arXiv 2501.136112025-01-23hep-lathep-thquant-ph

量子纠错与$Z(2)$格点规范理论

Quantum Error Correction and $Z(2)$ Lattice Gauge Theories

Seyong Kim

备注：: 10 pages, 8 figures, Proceedings of the 41st International Symposium on Lattice Field Theory (Lattice 2024), University of Liverpool, UK, 2024

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AI总结：

本文研究量子纠错阈值问题，通过将解码映射到$Z(2)$格点规范理论，并用蒙特卡洛模拟分析相图，以理解环面/表面码在噪声下的性能。

摘要：

$Z(2)$格点规范理论在量子纠错（QEC）码的阈值概率研究中扮演重要角色。对于某些QEC码，如著名的Kitaev环面/表面码，可以将QEC解码问题映射到给定噪声模型下的统计力学模型。阈值概率的研究则对应于映射后的统计力学模型的相图研究。这可以通过统计力学模型的蒙特卡洛模拟来进行。在文献[Rispler]中，我们研究了二维环面/表面码在真实噪声模型下的效应，包括综合征测量噪声，并引入了随机耦合plaquette规范模型，即三维$Z(2) \ imes Z(2)$格点规范理论。这个新的$Z(2)$规范理论模型捕捉了在退极化噪声和综合征噪声下环面/表面码的主要特征。在这些会议论文中，我们主要关注蒙特卡洛模拟的方面，并讨论映射后的$Z(2)$格点理论类的蒙特卡洛模拟的初步结果。

英文摘要：

$Z(2)$ lattice gauge theory plays an important role in the study of the threshold probability of Quantum Error Correction (QEC) for a quantum code. For certain QEC codes, such as the well-known Kitaev's toric/surface code, one can find a mapping of the QEC decoding problem onto a statistical mechanics model for a given noise model. The investigation of the threshold probability then corresponds to that of the phase diagram of the mapped statistical mechanics model. This can be studied by Monte Carlo simulation of the statistical mechanics model. In~\cite{Rispler}, we investigate the effects of realistic noise models on the toric/surface code in two dimensions together with syndrome measurement noise and introduce the random coupled-plaquette gauge model, 3-dimensional $Z(2) \times Z(2)$ lattice gauge theory. This new Z(2) gauge theory model captures main aspects of toric/surface code under depolarizing and syndrome noise. In these proceedings, we mainly focus on the aspects of Mont Carlo simulation and discuss preliminary results from Monte Carlo simulations of mapped classes of Z(2) lattice theories.

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