Abstract
Quantum information is prone to suffer from errors caused by the so-called decoherence, which describes the loss in coherence of quantum states associated to their interactions with the surrounding environment. This decoherence phenomenon is present in every quantum information task, be it transmission, processing or even storage of quantum information. Consequently, the protection of quantum information via quantum error correction codes (QECC) is of paramount importance to construct fully operational quantum computers. Understanding environmental decoherence processes and the way they are modeled is fundamental in order to construct effective error correction methods capable of protecting quantum information. Moreover, quantum channel models that are efficiently implementable and manageable on classical computers are required in order to design and simulate such error correction schemes. In this article, we present a survey of decoherence models, reviewing the manner in which these models can be approximated into quantum Pauli channel models, which can be efficiently implemented on classical computers. We also explain how certain families of quantum error correction codes can be entirely simulated in the classical domain, without the explicit need of a quantum computer. A quantum error correction code for the approximated channel is also a correctable code for the original channel, and its performance can be obtained by Monte Carlo simulations on a classical computer.
Highlights
Since Richard Feynman’s original and ground-breaking proposal in [1] of constructing computers that follow the laws of quantum mechanics to simulate physical systems that obey said laws, the scientific community has gone to extraordinary lengths in order to build an operational quantum computer
The rest of the paper is organized as follows: section II describes the typical decoherence models that are used for quantum error correction codes (QECC) design; section III approximates such models into channels that can be implemented efficiently on classical computers via twirling techniques; section IV discusses the use of memory in the twirled models in order to use more realistic approximations when correlations between different quantum information units exist; section V describes the basic theory of stabilizer coding and the way the so called Pauli-to-binary isomorphism is used to efficiently simulate such QECCs under the corruption of the presented twirled channels
By the Gottesman-Knill theorem, these quantum channel instances can be efficiently simulated on classical computers, and it is worthwhile to relate the decoherence processes described in section II, and mathematically modeled via the combined amplitude and phase damping channel, to the transformations described by the Pauli channel
Summary
Since Richard Feynman’s original and ground-breaking proposal in [1] of constructing computers that follow the laws of quantum mechanics to simulate physical systems that obey said laws, the scientific community has gone to extraordinary lengths in order to build an operational quantum computer. Which can be implemented with classical resources, and that the performance of quantum error correcting codes over the approximated channels is equivalent to that obtained for the original decoherence model. The rest of the paper is organized as follows: section II describes the typical decoherence models that are used for QECC design; section III approximates such models into channels that can be implemented efficiently on classical computers via twirling techniques; section IV discusses the use of memory in the twirled models in order to use more realistic approximations when correlations between different quantum information units exist; section V describes the basic theory of stabilizer coding and the way the so called Pauli-to-binary isomorphism is used to efficiently simulate such QECCs under the corruption of the presented twirled channels.
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