequency integration admits a power law w L x r , with >x 1, close to zero. Direct calculation then yields the following conditions: Being motivated by essentially the same physical requirements, these conditions are entirely analogous to those given in equation (47) for the single-qubit case. Accordingly, a plateau behavior may be engineered by choosing a base DD sequence such that all first-order have zeroes of sufficiently high order at each w = 0 L r . Suitable choices were described in section 3. We note, in particular, that collective as well as independent single-qubit sequences suffice as long as no direct Ising coupling is present ], otherwise sequences built out of composition (such as NUDD or multi-qubit CDD) will be required to ensure high CO The conditions in equation (53) contain, as a special case, the ones derived in [17] for a single qubit exposed to zero-mean stationary Gaussian noise, due to either a classical source or a quantum bosonic environment. , the plateau conditions in equation (53) apply, in fact, to the more general scenario where noise is non-Gaussian.
Paradigmatic examples of classical non-Gaussian dephasing arise when a qubit is exposed to random telegraph noise [42] or is nonlinearly coupled to a Gaussian noise source, e.g., Gaussian as for a qubit operated at an optimal point [43]. Consider, for illustration, the latter case. If the power spectra ( ) w 1 has a cutoff at L 2 for all k. To see this, can be written, by virtue of the Gaussianity of ( ) x t , as a sum of terms of the form ( ) ( ) . After Fourier transforming, direct calculation shows that each of these terms is of the form , and as long as , and so on. That is, any high-order spectra of ( ) z t has a cutoff of at most L 2 in all its frequency variables and, furthermore, is stationary, that is, , at order k. Also notice that from the form of the above expressions we may infer that the functional dependence with respect to the frequencies is such that å s r L r grows at most linearly with k, making it possible for the plateau constraints to be satisfied.
For bosonic noise, stationarity demands that , hence r B is diagonal in the multi-mode Fock basis. A simple example of non-Gaussian dephasing arises when r B is a mixture of thermal (Gaussian) components at different temperatures, e.g., r r [14]. While detailed analysis is beyond the scope of this work, and may be most meaningfully carried out for a concrete qubit device, our approach provides in each case sufficient conditions for a coherence plateau to be engineered in principle.

Fidelity plateau conditions for multi-qubit classical plus spin-boson dephasing
Multi-qubit dephasing arising from combined classical and quantum bosonic noise sources may be analyzed by generalizing the strategy of section 4.1. Again, given the convenient form of equation (32), it is possible to analyze the additional effect of the spin-boson interaction by analyzing the contribution of the factors resulting from a quantum average. In order to guarantee the existence of a fidelity plateau for this more general case, it is necessary to ensure that the exponents of the additional decay terms, arising from quantum bosonic noise sources, also do not diverge as  ¥ M . Hence, in the large M limit, we require that, at order k in the cumulant expansion (recall equation (8) and the stationarity assumption) Obtaining plateau conditions for equation (56) requires more work. Using the periodicity of ( ) ℓ y t over = T MT P , we may rewrite grows with M and, in the  ¥ M limit we are interested in, it diverges. One would like to show that under appropriate symmetry of ( ) ℓ y t , and consequently of the FFs in each T p interval, the terms linear in M vanish. As it turns out, the displacement anti-symmetry comes to our aid here as well, leading to the desired cancellations, as we show below. We stress that the linear M-dependence is a generic feature, and unless additional symmetry is built into the applied control sequence, it necessarily leads to a divergence, forbidding the existence of a plateau regime. For example, repeating a high-order multi-qubit NUDD sequence without demanding further structure will not lead to a fidelity plateau in the presence of quantum noise, whereas a displacement anti-symmetryenhanced NUDD sequence will achieve it in principle, provided that the appropriate conditions are obeyed.

Two-qubit setting
To illustrate how the displacement anti-symmetry comes into play, let us analyze the two-qubit case first. Recall that displacement anti-symmetry can be built into any control sequence by the construction detailed in equation (46). The specific choice of base sequence depends on the noise model: as noted, in the presence of time-dependent direct coupling multi-qubit high-order DD sequences, such as CDD or NUDD, are needed, while if this coupling vanishes we may employ the more efficient multi-qubit sequences described in equations (41) and (42). Regardless, imposing displacement anti-symmetry over T p , one finds that Representative results for the fidelity behavior when these are applied to two qubits subject to quantum-Gaussian noise with a sub-Ohmic spectrum are shown in figure 7, for a value of the transit time t 1,2 corresponding to generic (neither collective nor independent) coupling strengths. For any two-qubit initial pure state ( ) | | r º YñáY 0 , the fidelity is computed as in terms of the appropriate decay and phase terms (see appendix A for explicit expressions). Only one of the sequences satisfies the plateau conditions, and indeed the plateau is seen to appear only for such a sequence. Also, notice that the value at which the fidelity saturates, i.e., the quality of the plateau, deteriorates as τ (hence T p ) increases and approaches the upper bound imposed by equation (57). Figure 8 further illustrates how the control protocol incorporating displacement anti-symmetry is also the only one exhibiting model robustness, namely, the only one guaranteeing that the plateau may be achieved for arbitrary spin-boson couplings. Specifically, a protocol with displacement anti-symmetry (built out of ( ) ), one with mirror anti-symmetry (built out of ( ) , and a non-selective control protocol ( ( ) ( ) U T X X 2 1 2 ) are tested against the same Gaussian noise spectrum of figure 7 for different values of the transit time-equivalently, different spatial separation between the two qubits. In the  ¥ t 1,2 limit, which corresponds to private baths, all strategies work equally well, as expected. In the opposite limit of a collective , the non-selective control protocol fails to achieve a plateau since it cannot make ( In the general case ( < < ¥ t 0 1,2 ), only the displacement anti-symmetry enhanced protocol achieves a high-fidelity plateau.

Multi-qubit setting
In order to extend the approach to the N-qubit setting, the key step is to show that, once the generalized displacement anti-symmetry introduced in section 3.3 is imposed, a cancellation analog to equation (60) holds. Consider the terms proportional to M arising in ( ) for each qubit pair (see equation (59)), and let ℓ ℓ without loss of generality. Since, for the matrix defined in equation (44), 1 e e e e e 0. Consequently, for any ℓ ℓ ¹ ¢, we may re-express ( ) is finite. Thus, a plateau regime may still exist, provided that the conditions identified for the two-qubit case, equations (57) and (58), remain valid. Remarkably, this result does not depend on the specific choice of base sequence, but only on the displacement anti-symmetry, further highlighting its fundamental role in multi-qubit control protocols for dephasing noise.

Controlled entanglement generation and storage
Quantum entanglement is a crucial resource for QIP and hence devising ways to reliably generate and store it is a particularly important task. Entanglement may be achieved directly, via tunable or always-on couplings between qubits or, if the latter are not readily available, indirectly, with the aid of a common quantum environment. Various schemes for indirect generation of bipartite entanglement, as well as weaker quantum correlations quantified by discord, have been proposed [28,29,44]. Once created, such quantum correlations have to be stored (e.g., to be later used for quantum tasks of interest), and protected from unwanted decoherence. Several ways to do this have been invoked, e.g., employing the quantum Zeno effect [45], quantum feedback [46,47], and DD control [48].
While the emphasis of existing work is on two-qubit settings and bipartite entanglement, the formalism we've developed allows for the generation and storage of multi-partite entanglement with high-fidelity for an extended time. Let us assume a multi-qubit Hamiltonian that includes classical and bosonic dephasing, as in equations (2) and (3), with vanishing direct coupling, ℓ ℓ , (or else one could just use that to create entanglement), and no assumption on the initial bath state (in particular, no thermal equilibrium). Our proposed strategy consists of two steps: Stage 1: Entanglement generation. This can be achieved by using M g repetitions, each of duration T p , of a high-order multi-qubit DD sequence such as CDD or NUDD without displacement anti-symmetry. In this way, the contribution of all dephasing terms is suppressed to high order, except for the one stemming from the induced coupling term-which grows linearly with M g (recall equation (59)). As noted, the induced coupling is ruled by and, as such, it does not depend on r B but only on the actual dephasing Hamiltonian, in particular the spectral density function ( ) w J . The fact that the protocol is independent of r B is an important prerequisite for generating entanglement on demand, which may be achieved provided that accurate knowledge of the relevant power spectra is available. In principle, this may be obtained by extending noise spectroscopy protocols for single-qubit [14,49] and two-qubit classical dephasing [50] to general multi-qubit dephasing [51].
Stage 2: Entanglement storage. Once the state of the multi-qubit system after time = T M T g g p is sufficiently close to an entangled state of interest, we may switch to a DD sequence with displacement anti-symmetry, in order to achieve protection for long storage times, say, = ¢ T M T s s p , provided that the plateau conditions of equation (57) are satisfied, and where we allow for the duration of the storage base sequence to differ, in general. Basically, the displacement anti-symmetry acts like an on/off switch for entanglement generation. There is, however, an important subtlety in the analysis that must be pointed out. In our derivation of the the plateau conditions, we relied on the assumption that the initial joint state was of the form r r Ä ; S B this need not be the case at the end of the entanglement generation stage, with ( ) r T SB g involving entanglement between S and B in general (hence making ( ) r T S g mixed). To move forward, it is necessary to re-examine the derivation when the total evolution time T is divided in two consecutive stages, without and with displacement anti-symmetry, respectively. One can see that: enough for a linear approximation to be valid [74]. For example, the dephasing effects of current and voltage fluctuations in Josephson junction qubits are accurately described in terms of spin-boson interactions [75].
It is often the case that the noise is generated by a large number of independent, weakly coupled sources, and can therefore be assumed to be Gaussian. Generic noise, however, is non-Gaussian; in particular, as mentioned in section 4.1, the assumption of Gaussianity breaks down for qubits operated at an optimal point [43], as well as for strongly coupled bistable fluctuators of the sort that generate f 1 in Josephson nano-circuits [42,59]. For quantum environments, both the initial state and the nature of the interaction with the system will determine whether the noise is Gaussian or not [14]. In these cases, the interplay between generalized FFs and higher-order noise spectra needs to be carefully considered.
Despite the broad applicability of the dephasing noise model on which we have based our analysis, it necessarily neglects important aspects that will affect the evolution of the system in reality. In particular, it is generally true that interactions between a quantum system and its environment will generate both dephasing and relaxation effects, with associated characteristic time scales T 2 and T 1 , respectively. The multi-qubit DD protocols we have presented are designed to address dephasing effects only, and are therefore not expected to be effective if 'off-axis' noise effects are significant over the desired storage time. In such cases, multi-axis DD protocols are necessary to counter both dephasing and relaxation effects simultaneously, and it is an interesting open question to determine the extent to which suitable variants of generalized displacement anti-symmetry may possibly still prove beneficial. That being acknowledged, for most of the above-mentioned QIP settings, an appreciable separation between dephasing and relaxation times exists (that is,  T T 1 2 ), and dephasing effects provide the dominant source of error up to very long storage times. In these systems, relaxation will not be a significant factor limiting the validity of our results, including those pertaining to long-time quantum storage [17]. By contrast, if the dephasing noise is strongly Markovian, the lack of temporal correlations will limit the usefulness of the proposed protocols. However, as already remarked in section 2.1, this holds for DD techniques generally, and it is well understood that significant Markovian error contributions must be countered using closed-loop quantum error correction strategies [1].
Even if the above conditions are obeyed, the simplifying assumption of the existence of a hard noise cutoff w c L , r for each of noise spectrum frequencies w L r , made in section 4.1, will hardly be met in realistic scenarios. The purpose of this assumption was to avoid the singular behavior associated with the resonant frequencies defined by equation (50). In the absence of these cutoffs, the contribution of any noise at these frequencies will be amplified in a way that grows linearly with the number M of base sequence repetitions, ultimately making a longtime plateau unsustainable. However, building on the quantitative analysis carried out for the single-qubit case [17], if the the noise beyond the resonant frequencies decays sufficiently rapidly, one can expect the errors associated with high frequency noise to grow slowly with M. As in the single-qubit case, we expect that this may still allow for the maintenance of a fidelity plateau for storage times that are sufficiently long to be practically useful.
Beside deviations from the assumed decoherence model, realistic control scenarios will inevitably face a number of technological limitations and imperfections. While the precise nature and extent of the discrepancies resulting from the simplifying assumption of perfect control resources we have made will depend on the particular physical system being investigated, it is worthwhile highlighting those features that may have the greatest potential to limit the efficacy of the proposed quantum storage protocols across all potential QIP platforms. First, it is clear that any real control pulse has a non-zero duration. As a result, the application of each pulse in a DD sequence will introduce depolarization errors, in addition to a purely dephasing effect. It will not then be possible to describe the modulating influence of a pulse sequence in terms of uniaxial switching functions of the form exemplified in equation (20), and, consequently, the Magnus expansion for the controlled propagator˜( ) U T e will no longer exactly truncate. Strictly speaking, although it is possible to design sequences of finite-width pulses that achieve a non-zero CO [76], the (infinite-level-Magnus) FO [ ] F ¥ of any DD protocols for a two-axis decoherence model will be zero [19]. Generally, one may expect this problem to become more significant as the number of applied pulses increases. For short-term state preservation and relatively small numbers of qubits N, errors introduced by finiteduration pulses need not be a major concern, as the low-FO contributions may be negligible, so that the 'effective' FO [ ] F k for finite order κ remains sufficiently high [19,79]. For long-term memory and/or large N, however, the accumulation of pulse-induced errors may significantly reduce the attainable fidelity and the duration of any plateau. There are two readily apparent approaches to addressing this problem, which will likely have to be used in tandem in practical dephasing settings. The first, and a motivation for this work, is to use DD protocols that utilize the minimum number of pulses to achieve a desired level of error suppression for a given dephasing environment and system size N. The second is to replace 'primitive' DD pulses with dynamically corrected gates or composite pulses [7,8,[77][78][79], so that error cancellation is maintained for the duration of all pulses, up to a sufficiently high order. For single-qubit storage, this approach has been shown to successfully counter pulse-width effects which would otherwise prevent or degrade a coherence plateau [17]. While a quantitative (numerical) analysis of a specific multi-qubit system is beyond our current scope, we expect similar techniques to maintain their usefulness, with the caveat that different COs may be required for different qubits, in order for the overall sign patterns required by displacement anti-symmetry to be maintained. Lastly, another important source of control error arises from limited timing resolution and 'jitter'. This may take the form of imprecision in the timing of the individual pulses that comprise a DD sequence. As the analysis in [80] implies, even when pulses are assumed to have vanishing duration, and imprecision is restricted to small rounding errors in the timing of each pulse, the actual CO and FO will go to zero. For short-term storage, this issue is most serious for optimized sequences, such as those based on UDD, which are known to be particularly sensitive to pulse placement. For long-term storage, two main effects may be attributable to timing errors in the multi-qubit case. The first effect, similar to the single-qubit setting analyzed in [17], is premature or belated memory access time. The long-time memory protocols discussed in section 4 are designed to allow for highfidelity access to stored information only at integer multiples of the base sequence time, i.e., when = T MT p , with access latency capped at  T T p . If there is some small timing offset dt , so that d = + T MT t p , the suboptimal access time can have a similar effect on fidelity as pulse timing imprecision. The second effect, distinctive of the multi-qubit scenario, is breaking the generalized displacement anti-symmetry that is necessary for a plateau regime to exist. A full quantitative analysis of timing errors is necessarily highly system-dependent and, again, beyond our scope. We nevertheless include in appendix B a qualitative analysis of symmetry-breaking effects on plateau emergence, based on a timing error model which, despite its oversimplified nature, is analytically tractable and captures the essential idea. Ultimately, as anticipated at the single-qubit level [17], this analysis reinforces how it is essential that DD protocols, particularly those intended for long-time storage, be clocked by high-resolution (sub-ps) timing systems with minimum jitter.

Conclusion and outlook
We have provided criteria for the design of effective and efficient control protocols for the preservation of arbitrary multi-qubit states in a relevant class of dephasing models, that combines the effects of classical noise and a linear interaction with a bosonic bath, not necessarily in thermal equilibrium. Under the assumption that (nearly) instantaneous DD pulses may be selectively applied to arbitrary subsets of qubits, we showed that the reduced dynamics of a multi-qubit system can be expressed in terms of a hierarchy of noise spectra-that capture the statistical properties of quantum and classical dephasing sources in the frequency domain-and of a small set of firstand second-order FFs-that the describe the modulating effect of the applied control. These results allow for a relatively straightforward, exact characterization of the performance of an arbitrary DD protocol in both the time and frequency domains, and serve as the starting point for the derivation of conditions for the construction of resource-efficient high-order DD protocols.
Specifically, we showed that multi-qubit DD sequences may be constructed that are able to achieve highorder error suppression using exponentially fewer pulses than the most efficient existing protocols, so long as any direct qubit-qubit coupling is constant. This reduction in pulse number offers tremendous practical advantages in terms of the required minimum switching time and the cumulative effect of pulse errors. The improvement in efficiency derives from the property of displacement anti-symmetry with which the new protocols are endowed. Importantly, through the imposition of this form of temporal symmetry, it is also possible to ensure that a sequence achieves maximal filtering order in the frequency domain. This contrasts with previously proposed multi-qubit protocols based on nesting and concatenation, for which maximum filtering order cannot always be guaranteed.
We also demonstrated that the method of producing long-time quantum memory via DD sequence repetition, described previously for single-qubit systems under Gaussian noise, can be generalized to multiqubit systems possibly subject to general non-Gaussian noise. For strictly classical, Gaussian noise on N qubits, the conditions for engineering the required fidelity plateau are a natural extension of those derived for the singlequbit case [17]. However, when a quantum spin-boson interaction is included, the fidelity plateau cannot be maintained unless additional structure is imposed on the base sequence from the outset. We found that those sequences possessing displacement anti-symmetry have the necessary structure and, therefore, can be used to generate a fidelity plateau in a combined classical and quantum noise environment. On the basis of these observations, we have outlined a simple switched control protocol for the the generation and storage of entangled multi-qubit states.
The central role of that the displacement anti-symmetry plays in suppressing the effects of noise associated with the genuinely quantum (non-commuting) nature of the bath points to its relevance for characterizing bathinduced spatial correlations, along with their impact on the implementation of quantum technologies. By comparing the response of two qubits to different control sequences, for instance, it is now possible in principle to discriminate between a classical or a quantum bath with Gaussian statistics. This may have important implications for quantum verification and validation protocols and, ultimately, fault-tolerant quantum computing architectures in the presence of bath-induced spatial correlations. While exploring the usefulness of displacement anti-symmetry beyond the class of open quantum systems examined here is a natural direction for further investigation, we anticipate immediate applications of our enhanced DD sequences in the context of multi-qubit noise spectroscopy for correlated dephasing environments [50,51].  where · stands for an identity action. While the diagonal entries (populations) are unaffected by the dephasing noise, off-diagonal terms decay according to different controlled decoherence functions, in the absence of special symmetries 10 . Similarly, the phase evolution is not the same for all the off-diagonal coherence elements. 9 Classical non-Gaussian noise can also contribute to phase evolution, if odd-order cumulants are non-zero. For classical plus quantum non-Gaussian noise, i.e., when high-order cumulants of ( ) and/or ( ) ℓ z t exist, all factors in equation (32) may contribute to ( ) f t ab and it becomes hard to isolate classical versus quantum effects. 10 It is easy to verify that, for collective dephasing, qubit-permutation symmetry implies that c c c = = 2 1,1 2,2 1,2 , in agreement with decoherence-free subspace theory [1].
Rather, the diagonal and anti-diagonal entries do not exhibit phase evolution, whereas all other elements gain a time-dependent phase according to the following pattern: ) quantum bosonic noise (equation (59) in the main text) by using the noisy filters, thereby computing the corrections induced by a non-zero δ. One finds then the noisy FFs becomes difficult to identify a plateau at all. For sufficiently small violations, on the other hand, it is still possible to engineer a clear plateau that can be sustained for a significant number M of base-sequence repetitions. Thus, while the magnitude of any control errors plays an important role in determining the viability of an extended fidelity plateau, the mere existence of such imperfections need not, in itself, prohibit the generation and maintenance of a practically useful long-time quantum memory. As stressed in the main text, these results underline the need for high-precision control engineering in QIP.