论文标题
量子步行引起的电路
Electric circuit induced by quantum walk
论文作者
论文摘要
我们认为在图上的Szegedy步行将无限长度的尾部添加到有限的内部图中。我们假设在这些尾巴上，动力学是由自由量子步道给出的。我们设置$ \ ell^\ infty $ - 类别初始状态，以便内部图从尾部接收时间独立输入，例如$ \boldsymbolα__{in} $，在每个时间步骤。我们表明，szegedy步行对输入的响应是输出，例如$ \boldsymbolβ_{out} $，从内部图到尾部长期限制，这是根据基础随机步行的可逆性而发生的巨大变化。如果基础随机步行是可逆的，则我们有$ \boldsymbolβ_{out} = \ mathrm {sz}（\ boldsymbol {m} _ {δe}）\boldsymbolα__{in} $ $ \ mathrm {sz}（\ boldsymbol {m} _ {δe}）$是单位矢量$ \ boldsymbol {m} _ {Δee} $的反射矩阵，由内部图$δe$ $。然后，全局动力学，以便将内部图视为一个顶点在长期限制中恢复Szegedy Walk的局部动力学。此外，如果s Zegedy Walk的基础随机步行是可逆的，那么我们得到固定状态是通过可逆度量的线性组合和由内部图确定的电路和随机行走的可逆度量来表达的。另一方面，如果基础随机行走不是可逆的，则单位矩阵只是一个相位翻转。也就是说，$ \boldsymbolβ_{out} = - \boldsymbolα_{in} $，固定状态与当前流量相似，但满足了Kirchhoff法律的不同类型。
We consider the Szegedy walk on graphs adding infinite length tails to a finite internal graph. We assume that on these tails, the dynamics is given by the free quantum walk. We set the $\ell^\infty$-category initial state so that the internal graph receives time independent input from the tails, say $\boldsymbolα_{in}$, at every time step. We show that the response of the Szegedy walk to the input, which is the output, say $\boldsymbolβ_{out}$, from the internal graph to the tails in the long time limit, is drastically changed depending on the reversibility of the underlying random walk. If the underlying random walk is reversible, we have $\boldsymbolβ_{out}=\mathrm{Sz}(\boldsymbol{m}_{δE})\boldsymbolα_{in}$, where the unitary matrix $\mathrm{Sz}(\boldsymbol{m}_{δE})$ is the reflection matrix to the unit vector $\boldsymbol{m}_{δE}$ which is determined by the boundary of the internal graph $δE$. Then the global dynamics so that the internal graph is regarded as one vertex recovers the local dynamics of the Szegedy walk in the long time limit. Moreover if the underlying random walk of the Szegedy walk is reversible, then we obtain that the stationary state is expressed by a linear combination of the reversible measure and the electric current on the electric circuit determined by the internal graph and the random walk's reversible measure. On the other hand, if the underlying random walk is not reversible, then the unitary matrix is just a phase flip; that is, $\boldsymbolβ_{out}=-\boldsymbolα_{in}$, and the stationary state is similar to the current flow but satisfies a different type of the Kirchhoff laws.
