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arXiv:1208.6371

Equivalence and limitations of error suppression techniques for adiabatic quantum computing

Kevin C Young, Mohan Sarovar

Published Aug 1, 2012• 0 citations
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Abstract

While adiabatic quantum computation (AQC) possesses some intrinsic robustness to noise, it is expected that a form of error control will be necessary for large scale computations. Error control ideas developed for circuit-model quantum computation do not transfer easily to the AQC model and to date there have been two main proposals to suppress errors during an AQC implementation: energy gap protection and dynamical decoupling. Here we show that these two methods are essentially equivalent and can be analyzed within the same formalism. We analyze the effectiveness of such error suppression techniques and identify critical constraints on the performance of error suppression in AQC, suggesting that error suppression by itself is insufficient for fault-tolerant, large-scale AQC and that a form of error correction is needed.

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Two topics in the field of quantum information processing, optimized dynamical error suppression and quantum algorithms, are considered here. ? The computational errors induced by the surrounding environment is one of the main obstacles in building a quantum computer. Engineering powerful techniques to combat errors in quantum devices is highly demanding. In the first part of this thesis, I focus on one quantum error correction technique, dynamical decoupling (DD), introduced in Chapter 1. Chapter 2 is dedicated to nested UDD (NUDD), a highly efficient decoupling scheme that utilizes the decoupling characteristics of UDD by multi-layer nesting. UDD (1-layer NUDD) is an optimal DD method for eliminating single-qubit general dephasing, and QDD (2-layer NUDD) is a near-optimal DD method for eliminating one qubit general decoherence. I present a rigorous analytical proof of the performance and universality of QDD/NUDD, and obtain an explicit formula for the decoupling order of each error type, which elucidates the relationship between the error type and characteristics of NUDD. From the explicit formula, a NUDD scheme can be designed accordingly such that optimal efficiency of NUDD is achieved. Moreover, the highly efficient error cancellation mechanism is revealed by the analysis. The proof of QDD has been published in [31], and the proof of NUDD will be submitted for publication shortly. ? Chapter 3 is devoted to the Adiabatic Quantum Computation (AQC). In this work (published in [44]), a general time-optimal strategy, which in principle can optimize any quantum adiabatic algorithm for which the gap is known or can be estimated, is formulated. In addition, I present a natural differential-geometric framework for AQC.

•Jul 25, 2012•0 citations
While adiabatic quantum computation (AQC) possesses some intrinsic robustness to noise, it is expected that a form of error control will be necessary for large scale computations. Error control ideas developed for circuit-model quantum computation do not transfer easily to the AQC model and to date there have been two main proposals to suppress errors during an AQC implementation: energy gap protection and dynamical decoupling. Here we show that these two methods are essentially equivalent and can be analyzed within the same formalism. We analyze the effectiveness of such error suppression techniques and identify critical constraints on the performance of error suppression in AQC, suggesting that error suppression by itself is insufficient for fault-tolerant, large-scale AQC and that a form of error correction is needed.
While adiabatic quantum computing (AQC) has some robustness to noise and decoherence, it is widely believed that encoding, error suppression and error correction will be required to scale AQC to large problem sizes. Previous works have established at least two different techniques for error suppression in AQC. In this paper we derive a model for describing the dynamics of encoded AQC and show that previous constructions for error suppression can be unified with this dynamical model. In addition, the model clarifies the mechanisms of error suppression and allows the identification of its weaknesses. In the second half of the paper, we utilize our description of non-equilibrium dynamics in encoded AQC to construct methods for error correction in AQC by cooling local degrees of freedom (qubits). While this is shown to be possible in principle, we also identify the key challenge to this approach: the requirement of high-weight Hamiltonians. Finally, we use our dynamical model to perform a simplified thermal stability analysis of concatenated-stabilizer-code encoded many-body systems for AQC or quantum memories. This work is a companion paper to 'Error suppression and error correction in adiabatic quantum computation: techniques and challenges (2013 Phys. Rev. X 3 041013)', which provides a quantum information perspective on the techniques and limitations of error suppression and correction in AQC. In this paper we couch the same results within a dynamical framework, which allows for a detailed analysis of the non-equilibrium dynamics of error suppression and correction in encoded AQC.
Quantum computation (QC) relies on the ability to implement high-fidelity quantum gate operations and successfully preserve quantum state coherence. One of the most challenging obstacles for reliable QC is overcoming the inevitable interaction between a quantum system and its environment. Unwanted interactions result in decoherence processes that cause quantum states to deviate from a desired evolution, consequently leading to computational errors and loss of coherence. Dynamical decoupling (DD) is one such method, which seeks to attenuate the effects of decoherence by applying strong and expeditious control pulses solely to the system. Provided the pulses are applied over a time duration sufficiently shorter than the correlation time associated with the environment dynamics, DD effectively averages out undesirable interactions and preserves quantum states with a low probability of error, or fidelity loss. In this study various aspects of this approach are studied from sequence construction to applications of DD to protecting QC. First, a comprehensive examination of the error suppression properties of a near-optimal DD approach is given to understand the relationship between error suppression capabilities and the number of required DD control pulses in the case of ideal, instantaneous pulses. While such considerations are instructive for examining DD efficiency, i.e., performance vs the number of control pulses, high-fidelity DD in realizable systems is difficult to achieve due to intrinsic pulse imperfections which further contribute to decoherence. As a second consideration, it is shown how one can overcome this hurdle and achieve robustness and recover high-fidelity DD in the presence of faulty control pulses using Genetic Algorithm optimization and sequence symmetrization. Thirdly, to illustrate the implementation of DD in conjunction with QC, the utilization of DD and quantum error correction codes (QECCs) as a protection method for adiabatic quantum computing (AQC) is discussed. A performance comparison between two deterministic DD schemes is given, where preference towards one particular method is found due to sequence structure and procedure from which higher accuracy sequences are generated.
Abstract Incorporating protection against quantum errors into adiabatic quantum computing (AQC) is an important task due to the inevitable presence of decoherence. Here, we investigate an error-protected encoding of the AQC Hamiltonian, where qubit ensembles are used in place of qubits. Our Hamiltonian only involves total spin operators of the ensembles, offering a simpler route towards error-corrected quantum computing. Our scheme is particularly suited to neutral atomic gases where it is possible to realize large ensemble sizes and produce ensemble-ensemble entanglement. We identify a critical ensemble size Nc where the nature of the first excited state becomes a single particle perturbation of the ground state, and the gap energy is predictable by mean-field theory. For ensemble sizes larger than Nc, the ground state becomes protected due to the presence of logically equivalent states and the AQC performance improves with N, as long as the decoherence rate is sufficiently low.
Two topics in the field of quantum information processing, optimized dynamical error suppression and quantum algorithms, are considered here. ? The computational errors induced by the surrounding environment is one of the main obstacles in building a quantum computer. Engineering powerful techniques to combat errors in quantum devices is highly demanding. In the first part of this thesis, I focus on one quantum error correction technique, dynamical decoupling (DD), introduced in Chapter 1. Chapter 2 is dedicated to nested UDD (NUDD), a highly efficient decoupling scheme that utilizes the decoupling characteristics of UDD by multi-layer nesting. UDD (1-layer NUDD) is an optimal DD method for eliminating single-qubit general dephasing, and QDD (2-layer NUDD) is a near-optimal DD method for eliminating one qubit general decoherence. I present a rigorous analytical proof of the performance and universality of QDD/NUDD, and obtain an explicit formula for the decoupling order of each error type, which elucidates the relationship between the error type and characteristics of NUDD. From the explicit formula, a NUDD scheme can be designed accordingly such that optimal efficiency of NUDD is achieved. Moreover, the highly efficient error cancellation mechanism is revealed by the analysis. The proof of QDD has been published in [31], and the proof of NUDD will be submitted for publication shortly. ? Chapter 3 is devoted to the Adiabatic Quantum Computation (AQC). In this work (published in [44]), a general time-optimal strategy, which in principle can optimize any quantum adiabatic algorithm for which the gap is known or can be estimated, is formulated. In addition, I present a natural differential-geometric framework for AQC.