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 $$\ell ^\infty $$ ℓ ∞ -category initial state so that the internal graph receives time independent input from the tails, say $$\varvec{\alpha }_{in}$$ α in , at every time step. We show that the response of the Szegedy walk to the input, which is the output, say $$\varvec{\beta }_{out}$$ β 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 $$\varvec{\beta }_{out}=\mathrm {Sz}(\textit{\textbf{m}}_{\delta E})\varvec{\alpha }_{in}$$ β out = Sz ( m δ E ) α in , where the unitary matrix $$\mathrm {Sz}(\textit{\textbf{m}}_{\delta E})$$ Sz ( m δ E ) is the reflection matrix to the unit vector $$\textit{\textbf{m}}_{\delta E}$$ m δ E which is determined by the boundary of the internal graph $$\delta E$$ δ E . 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, $$\varvec{\beta }_{out}=-\varvec{\alpha }_{in}$$ β out = - α in , and the stationary state is similar to the current flow but satisfies a different type of the Kirchhoff laws.

中文翻译：

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量子行走诱导的电路

我们考虑在图上添加无限长度尾部到有限内部图的 Szegedy 游走。我们假设在这些尾巴上，动力学是由自由量子行走给出的。我们设置 $$\ell ^\infty $$ ℓ ∞ -category 初始状态，以便内部图从尾部接收与时间无关的输入，例如 $$\varvec{\alpha }_{in}$$ α in , at每一步。我们证明了 Szegedy walk 对输入的响应，也就是输出，比如 $$\varvec{\beta }_{out}$$ β out ，从内部图到长时间限制的尾部，是根据基础随机游走的可逆性发生了巨大变化。如果底层随机游走是可逆的，我们有 $$\varvec{\beta }_{out}=\mathrm {Sz}(\textit{\textbf{m}}_{\delta E})\varvec{\alpha }_{in}$$ β out = Sz ( m δ E ) α in , 其中酉矩阵 $$\mathrm {Sz}(\textit{\textbf{m}}_{\delta E})$$ Sz ( m δ E ) 是单位向量 $$\textit{\ textbf{m}}_{\delta E}$$ m δ E 由内部图 $$\delta E$$ δ E 的边界确定。然后全局动力学使得内部图被视为一个顶点，在很长的时间内恢复了Szegedy步行的局部动力学。此外，如果 Szegedy 游走的基础随机游走是可逆的，那么我们得到稳态由可逆测度和由内部图和随机游走的可逆测度确定的电路上的电流的线性组合表示。另一方面，如果底层的随机游走是不可逆的，那么酉矩阵只是一个相位翻转；那是，