Connection between the quantum-walk kernel and a physical model or Hamiltonian

Determine whether the kernel function $K_{\Delta t}$ arising from the quantum-walk evolution is connected with a physical model or with a Hamiltonian, in order to clarify the relationship between the finite-range quantum-walk formulation and Feynman’s path-integral picture.

Background

The paper compares the discrete-time quantum-walk evolution with Feynman’s path-integral formulation. After grouping multiple quantum-walk time steps into an interval of length $\Delta t$, the authors express the resulting evolution using a kernel function $K_{\Delta t}(x,y)$ generated by the quantum-walk operator.

The authors suggest that taking $\Delta t=\infty$ could correspond to Feynman’s path-integral picture, but they explicitly leave unresolved whether the kernel $K_{\Delta t}$ has a connection to an underlying physical model or Hamiltonian. Establishing such a connection would help determine whether the quantum walk provides not only a mathematical finite-range analogue but also a physically grounded realization of the path-integral framework.