Paul Dellar

Oxford Centre for Industrial and Applied Mathematics (OCIAM), Oxford University, UK



Quantum lattice algorithms originated with the Feynman checkerboard model for the Dirac equation that describes a spin-1/2 particle moving relativistically in one spatial dimension. Subsequent work has followed three parallel lines of development under the names quantum cellular automata, discrete time quantum walks, and quantum lattice Boltzmann or lattice Dirac algorithms. The first two strands emphasise algorithms for quantum computers, notably O(\sqrt{N}) search algorithms, while the last strand emphasises efficient simulation of quantum phenomena using conventional digital computers. This strand offers unitary and readily parallelisable algorithms that are free of the fermion-doubling problem found in conventional finite difference or finite element discretisations of quantum mechanical equations. In discrete quantum systems the finite-time evolution operator replaces the Hamiltonian as the principle object of interest. Unitary evolution implies that the expectation of this operator is itself conserved, offering an exact invariant of the discrete system consistent with energy conservation in the continuum.

This tutorial will cover all three strands, while devoting most attention to the development of second-order accurate quantum lattice algorithms for simulating the Dirac equation with scalar and vector potentials in multiple dimensions. Such algorithms are finding new applications beyond quantum electrodynamics and laser-plasma interactions. The Dirac equation offers a large-scale description of charge carriers within graphene, a two-dimensional hexagonal lattice of carbon atoms, with a vector potential arising from uneven strain within the lattice. Other experimental replications of the Dirac equation use trapped ions in optical lattices. These analogous systems are bringing previously inaccessible phenomena
from quantum electrodynamics such as Zitterbewegung and the Klein paradox within experimental and technological reach.

Finally, we will draw parallels with hydrodynamic lattice Boltzmann algorithms, notably through quantum analogues of the diffusive and acoustic scalings for hydrodynamics, and a reformulation of the hydrodynamic algorithm as a second-order accurate splitting in time between streaming and collisions.