对数深度费米子采样：反集中性与平均情形困难性

中文摘要 AI 辅助

采样问题给出了量子计算与经典计算之间一些最清晰的条件性分离。在费米子采样中，无源线性光学作用于非高斯魔幻输入。全局Haar随机变换给出反集中性和高精度的平均情形困难性，但需要线性深度和二次数量的门。一个自然的开放问题是线性深度是否必要？我们证明在对数深度下，同一系综中两个保证均成立。对于具有独立Haar双模门的新鲜均匀匹配，碰撞在$\nlog n/\nlog(9/4)\napprox0.855\nlog_2 n$层的尖锐阈值处达到无源Haar尺度，具有显式的极限过渡轮廓和来自双粒子关联的匹配下界。该机制依赖于输入：魔幻输入抑制了最慢弛豫模式的广泛权重，留下下一个模式来设定尺度。基于占据数基的输入即使在完全Haar随机性下也保持大碰撞。在更大的对数深度下，我们证明在实RAM模型中，估计输出概率至加性误差$2^{-O(n\nlog^2 n)}$在超过$3/4$的任何固定比例实例上是平均情形$\\#\mathsf{P}$-困难的。我们在四个原生层中构造困难实例，将其嵌入典型调度中，并沿Cayley路径使用有理线性规划解码器进行插值。一个有限的$192$门字母表精确保持碰撞定律。两个保证均使用任意模式对之间的对数原生费米子深度和$O(n\nlog n)$个门，取代了全局Haar构造的二次预算。所证明的精度比采样到计数所需的$1/N$尺度更精细，其中$N=\binom{n}{n/2}$；在常数全变差距离内的采样困难性仍然开放。

英文摘要

Sampling problems give some of the clearest conditional separations between quantum and classical computation. In Fermion Sampling, passive linear optics acts on a non-Gaussian magic input. Globally Haar-random transformations give anticoncentration and high-precision average-case hardness, but require linear depth and quadratically many gates. A natural open question is whether linear depth is necessary? We show that logarithmic depth suffices for both guarantees in the same ensemble. For fresh uniform matchings with independent Haar two-mode gates, the collision reaches the passive-Haar scale at a sharp threshold of $\log n/\log(9/4)\approx0.855\log_2 n$ layers, with an explicit limiting transition profile and a matching lower bound from two-particle correlations. The mechanism is input-dependent: the magic input suppresses the extensive weight of the slowest relaxation mode, leaving the next to set the scale. An occupation-basis input retains large collision even under full Haar randomness. At a larger logarithmic depth, we prove, in the real-RAM model, average-case $\#\mathsf{P}$-hardness of estimating output probabilities to additive error $2^{-O(n\log^2 n)}$ on any fixed fraction of instances above $3/4$. We construct hard instances in four native layers, embed them in typical schedules, and interpolate along Cayley paths using a rational linear-program decoder. A finite $192$-gate alphabet preserves the collision law exactly. Both guarantees use logarithmic native fermionic depth between arbitrary mode pairs and $O(n\log n)$ gates, replacing the global-Haar construction's quadratic budget. The accuracy proved is finer than the $1/N$ scale, where $N=\binom{n}{n/2}$, required by sampling-to-counting; hardness of sampling within constant total-variation distance remains open.

发表机构

• Quantinuum, Singapore（Quantinuum）
• Quantum Signals, Paris, France（Quantum Signals）
• IRIF, CNRS and Université Paris Cité（IRIF、法国国家科学研究中心和巴黎西岱大学）

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