AI 中文总结
研究量子纠错中错误阈值估计问题，通过傅里叶 - 沃尔什投影方案和最小副本理论方法，对多种稳定码进行解析估计，与数值结果精确一致，满足自对偶熵关系，揭示了清洁与无序模型的联系。
AI 中文摘要
在量子纠错中，错误阈值为通过解码非相干噪声、弱测量或推理的影响来实现容错能力提供了重要的定量指导。然而，错误阈值的数值通常只能通过对基础噪声模型的大规模数值模拟来获得。在此，我们通过傅里叶 - 沃尔什投影方案引入了对落入西森普遍性类别的错误阈值的解析估计，该方案将基础无无序统计力学模型的临界点映射到玻恩无序的西森临界点。使用最小副本理论方法，这种封闭形式的估计是从精确复制的单键权重的投影中获得的，我们发现它在空间维度\(d = 2 - 5\)中能（在一个百分点内）重现随机键和随机格点伊辛模型/\(\mathbb{Z}_2\)稳定码的已知数值阈值，并扩展到\(q\leq4\)的Potts变量。我们投影方案的主要应用是\(\mathbb{Z}_q\)表面码，其解码问题映射到无序的\(q\)态时钟模型。对于\(q\geq5\)，清洁时钟模型有两个贝雷津斯基 - 科斯特利茨 - 托尔斯过渡，投影将其映射到两个界定中间信息临界相的西森温度。所得阈值不仅与最近的去相干\(\mathbb{Z}_q\)环面码数值精确一致，而且发现满足吉尔伯特 - 瓦尔沙莫夫自对偶熵关系\(\ln q\simeq H_q(T_1^\ast)+H_q(T_2^\ast)\)，尽管在构建中未施加对偶条件。我们的方法指出了清洁模型和玻恩无序模型之间更深层次的联系，同时允许对各种稳定码的错误阈值进行即时解析估计。
英文摘要
In quantum error correction, the error threshold provides essential quantitative guidance for the ability to bring about fault-tolerance through decoding the effects of incoherent noise, weak measurement or inference. However, the numerical value of an error threshold is typically only accessible through large-scale numerical simulations of the underlying noise model. Here we introduce an analytical estimate of error thresholds falling into the Nishimori universality class via a Fourier--Walsh projection scheme that maps the critical point of the underlying disorder-free statistical-mechanics model to the Born-disordered Nishimori critical point. Using a minimal replica theory approach, this closed-form estimate is obtained from a projection of the exact replicated single-bond weight which we find to reproduce (within a percentage point) the known numerical thresholds of random-bond and random-plaquette Ising models / $\mathbb Z_2$ stabilizer codes in spatial dimensions $d=2-5$, and extends to Potts variables with $q\le4$. The main application of our projection scheme is to $\mathbb Z_q$ surface codes, whose decoding problem maps to the disordered $q$-state clock model. For $q\ge5$ the clean clock model has \textit{two} Berezinskii--Kosterlitz--Thouless transitions, which the projection maps to two Nishimori temperatures that bound an intermediate information-critical phase. The resulting threshold values not only accurately agree with recent decohered-$\mathbb Z_q$-toric-code numerics, but are found to satisfy the Gilbert--Varshamov self-dual entropy relation $\ln q \simeq H_q(T_1^\ast)+H_q(T_2^\ast),$ although no duality condition is imposed in the construction. Our approach thereby points to a deeper connection between the clean and Born-disordered models, while allowing for instant analytical estimates of error thresholds for a variety of stabilizer codes.
Comments22 pages, 5 figures
