arXiv 2203.017062022-03-03quant-phmath-phmath.MP

具有相干误差的五量子比特码的精确性能

Exact performance of the five-qubit code with coherent errors

Chaobin Liu

摘要：

为了更好地理解量子纠错码（QECC）在噪声环境中的行为，我们需要得到QECC的显式编码映射。由于它们需要大量的计算工作，显式编码映射鲜为人知。事实上，这甚至适用于最常见的量子码之一-五量子比特码，也被称为允许纠正一般单量子比特错误的最小完美码。通过直接但复杂的计算，我们得到了五量子比特码的具有单位误差通道的编码映射的显式过程矩阵。过程矩阵允许我们对量子码的性能进行精确的分析。我们证明了代码可以纠正一个通用的错误，在这个意义上说，在重复级联的编码映射与自身，代码不做任何假设的错误模型以外，它是弱的，从而可以消除错误（它可以转换/采取的错误通道的身份通道，如果错误是足够小的。本文重点研究了近年来文献中研究的几种相干误差模型（非对角信道）。我们数值推导出一个下限阈值的收敛代码。此外，我们分析了代码如何影响平均门失真和钻石距离的错误通道。推导了差错前信道和差错后信道的两种测量值的显式表达式，并分析了上述量子码的逻辑误码率。我们的研究结果收紧了钻石距离的噪声通道的误差修正后，在文献中获得的上限。

英文摘要：

To well understand the behavior of quantum error correction codes (QECC) in noise processes, we need to obtain explicit coding maps for QECC. Due to extraordinary amount of computational labor that they entails, explicit coding maps are a little known. Indeed this is even true for one of the most commonly considered quantum codes-the five-qubit code, also known as the smallest perfect code that permits corrections of generic single-qubit errors. With direct but complicated computation, we obtain explicit process matrix of the coding maps with a unital error channel for the five-qubit code. The process matrix allows us to conduct exact analysis on the performance of the quantum code. We prove that the code can correct a generic error in the sense that under repeated concatenation of the coding map with itself, the code does not make any assumption about the error model other than it being weak and thus can remove the error(it can transform/take the error channel to the identity channel if the error is sufficiently small.). We focus on the examination of some coherent error models (non diagonal channels) studied in recent literatures. We numerically derive a lower bound on threshold of the convergence for the code. Furthermore, we analytically show how the code affects the average gate infidelity and diamond distance of the error channels. Explicit formulas of the two measurements for both pre-error channel and post-error channel are derived, and we then analyze the logical error rates of the aforesaid quantum code. Our findings tighten the upper bounds on diamond distance of the noise channel after error corrections obtained in literature.

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