有噪声随机电路采样的硬度与复杂度转变
Hardness and Complexity Transition of Noisy Random Circuit Sampling
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中文总结 AI 辅助
研究有噪声随机电路采样中经典可模拟与困难区域的噪声强度边界，基于理想RCS硬度猜想，通过低阶多项式外推和单调性约简，得出γ = Θ(log n/(nd))是渐近复杂度转变尺度，确定了该尺度的通用硬度界。
中文摘要 AI 辅助
随机电路采样（RCS）是证明量子优势的主要候选方法，在理想情况下有强大的复杂性理论证据支持其硬度，且实验进展迅速。但实际中噪声不可避免，核心问题是确定经典可模拟和经典困难区域之间的噪声强度边界。本文为强度为γ的标准局部去极化噪声建立了该边界的通用硬度界。假设理想RCS的标准平均情况#P - 硬度猜想成立，对于满足该猜想的任何电路架构，当γ = O(log n/(nd))时，n量子比特深度为d的电路上的有噪声RCS在任何逆多项式总变差距离内都难以经典模拟，除非多项式层次结构崩溃。证明结合了低阶多项式外推和单调性约简。最后，结合Dalzell等人的收敛到均匀性结果，得出在分层、规则连接架构上γ = ω(log n/(nd))时的高效经典模拟。因此，确定了γ = Θ(log n/(nd))为渐近复杂度转变尺度。
英文摘要
Random circuit sampling (RCS) is a leading candidate for demonstrating quantum advantage, supported by strong complexity-theoretic evidence of hardness in the ideal setting and by rapid experimental progress to date. In practice, however, noise is unavoidable, and a central problem is to identify the noise-strength boundary between classically simulable and classically hard regimes. In this work, we establish an architecture-general hardness bound for this boundary for the standard local depolarizing noise of strength $γ$. Assuming the standard average-case #P-hardness conjecture for ideal RCS, we show that, for any circuit architecture satisfying this conjecture, noisy RCS on the same architecture remains hard to simulate classically within any inverse-polynomial total variation distance whenever $γ=O(\log n/(nd))$ for $n$-qubit circuits of depth $d$, unless the polynomial hierarchy collapses. Crucially, noisy-RCS hardness follows without any additional conjectural or architecture-specific assumption beyond those already entering the ideal-RCS hardness framework. Our proof combines a low-degree polynomial extrapolation with a monotonicity reduction showing that efficient classical simulation at one depolarizing noise strength implies efficient simulation at every larger strength. Together, these ingredients transfer the standard ideal-RCS hardness conjecture to sampling hardness at a prespecified noise strength. Finally, combining the convergence-to-uniformity result of Dalzell et al. [Commun. Math. Phys. 405, 78 (2024)] with our monotonicity reduction yields efficient classical simulation for $γ=ω(\log n/(nd))$ on layered, regularly connected architectures. Thus, wherever the two architectural settings overlap, this identifies $γ=Θ(\log n/(nd))$ as the asymptotic complexity-transition scale.
