EPJ Quantum Technol. (2015) 2: 12
https://doi.org/10.1140/epjqt/s40507-015-0025-1

Research

Quantum lattice Boltzmann is a quantum walk

¹ IAC-CNR, Istituto per le Applicazioni del Calcolo ‘Mauro Picone’, Via dei Taurini, 19, Rome, 00185, Italy
² INRS-Énergie, Matériaux et Télécommunications, Université du Québec, 1650, boulevard Lionel-Boulet, Varennes, J3X 1S2, Canada
³ Numidia s.r.l., via G. Peroni, 130, Rome, 00131, Italy

^* e-mail: succi@iac.rm.cnr.it

Received: 1 August 2014
Accepted: 30 April 2015
Published online: 16 May 2015

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.