Quantum Lattice Boltzmann is a quantum walk

TL;DR

This paper demonstrates that quantum lattice Boltzmann methods for the Dirac equation are equivalent to quantum walks, establishing a link that enables quantum simulation of relativistic quantum dynamics.

Contribution

It shows that QLB schemes for the Dirac equation are a form of quantum walks, connecting numerical methods, quantum computation, and relativistic quantum physics.

Findings

QLB schemes are equivalent to quantum walks.

The quantum walk structure is preserved in curved space extensions.

This link enables potential quantum computer simulations of relativistic quantum dynamics.

Abstract

Numerical methods for the 1-D Dirac equation based on operator splitting and on the quantum lattice Boltzmann (QLB) schemes are reviewed. It is shown that these discretizations fall within the class of quantum walks, i.e. discrete maps for complex fields, whose continuum limit delivers Dirac-like relativistic quantum wave equations. The correspondence between the quantum walk dynamics and these numerical schemes is given explicitly, allowing a connection between quantum computations, numerical analysis and lattice Boltzmann methods. The QLB method is then extended to the Dirac equation in curved spaces and it is demonstrated that the quantum walk structure is preserved. Finally, it is argued that the existence of this link between the discretized Dirac equation and quantum walks may be employed to simulate relativistic quantum dynamics on quantum computers.