量子随机游走

符号定义

考虑随机矩阵\(P \in \mathbb R^{N \times N}\),这里规定列和为\(1\).故对于任意状态\(x \in \mathbb{R}^n\), 游走一步后为

\[ x^\prime = Px. \]

在量子离散随机游走当中,我们可以用一个\(\mathbb{C}^{N}\otimes \mathbb{C}^N\)中的酉变换描述.引入状态

\[ \ket{\psi_j} = \ket{j} \otimes \sum_{k=1}^{N} \sqrt{p_{kj}}\ket{k} = \sum_{k=1}^{N} \sqrt{p_{kj}}\ket{j,k} \]

由随机矩阵性质可知\(\ket{\psi_{j}}\)是归一化的.令

\[ \Pi = \sum_{j=1}^{N} \ket{\psi_{j}}\bra{\psi_{j}} \]

为空间\(\text{span}\{\ket{\psi_j}\}\)上的投影,并定义交换算子

\[ S = \sum_{j, k = 1}^N \ket{j, k}\bra{k, j}. \]

上述矩阵的性质:\(S^2 = I, \Pi^2 = \Pi, S = S^\dagger, \Pi = \Pi^\dagger\).

我们将量子游走的一步定义为\(U = S(2\Pi -I)\).

首先不难看出\(U\)是酉变换.因为

\[\begin{aligned} UU^\dagger &= S(2\Pi -I)[S(2\Pi -I)]^\dagger \\ &= S[(2\Pi -I)]^2S \\ &= S^2 = I \end{aligned} \]

考虑一下游走两步的效果,即

\[\begin{aligned} U^2 &= S(2\Pi-I)S\cdot(2\Pi-I) \\ & = [2S\Pi(S\Pi)^\dagger-I](2\Pi-I) \end{aligned} \]

这里可以视为先进行\(\text{span}\{\ket{\psi_j}\}\)上的投影, 后进行\(\text{span}\{S\ket{\psi_j}\}\)上的投影.这是因为

\[S\Pi(S\Pi)^\dagger = \sum_{j=1}^{N} S\ket{\psi_{j}}\bra{\psi_{j}} \sum_{l=1}^{N} \ket{\psi_{l}}\bra{\psi_{l}}S = \sum_{j=1}^{N} S\ket{\psi_{j}}\bra{\psi_{j}}S^\dagger \]