Continuum Limits of Lazy Open Quantum Walks

TL;DR

This paper derives explicit continuous spacetime equations for a three-state lazy open quantum walk, incorporating decoherence effects, and reveals how internal symmetry and noise influence large-scale quantum transport.

Contribution

It provides the first explicit continuum limit formulation for lazy three-state quantum walks with noise, using SU(3) and Lindblad formalisms.

Findings

Unitary limit described by a Dirac-type SU(3) Hamiltonian

Coin dephasing dampens internal coherences while preserving transport

Spatial dephasing suppresses interference and leads to classical behavior

Abstract

We derive the continuous spacetime limit of the one dimensional lazy discrete time quantum walk, obtaining explicit macroscopic evolution equations for a three state model in the presence of decoherence. While continuum limits of two state quantum walks are well established, an explicit continuous spacetime formulation for the lazy three state walk, particularly including noise, has not previously been constructed. Using an SU(3) representation of a Grover type coin together with a Lindblad formulation of decoherence acting either on the coin or the spatial subspace, we systematically expand the discrete dynamics in both space and time to obtain continuum master equations governing the coarse grained evolution. The resulting generators yield a genuine partial differential equation description of the walk, going beyond purely probabilistic or spectral correspondences. We show that the unitary limit is governed by a Dirac-type SU(3) Hamiltonian describing ballistic advection of left and right moving modes coupled by local symmetric mixing, with the rest state acting as an additional internal degree of freedom. Coin dephasing selectively damps internal coherences while preserving coherent spatial transport, whereas spatial dephasing suppresses long range spatial interference and rapidly drives the dynamics toward classical behaviour. This continuum framework clarifies how internal symmetry, rest state coupling, and distinct decoherence channels shape large scale transport in lazy open quantum walks, and provides a foundation for future extensions toward multichannel quantum transport models and quantum-inspired algorithms.