We introduce a minimal set of physically motivated postulates that the Hamiltonian $H$ of a continuous-time quantum walk should satisfy in order to properly represent the quantum counterpart of the classical random walk on a given graph. We found that these conditions are satisfied by infinitely many quantum Hamiltonians, which provide novel degrees of freedom for quantum enhanced protocols, In particular, the on-site energies, i.e., the diagonal elements of $H$, and the phases of the off-diagonal elements are unconstrained on the quantum side. The diagonal elements represent a potential-energy landscape for the quantum walk and may be controlled by the interaction with a classical scalar field, whereas, for regular lattices in generic dimension, the off-diagonal phases of $H$ may be tuned by the interaction with a classical gauge field residing on the edges, e.g., the electromagnetic vector potential for a charged walker.

中文翻译：

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图上随机游走的广义量子经典对应关系

我们引入了一组最小的物理动机假设，即哈密顿量 $H$为了在给定的图上正确表示经典随机游走的量子对应物，连续时间量子游走的 应该满足。我们发现无限多个量子哈密顿量满足这些条件，它们为量子增强协议提供了新的自由度，特别是现场能量，即$H$，并且非对角元素的相位在量子侧不受约束。对角线元素代表量子行走的势能景观，可以通过与经典标量场的相互作用来控制，而对于一般维度的规则晶格，非对角线相$H$可以通过与驻留在边缘上的经典规范场相互作用来调谐，例如带电步行者的电磁矢量势。