Models of quantum walks which admit continuous time and continuous spacetime limits have recently led to quantum simulation schemes for simulating fermions in relativistic and nonrelativistic regimes (Molfetta GD, Arrighi P. A quantum walk with both a continuous-time and a continuous-spacetime limit, 2019). This work continues the study of relationships between discrete time quantum walks (DTQW) and their ostensive continuum counterparts by developing a more general framework than was done in Molfetta and Arrighi (A quantum walk with both a continuous-time and a continuous-spacetime limit, 2019) to evaluate the continuous time limit of these discrete quantum systems. Under this framework, we prove two constructive theorems concerning which internal discrete transitions (“coins”) admit nontrivial continuum limits. We additionally prove that the continuous space limit of the continuous time limit of the DTQW can only yield massless states which obey the Dirac equation. Finally, we demonstrate that for general coins the continuous time limit of the DTQW can be identified with the canonical continuous time quantum walk when the coin is allowed to transition through the continuous limit process.

允许连续时间和连续时空限制的量子行走模型最近导致了用于在相对论和非相对论状态下模拟费米子的量子模拟方案（Molfetta GD，ArrighiP。具有连续时间和连续时空限制的量子行走， 2019）。这项工作通过开发比Molfetta和Arrighi（具有连续时间和连续时空限制的量子行走，更复杂的框架）的通用框架，继续研究离散时间量子行走（DTQW）与表面上连续的量子行走之间的关系。 2019）评估这些离散量子系统的连续时间限制。在此框架下，我们证明了两个建设性定理，涉及哪些内部离散转变（“硬币”）允许非平凡的连续极限。我们还证明了DTQW的连续时间限制的连续空间限制只能产生服从Dirac方程的无质量状态。最后，我们证明了对于普通硬币，当允许硬币通过连续限制过程过渡时，可以使用规范的连续时间量子游动来确定DTQW的连续时间限制。