发表机构
International School for Advanced Studies (SISSA); INFN Sezione di Trieste（高级研究国际学院; 意大利国家核物理研究所的里雅斯特分部）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究通过洗牌问题研究排列群上的量子随机游走，证明量子相干性可加速经典混合过程，并给出加速判据及混合时间比值的标度律。
AI 中文摘要
有序性以多快的速度让位于随机性，量子相干性能否加速这一过程？我们通过洗牌这一范式问题来探讨这些问题，将其表述为对称群 $S_n$ 上的随机游走。我们首先将Diaconis和Shahshahani研究的随机转置游走，以及由$S_n$的共轭类生成的更一般的游走，重新表述为连续时间形式。然后，我们将每个经典游走的转移矩阵与一个置换哈密顿量等同起来，该哈密顿量生成相应的酉量子游走。然而，纯粹的酉演化通常不会在经典的混合意义上收敛到均匀分布：相干性保留信息而非抹除信息。因此，我们将问题嵌入到量子随机游走中，其中相干动力学与负责经典混合的耗散过程相互竞争。在这种背景下，量子相干性有助于随机化。我们证明，它只能减小计算基下与均匀分布的距离，从而能够加速混合。对最慢模式的分析给出了产生显著加速所需的耦合强度的判据。最后，数值结果揭示了量子与经典混合时间之比在一维参数形式下的标度坍缩。我们的结果阐明了在排列群上的游走中，相干性和耗散如何在随机性的涌现中协同作用。
英文摘要
How rapidly does order give way to randomness, and can quantum coherence accelerate this process? We address these questions through the paradigmatic problem of card shuffling, formulated as a random walk on the symmetric group $S_n$. We first recast the random-transposition walk studied by Diaconis and Shahshahani, as well as more general walks generated by conjugacy classes of $S_n$, in continuous time. We then identify the transition matrix of each classical walk with a permutation Hamiltonian generating a corresponding unitary quantum walk. Purely unitary evolution, however, does not generically converge to the uniform distribution in the classical sense of mixing: coherence preserves information rather than erasing it. We therefore embed the problem into a quantum stochastic walk, where coherent dynamics competes with the dissipative process responsible for classical mixing. In this setting, quantum coherence assists randomization. We prove that it can only decrease the distance from the uniform distribution in the computational basis and can therefore accelerate mixing. An analysis of the slowest mode yields a criterion for the coupling strength required to produce an appreciable speedup. Finally, numerical results reveal a scaling collapse of the ratio between quantum and classical mixing times onto a simple one-parameter form. Our results illustrate how coherence and dissipation can cooperate in the emergence of randomness in walks on permutation groups.
Commentsv2: minor corrections. 26+7 pages; 9+1 figures
