并行 Kac 行走的快速混合：从球面到 Stiefel 流形

中文摘要 AI 辅助

Kac 行走是一种经典的局部随机行走，其在维度 $d$ 中对单个实单位向量的作用在 $\Theta(d\log d)$ 顺序步骤内以全变差距离混合~\cite{PS17}。Lu、Qin、Song、Yao 和 Zhao 引入了 Kac 行走的并行版本，该版本在 $O(\log d)$ 轮内混合单个量子态~\cite{LQSY+26}。在将随机性离散化并替换为合适的伪随机原语后，这种并行行走产生了伪随机态扰码器，并随后被证明能产生伪随机酉算子~\cite{LQSY+25}。我们研究并行 Kac 行走同时作用于 $k$ 个正交量子态时会发生什么。我们证明，对于任意 $1\leq k < d$，在 $O\\!\left((k+\log d)\log(d/\varepsilon)\right)$ 步之后，$k$ 个输出态的联合分布在 Wasserstein 距离和全变差距离下都 $\varepsilon$-接近通过将相同的 Haar 随机酉算子应用于相同输入所得到的分布。这推广了并行 Kac 行走从单个量子态到多个正交量子态的分散性质。等价地，将 $k$ 个有序正交态视为复 Stiefel 流形 $V_{d,k}=\{X\in\mathbb C^{d\times k}:X^\dagger X=I_k\}$ 上的一个点，我们证明并行 Kac 行走在 $V_{d,k}$ 上快速混合，其 Wasserstein 和全变差混合时间都以 $O\\!\left((k+\log d)\log(d/\varepsilon)\right)$ 为界。这扩展了 Pillai、Smith 和 Vaikuntanathan 关于实 Stiefel 流形上标准 Kac 行走的 Wasserstein 混合结果~\cite{PSV26}。

英文摘要

Kac's walk is a classical local random walk whose action on a single real unit vector in dimension $d$ mixes in total variation in $Θ(d\log d)$ sequential steps~\cite{PS17}. Lu, Qin, Song, Yao, and Zhao introduced a parallel version of Kac's walk that mixes a single quantum state in $O(\log d)$ rounds~\cite{LQSY+26}. After discretizing the randomness and replacing it by suitable pseudorandom primitives, this parallel walk gives rise to pseudorandom state scramblers, and was subsequently shown to yield pseudorandom unitaries~\cite{LQSY+25}. We study what happens when the parallel Kac's walk acts simultaneously on $k$ orthonormal quantum states. We prove that, for any $1\leq k < d$, after $O\!\left((k+\log d)\log(d/\varepsilon)\right)$ steps, the joint distribution of the $k$ output states is $\varepsilon$-close, in both Wasserstein and total variation distance, to that obtained by applying a common Haar-random unitary to the same inputs. This generalizes the dispersing property of the parallel Kac's walk from a single quantum state to multiple orthonormal quantum states. Equivalently, viewing an ordered collection of $k$ orthonormal states as a point on the complex Stiefel manifold $V_{d,k}=\{X\in\mathbb C^{d\times k}:X^\dagger X=I_k\}$, we show that the parallel Kac's walk mixes rapidly on $V_{d,k}$, with both Wasserstein and total variation mixing times bounded by $O\!\left((k+\log d)\log(d/\varepsilon)\right)$. This extends the Wasserstein mixing result of Pillai, Smith, and Vaikuntanathan for the standard Kac's walk on real Stiefel manifolds~\cite{PSV26}.

发表机构

• The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））
• National University of Singapore（新加坡国立大学）
• Portland State University（波特兰州立大学）
• Nanjing University（南京大学）
• Hefei National Laboratory（合肥国家实验室）

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