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Experimental protection of quantum gates against decoherence and control errors

One of the biggest challenges for implementing quantum devices is the requirement to perform accurate quantum gates. The destructive effects of interactions with the environment present some of the most difficult obstacles that must be overcome for precise quantum control. In this work we implement a proof of principle experiment of quantum gates protected against a fluctuating environment using dynamical decoupling techniques. We show that decoherence can be reduced during the application of quantum gates. High fidelity quantum gates can be achieved even if the gate time exceeds the decoherence time by one order of magnitude.

One of the biggest challenges for implementing quantum devices is the requirement to perform accurate quantum gates. The destructive effects of interactions with the environment present some of the most difficult obstacles that must be overcome for precise quantum control. In this work we implement a proof of principle experiment of quantum gates protected against a fluctuating environment using dynamical decoupling techniques. We show that decoherence can be reduced during the application of quantum gates. High fidelity quantum gates can be achieved even if the gate time exceeds the decoherence time by one order of magnitude. Introduction.-Quantum information processing (QIP) [1] can lead to a dramatic computational speed-up over classical computers for certain problems. However, any physical QIP device is subject to errors arising from unavoidable interactions with the environment or from control imperfections. Therefore, scalable QIP needs methods for avoiding or correcting those errors. The theory of quantum error correction (QEC) states that it is possible to stabilize a quantum computation provided that the error per gate is below some threshold [2] and high fidelity initial states are prepared [3,4]. However, QEC needs many auxiliary qubits, generating a significant overhead in additional resources. Therefore, it is highly desirable to develop methods for reducing the perturbation both between and inside the quantum gates without requiring additional qubits. In addition, reaching the error threshold required for the implementation of QEC requires that the fidelity of the individual gate operations must be very high [5].
A simple way to avoid decoherence and thus reduce the error per gate consists in choosing qubit systems whose decoherence times are long compared to the duration of a gate operation. However, this is not always possible; in particular in systems that combine different types of qubits, such as electronic and nuclear spins, the decoherence of the electron spin can be faster than the possible gate durations of the nuclear spins. Furthermore, the reduction of the decoherence time of a quantum register with the number of qubits [6,7] imposes an additional difficulty for implementing quantum computation in large systems.
Dynamical decoupling (DD) [8][9][10] is a promising method developed to reduce decoherence by attenuating the system-environment interaction with a sequence of inversion pulses periodically applied to the qubits. Recent experiments have successfully implemented DD methods and demonstrated the resulting increase of the storage times in different systems [11][12][13][14][15][16][17]. In all these implementations, the goal was to preserve a given input state, i.e. to protect a quantum memory against environmental perturbations. The necessity of protecting qubits against environmental noise occurs also in the context of quantum information processing. In a structured environment that induces well-characterized relaxation pathways, it is possible to design protected gate operations by optimal control techniques [18]. If the relaxation mechanism is not known or it affects all modes of the system, it may still be possible to use DD techniques, provided the effect of the DD sequence is compatible with the gate operations used for information processing. In the simplest case, gate operations can be made insensitive to static environmental perturbations by refocusing them [19,20], in a manner quite similar to a Hahn echo. In the more general case of a fluctuating environment, the Hahn echo has to be replaced by multiple-pulse DD sequences. Initial experiments in this direction have been made recently by applying gate operations between cycles of DD sequences acting on the electron spin qubit of an NV center [21] or an effective qubit in a semiconductor quantum dot [22]. In these schemes, the DD protection is not active during the actual gate operation. Possible scheme for maintaining DD protection during the gate operation were also suggested [23][24][25][26][27][28]. These schemes were developed under the assumption of perfect controls, i.e. the controls operations used e.g. for DD should not introduce any additional errors. Here, we propose a general scheme for protection against a fluctuating environment that does not rely on this assumption but is robust against experimental errors and can therefore be implemented in an experimental scheme with realistic control operations.
We show that the decoherence can be reduced during the application of quantum gates for a single qubit in a solid state system. High fidelity quantum gates are achieved even if the gate time exceeds the decoherence time by one order of magnitude. Since the protection scheme introduces many additional control operations, we design the protected gate operations in such a way that the effect of control imperfections on the fidelity of the system is minimized.
Protection scheme.-Quantum logical gates are achieved by using time-varying control Hamiltonians H c (t) for the relevant qubit system. Their propagators can be represented by unitary operators where T is the Dyson time ordering operator and T is the gate time. The propagator can also be expressed in terms of the time-independent average Hamiltonian H G , which would generate the same operation if it were active for the time T . For single qubit gates, this Hamiltonian can be expressed as: where S = (S x , S y , S z ) is the spin vector operator of the system qubit, n is a 3D vector and the strength ω is a real parameter.
In any physical implementation, the system also interacts with the environment, which introduces decoherence reducing the gate fidelity. If the environmental effects become too strong, the quantum computation cannot be stabilized by QEC codes [2]. One approach to avoid this is to use DD for reducing the effects of the environment. Consider the Hamiltonian describing a one qubit system and its environment where H S = ω S S z is the system Hamiltonian and ω S is the Larmor frequency of the system. H E is the environmental Hamiltonian and H SE the system-environment coupling. Here, we describe the environment as a spin bath and the coupling as a pure dephasing interaction where I k z is the spin operator of the k th environment spin, b k is the coupling constant between the system and the k th spin of the environment.
The identity-operation (H G = 0) can be implemented by just applying DD sequences, such as the XY-4 sequence. This sequence was initially introduced in the context of nuclear magnetic resonance (NMR) [29,30]. A similar sequence is the PDD sequence [8,16,31]. Neglecting pulse errors, we can write the zeroth and first order terms of the average Hamiltonian of the XY-4 sequence as H 0 = H E and H 1 = 0. The system-environment Hamiltonian, which causes decoherence, as well as the internal Zeeman Hamiltonian of the system qubit are removed up to the first order approximation. The only remaining term is the environmental Hamiltonian H E , which has no effect on the system qubit.
A simple way to introduce gates with real pulses during decoupling consists in dividing the gate into two equal parts √ U = e −iHGT /2−iHT /2 . Here, e −iHGT is the target control operation, while the drift term HT /2 is an unwanted contribution that we cancel in the protected control operation. We insert the two half-gates Figure 1: Pulse sequence for protecting quantum gates using the XY-4 sequence.
the initial and final free precession periods of the DD sequence, as shown in Fig. 1. Adjusting T = τ , the delay between the DD pulses, the propagator for the full cycle (duration τ c = 4τ , neglecting the pulse duration) becomes, to first order in the cycle time, The free evolution terms ∝ H S + H SE are cancelled to first order by DD. This scheme resembles the theoretical approach of Refs. [23,26], except that our scheme has higher symmetry, which helps to eliminate some control errors [31] . The remaining terms are H E , which acts only on the environment, and the gate operator H G . Since the effect of the SE-coupling is absent here, decoherence has been eliminated in this first order approximation.
Robust implementation.-In real implementations, experimental imperfections must also be taken into account. In most cases, the dominant imperfection is a deviation between the actual and the ideal amplitude of the control field. The result of this amplitude error is that the rotation angle deviates from the target angle typically by a few percent. The imperfect control can affect both the implementation of H G as well the π pulses of the dynamical decoupling sequence and its effect can be particularly devastating when the number of gate operations is large and the errors accumulate.
The systematic control errors of the DD pulses can be reduced by choosing robust DD sequences [10,16]. A well established method for eliminating control field errors is the use of composite pulses [32]. Composite pulses are sequences of consecutive pulses designed such that the resulting total operation remains close to the ideal target operation even in the presence of some experimental imperfections. A good choice for correcting amplitude errors in a general single qubit rotation R φ (θ) is the BB1 composite pulse [33]: where φ describes the rotation axis, θ the rotation angle, and cos ψ = −θ/4π. Figure 2 shows how a general rotation can be made robust against amplitude errors and protected against decoherence from a fluctuating environment. For this purpose, we replace each rotation R φ (θ) in the BB1 pulse by the protected rotation R φ (θ) according to the scheme of Fig. 1. This scheme can obviously be extended to other DD sequence with symmetrical timing simply by replacing the XY-4 cycle with a different cycle.
Experimental performance.-For the experimental tests we used natural abundance 13 C nuclear spins in the CH 2 groups of a polycrystalline adamantane sample as the system qubit. In this system, the carbon spins are coupled to 1 H nuclear spins by heteronuclear magnetic dipole interactions. The protons are coupled to each other by homonuclear dipolar interactions. Under our conditions, the couplings between the carbon nuclei can be neglected. The experiments were performed on a homebuilt 300 MHz solid-state NMR spectrometer.
In the context of quantum information processing, it is important that the performance of gate operations be independent of the initial conditions. For quantifying the performance of a general quantum operation, the fidelity F can be used [34]: Here, A is the target propagator for the process and B the actual propagator. The actual operations are not always unitary. We therefore write the process as where ρ i and ρ f are the density matrices at the beginning and end of the process. The operators E m must form a basis. For the present case, we choose them as E m = (I, σ x , iσ y , σ z ), with the Pauli matrices σ α . The ideal and actual processes can therefore be quantified by the matrix elements χ mn and experimentally determined by quantum process tomography [35].
In the experimental implementation of this concept, we tested the Hadamard (H), NOT and π/8 gates. For protection, we used the DD sequences XY-4 [29], XY-8 [30] and KDD [16]. The gates were decomposed into sequences of rotations around axes in the xy-plane (see table I) and each rotation was implemented as shown in Fig. 2. Fig. 3 shows the results of the quantum process tomography for the case of the Hadamard gate protected by the XY-8 sequence. Without dynamical decoupling, the system coherence decays on a timescale T * 2 ≈ 370 µs. The decay time only due to the interaction with a fluctuating environment (measured by the Hahn echo) is T 2 ≈ 750 µs. As shown in Table I, the gate fidelities for XY-8 are > 0.95, although the durations of the gate operations are T 2 . All the fidelities were calculated from the process matrices obtained directly from the raw data, without optimization methods as used in previous experiments (see for example [36]). The obtained fidelity values close to one indicate that the accumulation of incoherent errors is well compensated even for a very large number of pulses.
In Fig. 4, we show the achieved gate fidelities for four gate operations: identity = NOOP, NOT, Hadamard and π/8 phase gate. For each type of gate, we plot the achieved fidelity against the total operation time for the gate, using the direct implementation, labeled 'simple' in the figure, using only BB1 pulses and protected gates with different DD sequences. The three DD sequences differ in cycle time and robustness, with XY-4 having the shortest cycle and KDD being the most robust sequence [16]. In the top panel, we plot the fidelity of the NOOP gate for different gate durations, while the lower three panels only show the fidelity for the shortest cycle with each DD sequence. In all cases, the measured gate fidelities were > 0.928. This result is very gratifying, since it shows that high gate fidelities can be obtained even if the gate duration exceeds the decoherence time (T * 2 ) by an order of magnitude. Additionally, they exceed the T 2 time, given by the Hahn echo decay, being a clear indication that the fluctuating environment has been decoupled during the gate execution.
Discussion and Conclusion.-In summary, we have presented a proof of principle demonstration of decoherence suppression during quantum logical gate operations by dynamical decoupling. For this purpose, we inserted robust gate operations into different DD sequences in such a way that they do not interfere destructively with the DD. Using quantum process tomography, we have shown that high fidelity single qubit quantum gates can be achieved even if the gate time exceeds the decoherence time by one order of magnitude. We carefully designed the protection scheme to be robust against deviations of the control fields. As a result, even for protected operations consisting of up to 330 individual control pulses, the resulting fidelity is not significantly reduced compared to a gate implemented with a single pulse, and often it is higher. For some systems, slower gates promise higher fidelities than short gates [37,38]. In these cases, the approach that we have demonstrated here, appears particularly appealing for further increasing the robustness and precision of the gate operations.
This result indicates that quantum computation can be made reliable even for systems in which the gate time is comparable to or even greater than the decoherence time of the individual qubits. In the present study, we have tested the scheme with three different robust DD sequences. Our results are an additional demonstration that dynamical decoupling can be a useful tool that complements quantum error correction. We expect that the scheme is equally applicable to other types of qubit systems as well as to other types of gate operations. In particular, it will be interesting to apply this concept also to multi-qubit systems. In this case we need to refocus the undesired system-environment interactions without eliminating the desired qubit-qubit interactions. This leads to more complex pulse sequences but can be achieved in principle. Efficient methods for selectively turning "on" and "off" specific Hamiltonian terms have been proposed in [39,40]. While the evolution-time overhead grows linearly in these schemes, a particular scheme [41] designed for a networks of dipolar-coupled spins leads to an evolution-time that is independent of the number of qubits. A proposal to combine this scheme with dynamical decoupling was made in [42]. We acknowledge useful discussions with Daniel Lidar. This work is supported by the DFG through Su 192/24-1.  Figure 4: Gate fidelity as a function of gate time for different gate operations protected by different DD sequences. "Simple" indicates gates that were implemented by unprotected rotations. The delay between the pulses for the NOOP was ≈ 3µs.