A quantum lattice algorithm with fourth-order accuracy for the one-dimensional Dirac equation

The discrete time quantum walk is a quantum cellular automaton whose wavefunction comprises pairs of complex numbers assigned to uniformly spaced points on a line. The wavefunction evolves through the application of an alternating sequence of unitary operators: streaming of wavefunction values to adjacent points, and a Hadamard-type unitary matrix to blend pairs of values at individual points. Each operator generates the exact evolution due to part of the Hamiltonian for the one-dimensional Dirac equation over a finite time step. Composing these operators thus creates a discrete approximation to the Dirac equation. However, the composition of two non-commuting operators creates a global splitting error proportional to the length of the time step. The global error can be reduced from first order to second order in the time step by a unitary pre- and post-processing of the initial conditions and final output. The algorithm then becomes equivalent to a symmetric composition, a Strang splitting, between the two operators. This paper describes a fourth-order accurate composition scheme using nine stages, the fewest possible when the lengths of the time steps employed in the different stages are constrained to be integer multiples of some base time step. Each stage is itself a symmetric composition between two operators. This fourth-order scheme produces quantitatively smaller errors for a typical benchmark problem on spatial lattices with 1024 or more points, and shows the expected fourth-order convergence on sufficiently fine lattices. It has greater accuracy, over sufficiently long times, than three better-known fourth-order composition schemes using fewer stages, but with lengths related by irrational coefficients. The truncation error for plane-wave solutions is due to an operator that separates into a resonant part proportional to the Hamiltonian, and a non-resonant part orthogonal to the Hamiltonian. The resonant part commutes with the exact evolution operator, so its error accumulates to grow linearly with time. The orthogonal part produces oscillations that remain bounded over many time steps. The nine-stage integer scheme has the smallest resonant truncation error of the four schemes, despite being the only scheme that can be implemented using local operations. The other schemes implement streaming by irrational fractions of the lattice spacing using discrete Fourier transforms.