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International Journal of Quantum Information

QUANTUM ERROR-CORRECTION FOR SPATIALLY CORRELATED ERRORS

Toshihico Arimitsu, Tsuyoshi Hayashi, Sachiko Kitajima, Fumio Shibata

Published Jul 1, 2008• 0 citations
Publisher

Abstract

It is shown that errors due to spatially correlated noises can be corrected by the quantum error-correction code and error-correction procedure prepared for those for independent noises. A model of noisy-channel which is under the influence of spatially correlated quantum Brownian motion is investigated within the framework of non-equilibrium thermo field dynamics that is a canonical operator formalism for dissipative quantum systems.

DOI
10.1142/s0219749908003803

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It is shown that errors due to spatially correlated noises can be corrected by the quantum error-correction code and error-correction procedure prepared for those for independent noises. A model of noisy-channel which is under the influence of spatially correlated quantum Brownian motion is investigated within the framework of non-equilibrium thermo field dynamics that is a canonical operator formalism for dissipative quantum systems.
The framework of the canonical operator formalism for dissipative and/or stochastic quantum systems named non-equilibrium thermo field dynamics (NETFD) is presented in a self-contained manner. As an application of the formalism, a model of noisy-channel which is under the influence of spatially correlated quantum Brownian motion is investigated within the framework of NETFD. It is shown that errors due to spatially correlated noises can be corrected by the quantum error-correction code and error-correction procedure prepared for those for independent noises.
It is shown that errors on qubits caused by spatially correlated noise of quantum Brownian motion can be corrected with a stabilizer code (a quantum error correction code) and recovery operations prepared for uncorrelated noise without modifications. The analysis is made by means of the quantum stochastic Liouville equation approach which has been developed within the canonical operator formalism for dissipative systems called non-equilibrium thermo field dynamics. This approach yields a transparent procedure to derive completely positive maps describing errors on qubits under given statistics of noise.
The effects of quantum error correction (QEC) on the dynamics of entanglement between logical qubits in the presence of a dephasing interaction with a correlated environment is investigated. Such a correlated reservoir introduces entanglement between physical qubits which, for short times, is interpreted as error and suppressed by the QEC routine. However for longer times, although QEC is no longer able to correct errors, it enhances the rate of entanglement production due to the interaction with the environment.
Quantum error correction will be an indispensable ingredient of large-scale quantum computations. Conventional quantum error correction codes (QECC) have been devised with an independent-error model in mind, but one may expect that the noise affecting a system of qubits will, in general, exhibit nonzero correlations in time, or space, or both. This talk will present a brief introduction to the principles of quantum error correction, followed by a discussion of the performance of conventional QECCs in the presence of correlated noise.
We study the performance of simple error correcting and error avoiding quantum codes together with their concatenation for correlated noise models. Specifically, we consider two error models: (i) a bit-flip (phase-flip) noisy Markovian memory channel (model I); (ii) a memory channel defined as a memory degree dependent linear combination of memoryless channels with Kraus decompositions expressed solely in terms of tensor products of X-Pauli (Z-Pauli) operators (model II). The performance of both the three-qubit bit flip (phase flip) and the error avoiding codes suitable for the considered error models is quantified in terms of the entanglement fidelity. We explicitly show that while none of the two codes is effective in the extreme limit when the other is, the three-qubit bit flip (phase flip) code still works for high enough correlations in the errors, whereas the error avoiding code does not work for small correlations. Finally, we consider the concatenation of such codes for both error models and show that it is particularly advantageous for model II in the regime of partial correlations.