Continuous Time Limit of the DTQW in 2D+1 and Plasticity

TL;DR

This paper extends the concept of plastic quantum walks to two dimensions, providing conditions under which discrete-time quantum walks can admit continuous limits and deriving the associated Hamiltonians.

Contribution

It introduces necessary and sufficient conditions for 2D quantum walks to admit plasticity, generalizing previous 1D results and considering broad classes of coin operators dependent on lattice step size.

Findings

Derived conditions for 2D quantum walks to admit continuous limits.

Identified the form of Hamiltonians associated with these walks.

Generalized the framework to include broad classes of coin operators.

Abstract

A Plastic Quantum Walk admits both continuous time and continuous spacetime. The model has been recently proposed by one of the authors in \cite{molfetta2019quantum}, leading to a general quantum simulation scheme for simulating fermions in the relativistic and non relativistic regimes. The extension to two physical dimensions is still missing and here, as a novel result, we demonstrate necessary and sufficient conditions concerning which discrete time quantum walks can admit plasticity, showing the resulting Hamiltonians. We consider coin operators as general $4$ parameter unitary matrices, with parameters which are function of the lattice step size $\varepsilon$. This dependence on $\varepsilon$ encapsulates all functions of $\varepsilon$ for which a Taylor series expansion in $\varepsilon$ is well defined, making our results very general.