离散时间量子游走与其在图上的内禀随机游走之间的插值游走

中文摘要 AI 辅助

我们考虑在有限连通图上，参数 $p\in [0,1]$ 的离散时间量子游走（$p=0$）与其内禀随机游走（$p=1$）之间的插值。其时间演化由量子游走和随机游走的Kraus（CPTP）映射的凸组合定义。随机游走的Kraus映射通过对量子游走的Kraus映射进行某种投影来表示。插值的时间演化被解释为在同一图上相互关联的$2$个游走者的以下两种动力学的组合：$2$个游走者以概率$1-p$彼此独立移动（无关联），而$2$个游走者以概率$p$在每个时间步总是移动到相同位置（最大关联）。在本文中，我们将随机游走推广到一种新的游走，即关联游走，它保留了内禀性质并放宽了随机游走的最大关联。这种关联由底层图中弧集的一个划分决定。我们证明了关联游走与量子游走之间的插值游走的特征值位于$\{ z\in \mathbb{C}\\;|\\;1-p\leq |z|\leq 1 \}$中，并且吸收态与关联游走的吸收态一致，该吸收态由底层图结构刻画。

英文摘要

We consider an interpolation with the parameter $p\in [0,1]$ between the discrete-time quantum ($p=0$) and its intrinsic random ($p=1$) walks on a finite connected graph. Its time evolution is defined by the convex combination of the Kraus (CPTP) maps of the quantum and random walks. The Kraus map of the random walk is represented by taking a kind of projection of that of the quantum walk. The time evolution of the interpolation is interpreted as the combination of the following two dynamics of $2$ walkers on the same graph correlated with each other: $2$ walkers move independently of each other (no-correlation) with probability $1-p$, while $2$ walkers always move into the same position (the maximal correlation) with probability $p$ at each time step. In this paper, we generalize the random walk to a new walk, namely, the correlated walk, which preserves the intrinsic property and relaxes the maximal correlation of the random walk. This correlation is determined by a partition of the arc sets in the underlying graph. We show that the eigenvalues of the interpolating walk between the correlated and quantum walks live in $\{ z\in \mathbb{C}\;|\;1-p\leq |z|\leq 1 \}$ and that the absorption state coincides with that of the correlated walk, which is characterized by the underlying graph structures.

发表机构

• Gakushuin University（学习院大学）
• Yokohama National University（横滨国立大学）
• The University of Tokyo（东京大学）

机构由 AI 辅助整理，请以论文原文为准。

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32 pages, 5 figures