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arXiv:quant-ph/0607086 · Physical Review A

Performance of deterministic dynamical decoupling schemes: Concatenated and periodic pulse sequences

Kaveh Khodjasteh, Daniel A. Lidar

Published Jun 8, 2007• 215 citations
arXivHTMLPDF

Abstract

Dynamical decoupling can be used to preserve arbitrary quantum states despite undesired interactions with the environment, using control Hamiltonians affecting the system only. We present a system-independent analysis of dynamical decoupling based on leading order decoupling error estimates, valid for bounded-strength environments. Using as a key tool a renormalization transformation of the effective system-bath coupling Hamiltonian, we delineate the reliability domain of dynamical decoupling used for quantum state preservation, in a general setting for a single qubit. We specifically analyze and compare two deterministic dynamical decoupling schemes---periodic and concatenated---and distinguish between two limiting cases of fast versus slow environments. We prove that concatenated decoupling outperforms periodic decoupling over a wide range of parameters. These results are obtained for both ``ideal'' (zero-width) and realistic (finite-width) pulses This work extends and generalizes our earlier work [Phys. Rev. Lett. 95, 180501 (2005)].

DOI
10.1103/physreva.75.062310

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Dynamical decoupling can be used to preserve arbitrary quantum states despite undesired interactions with the environment, using control Hamiltonians affecting the system only. We present a system-independent analysis of dynamical decoupling based on leading order decoupling error estimates, valid for bounded-strength environments. Using as a key tool a renormalization transformation of the effective system-bath coupling Hamiltonian, we delineate the reliability domain of dynamical decoupling used for quantum state preservation, in a general setting for a single qubit. We specifically analyze and compare two deterministic dynamical decoupling schemes---periodic and concatenated---and distinguish between two limiting cases of fast versus slow environments. We prove that concatenated decoupling outperforms periodic decoupling over a wide range of parameters. These results are obtained for both ``ideal'' (zero-width) and realistic (finite-width) pulses This work extends and generalizes our earlier work [Phys. Rev. Lett. 95, 180501 (2005)].
Dynamical decoupling pulse sequences have been used to extend coherence times in quantum systems ever since the discovery of the spin-echo effect. Here we introduce a method of recursively concatenated dynamical decoupling pulses, designed to overcome both decoherence and operational errors. This is important for coherent control of quantum systems such as quantum computers. For bounded-strength, non-Markovian environments, such as for the spin-bath that arises in electron- and nuclear-spin based solid-state quantum computer proposals, we show that it is strictly advantageous to use concatenated pulses, as opposed to standard periodic dynamical decoupling pulse sequences. Namely, the concatenated scheme is both fault tolerant and superpolynomially more efficient, at equal cost. We derive a condition on the pulse noise level below which concatenation is guaranteed to reduce decoherence.
Dynamical decoupling (DD) refers to a well-established family of methods for error mitigation, comprising pulse sequences aimed at averaging away slowly evolving noise in quantum systems. Here we revisit the question of its efficacy in the presence of noisy pulses in scenarios important for quantum devices today: pulses with gate control errors, and the computational setting where DD is used to reduce noise in every computational gate. We focus on the well-known schemes of periodic (or universal) DD and its extension, concatenated DD, for scaling up its power. The qualitative conclusions from our analysis of these two schemes nevertheless apply to other DD approaches. In the presence of noisy pulses, DD does not always mitigate errors. It does so only when the added noise from the imperfect DD pulses does not outweigh the increased ability in averaging away the original background noise. We present break-even conditions that delineate when DD is useful, and further find that there is a limit in the performance of concatenated DD, specifically in how far one can concatenate the DD pulse sequences before the added noise no longer offers any further benefit in error mitigation.
For a generic dynamical decoupling sequence employing a single-axis control, we study its efficiency in the presence of small errors in the direction of the controlling pulses. In the case that the corresponding ideal dynamical decoupling sequence produces sufficiently good results, the impact of the errors is found to scale as $\ensuremath{\xi}{}^{2}$, with negligible first-order effect, where $\ensuremath{\xi}$ is the dispersion of the random errors. This analytical prediction is numerically tested in a model in which the environment is modeled by one qubit coupled to a quantum kicked rotator in chaotic motion. In this model, with periodic pulses applied to the qubit in the environment, it is found numerically that periodic bang-bang control may outperform Uhrig dynamical decoupling.
We study how dynamical decoupling (DD) pulse sequences can improve the reliability of quantum computers. We prove upper bounds on the accuracy of DD-protected quantum gates and derive sufficient conditions for DD-protected gates to outperform unprotected gates. Under suitable conditions, fault-tolerant quantum circuits constructed from DD-protected gates can tolerate stronger noise and have a lower overhead cost than fault-tolerant circuits constructed from unprotected gates. Our accuracy estimates depend on the dynamics of the bath that couples to the quantum computer and can be expressed either in terms of the operator norm of the bath's Hamiltonian or in terms of the power spectrum of bath correlations; we explain in particular how the performance of recursively generated concatenated pulse sequences can be analyzed from either viewpoint. Our results apply to Hamiltonian noise models with limited spatial correlations.
One of the fundamental problems of chemistry is the difficulty in determining full quantum mechanical descriptions of molecules. Molecules contain many degrees of freedom and as a result the classical description scales factorially with the number of electrons. One way to overcome this mismatch between quantum and classical pictures is to use one quantum system, a quantum computer, to simulate another. Quantum computation is based upon a two-level system, a qubit, which is analogous to a bit in classical computation. Unlike a classical bit, a qubit can exist in a superposition of quantum states. The interaction between a quantum system and its environment results in the decay of the superposition state of the system into a probability density of classical states [1]. This is known as decoherence. A realistic quantum computer requires qubits with long coherence times; with respect to the achievable gating times, for a formal relation refer to section 1.2.2. The spin echo effect in nuclear magnetic resonance (NMR) offers a way to increase the coherence time of a qubit and has been studied as a way to mitigate decoherence in quantum computation[2][3]. By applying an appropriate Π pulse to a system under the influence of a large external field, as in NMR, the interaction term of the Hamiltonian between the qubit and environment can be eliminated[3][4]. Periodic implementation of such pulses is known as periodic dynamic decoupling (PDD). Similar to PDD, concatenated dynamic decoupling (CDD) is a periodic pulse sequence which removes the interaction term of the Hamiltonian, however in CDD pulse sequences are recursively embedded within the overall pulse train. Such pulse sequences have been shown theoretically to work for model quantum systems such as GaAs quantum dots, and theoretically such pulse sequences vastly improve the coherence time of a qubit [5]. There have been many calculations showing that by progressively increasing the number of concatenated pulse sequences increasingly longer coherence times can be achieved [6][7]. These techniques have not been tested in the laboratory and experimental implementation of these promising techniques motivates current research. There also continues to be extensive work on developing novel pulse sequences which may improve coherence times even further. The goal of this research is to experimentally determine the coherence times which can be achieved with a trapped calcium atom by applying these pulse sequences.
We investigate the feasibility for the preservation of coherence and entanglement of one- and two-spin qubits coupled to an interacting quantum spin-1/2 chain within the dynamical decoupling (DD) scheme. The performance is examined by counting the number of computing pulses that can be applied periodically with periods of TTT before qubits become decoherent, while an identical decoupling pulse sequence is applied within each cycle. By considering pulses with mixed directions and finite width controlled by magnetic fields, it is shown that pulse-width accumulation degrades the performance of sequences with larger numbers of pulses and that feasible magnetic fields in practice restrict consideration to sequences with few than 10 decoupling pulses within each cycle. Furthermore, within each cycle TTT, exact nontrivial pulse sequences are found to suppress the qubit-bath coupling to O(TN+1)O({T}^{N+1})O(TN+1) progressively with the minimum number of pulses being 4, 7, and 12 for N=1,2,3N=1,2,3N=1,2,3. These sequences, when applied to all qubits, are shown to preserve both the entanglement and coherence. Based on time-dependent density matrix renormalization, our numerical results show that for modest magnetic fields (10--40 T) available in laboratories, the overall performance is optimized when the number of pulses in each cycle is 4 or 7 with pulse directions alternating between xxx and zzz. Our results provide useful guides for the preservation of coherence and entanglement of spin qubits in the solid state.