Quantum lattice algorithms: similarities and connections to some classic finite difference algorithms

Abstract

Quantum lattice algorithms originated with the Feynman checkerboard model for the one-dimensional Dirac equation. They offer discrete models of quantum mechanics in which the complex numbers representing wavefunction values on a discrete spatial lattice evolve through discrete unitary operations. This paper draws together some of the identical, or at least unitarily equivalent, algorithms that have appeared in three largely disconnected strands of research. Treated as conventional numerical algorithms, they are all only first order accurate under refinement of the discrete space/time grid, but may be raised to second order by a unitary change of variables. Much more efficient implementations arise from replacing the evolution through a sequence of unitary intermediate steps with a short path integral formulation that expresses the wavefunction at each spatial point on the most recent time level as a linear combination of values at immediately preceding time levels and neighbouring spatial points. In one dimension, a particularly elegant reformulation replaces two variables at two time levels with a single variable over three time levels. The resulting algorithm is a variational integrator arising from a discrete action principle, and coincides with the Ablowitz–Kruskal–Ladik finite difference scheme for the Klein–Gordon equation.