发表机构
The University of Texas at Austin; University of Chicago（德克萨斯大学奥斯汀分校; 芝加哥大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出一种无需采样的预测器，通过稳定子和逻辑算子计算级联量子纠错码逻辑错误率的首项，实现快速评估与比较。
AI 中文摘要
通过蒙特卡洛采样估计量子纠错码（QEC）的逻辑错误率（LER）需要 $O(p^{-\lceil d/2 \rceil}/\varepsilon^2)$ 个样本，其中物理错误率为 $p$，码距为 $d$，相对误差为 $\varepsilon$。在大距离和低错误率下，尤其是在级联码中（其距离随级数 $\ell$ 按 $d^{\ell}$ 增长），这种采样变得难以处理。快速失败方法通过分别对每个权重的失败进行采样来降低这一成本，但它仅适用于非级联码，并且不模拟级联码各级之间传递的软信息。我们提出了一种预测器，它利用稳定子、最小权重逻辑算子，以及对于奇数 $d$ 的权重为 $d+1$ 的逻辑算子，在不进行采样的情况下，计算 LER 作为 $p$ 的多项式时的首项单项式 $c\\,p^{o}$，在级联的每一级，时间复杂度为码参数的多项式，并且在固定预算下对 $\ell$ 是线性的。我们证明，对于任何非级联稳定子码，对于每个逻辑可观测量和任意的每量子比特错误率，只要没有预算耗尽，我们的预测就是精确的。此外，对于级联码，我们构建了一种递归方法，可以在指数时间内精确计算首项单项式，而我们的预测器近似于这种方法。我们的预测在级别 1 上与我们评估的每个码的精确枚举相匹配，对于冰山码（iceberg code）则匹配到三个级别。使用我们的预测器，我们在 48 个核心小时内评估并比较了四种噪声偏置下的 37 种码配置，而蒙特卡洛采样在 $p = 10^{-3}$ 时对于这 148 种情况中的 113 种需要超过 $10^{12}$ 次采样。
英文摘要
Estimating the logical error rate (LER) of a quantum error-correcting (QEC) code by Monte Carlo sampling takes $O(p^{-\lceil d/2 \rceil}/\varepsilon^2)$ samples at physical error rate $p$, code distance $d$ and relative error $\varepsilon$, which becomes intractable at large distances and low error rates, and especially for concatenated codes, whose distance grows as $d^{\ell}$ with the number of levels $\ell$. The fail fast method reduces this cost by sampling the failures of each weight separately, but it only works for unconcatenated codes, and does not model the soft information that the levels of a concatenated code pass to each other. We propose a predictor that takes the stabilizers, the minimum-weight logical operators and, for odd $d$, the logical operators of weight $d + 1$ of a code, and computes, without sampling, the leading monomial $c\,p^{o}$ of the LER when expressed as a polynomial in $p$, at every level of concatenation, in time polynomial in the code parameters and linear in $\ell$ for fixed budgets. We prove that our prediction is exact for any unconcatenated stabilizer code, for every logical observable and arbitrary per-qubit error rates, whenever no budget is exhausted. Furthermore, for concatenated codes, we construct a recursive approach that computes the leading monomial exactly in exponential time, and our predictor approximates this approach. Our predictions match exact enumeration at level 1 on every code we evaluate, and up to three levels for the iceberg code. Using our predictor, we evaluate and compare $37$ code configurations under four noise biases in $48$ core-hours, where Monte Carlo sampling would need more than $10^{12}$ shots at $p = 10^{-3}$ for $113$ of these $148$ cases.
Comments25 pages, 10 figures
