量子采样的紧致查询下界，及其在认证随机性中的应用
Tight Query Lower Bounds for Quantum Sampling, with an Application to Certified Randomness
- Virginia Tech（弗吉尼亚理工大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究证明量子采样中XEB分数超过理想值需Ω(N^{1/3})次查询，并揭示高分数可认证近最优的随机性，为量子优势验证提供紧致下界。
AI中文摘要：
随机电路采样实验是量子计算优势的主要演示，通过线性交叉熵基准（XEB）进行测试，该基准根据结果的理想概率对输出进行评分。我们研究了高XEB分数对不可信量子设备的认证意义。在量子查询模型中，我们证明了诚实采样器的理想分数对于高效设备和认证随机性起到Tsirelson界的作用。首先，超过理想分数一个常数需要对于具有N个结果的Haar和Fourier预言机集合进行Ω(N^{1/3})次查询，且该界是紧致的；这证明了先前关于Haar随机态的猜想，并将其扩展到Fourier采样。其次，对于Fourier采样以及通过标准态制备预言机访问的Haar随机态，进行多项式次查询且分数在理想分数的o(1)范围内的设备的n比特输出具有n-O(log n)比特的平滑最小熵，即使对手与设备纠缠并在之后得知预言机；这几乎是紧致的。作为推论，采样碰撞分布（与理想分布的平方成正比）的查询复杂度为Θ(N^{1/3})，同一算法的相干版本近似制备具有均匀振幅幅度的态的量子Hadamard积。两个下界都依赖于进展度量，该度量限制超过理想分数的超额，并且每次查询仅略微增加。对于Fourier采样，该度量是随机函数纯化中移除对的相干振幅的范数，结合了多项式方法和压缩预言机技术；对于Haar随机态，它是Dirichlet正交多项式展开中平均水平的平方根，由一次观察效果的精确恒等式控制。
英文摘要：
Random circuit sampling experiments, the leading demonstrations of quantum computational advantage, are tested with the linear cross-entropy benchmark (XEB), which scores outcomes by their ideal probabilities. We study what a high XEB score certifies about an untrusted quantum device. In the quantum query model, we show that the honest sampler's ideal score acts as a Tsirelson bound for efficient devices and certified randomness. First, exceeding the ideal score by a constant requires $Ω(N^{1/3})$ queries for the Haar and Fourier oracle ensembles with $N$ outcomes, and this bound is tight; this proves an earlier conjecture for Haar-random states and extends it to Fourier sampling. Second, for Fourier sampling and for Haar-random states accessed through the canonical state-preparation oracle, the $n$-bit output of a device that makes polynomially many queries and scores within $o(1)$ of the ideal score has $n-O(\log n)$ bits of smooth min-entropy, even against an adversary who is entangled with the device and later learns the oracle; this is nearly optimal. As a corollary, sampling the collision distribution, which is proportional to the square of the ideal distribution, has query complexity $Θ(N^{1/3})$, and a coherent version of the same algorithm approximately prepares quantum Hadamard products of states with uniform amplitude magnitudes. Both lower bounds rest on progress measures that bound the excess over the ideal score and increase only slightly with each query. For Fourier sampling the measure is the norm of the coherent amplitudes for removing pairs in a purification of the random function, combining the polynomial method with the compressed oracle technique; for Haar-random states it is the square root of the average level in an expansion in Dirichlet orthogonal polynomials, controlled by an exact identity for the effect of one observation.
