Continuum Limits of Lazy Open Quantum Walks

TL;DR

This work derives the explicit continuum spacetime master equation for a lazy one-dimensional three-state open quantum walk by embedding the three internal states in an SU(3) framework and using a Lindblad description of decoherence. The unitary backbone is shown to be a Dirac-type SU(3) Hamiltonian, with an effective mass set by the coin parameter θ, yielding ballistic advection and internal state mixing. They analyze two decoherence channels—coin and spatial—and show they imprint qualitatively different macroscopic dynamics: coin dephasing preserves spatial coherence and advection while attenuating internal coherences, whereas spatial dephasing suppresses long-range spatial interference and drives the system toward classical diffusion. The resulting continuum PDEs provide a principled bridge from discrete lazy quantum walks to multichannel transport models and lay groundwork for hydrodynamic or quantum-inspired kinetic extensions.

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.

Figures (2)

• Figure 1: Probability distribution of the three–state lazy DTQW and the standard two–state DTQW after $t=100$ steps. The standard walk uses an initial state $\ket{0}\otimes\frac{1}{\sqrt{2}}(\ket{L}+i\ket{R})$. The initial state of the lazy walk is an equal magnitude Fourier–symmetric superposition, $|\psi_0\rangle = |0\rangle \otimes \frac{1}{\sqrt{3}}\bigl(|L\rangle + \omega\,|S\rangle + \omega^{2}\,|R\rangle\bigr),$ with $\omega = e^{2\pi i/3}$. The lazy walk exhibits ballistic side peaks together with a pronounced central rest state peak, illustrating the three channel structure of the dynamics.
• Figure 2: Effect of dephasing on the lazy DTQW for different coin angles $\theta$ and decoherence strengths $\gamma$. Panels (a,c) show coin dephasing and panels (b,d) show spatial dephasing, for $\theta=\pi/2$ (top row) and $\theta=\pi$ (bottom row). The initial state is a localised position state with a Fourier symmetric coin superposition, $|\psi_0\rangle = |0\rangle \otimes \frac{1}{\sqrt{3}}\bigl(|L\rangle + \omega\,|S\rangle + \omega^{2}\,|R\rangle\bigr)$, where $\omega = e^{2\pi i/3}$.