Quantum algorithms for grid-based variational time evolution

TL;DR

This paper introduces a variational quantum algorithm for simulating quantum dynamics in first quantized grid encodings, reducing circuit depth and measurement complexity, and demonstrating its effectiveness on systems in one and two dimensions.

Contribution

The paper presents a novel variational quantum algorithm tailored for first quantized grid-based quantum dynamics, addressing measurement and Hamiltonian decomposition challenges.

Findings

Efficient measurement in position and momentum bases regardless of system size.

Significant attenuation of numerical instabilities in variational time propagation.

Feasible application to systems in one and two dimensions with manageable additional gates.

Abstract

The simulation of quantum dynamics calls for quantum algorithms working in first quantized grid encodings. Here, we propose a variational quantum algorithm for performing quantum dynamics in first quantization. In addition to the usual reduction in circuit depth conferred by variational approaches, this algorithm also enjoys several advantages compared to previously proposed ones. For instance, variational approaches suffer from the need for a large number of measurements. However, the grid encoding of first quantized Hamiltonians only requires measuring in position and momentum bases, irrespective of the system size. Their combination with variational approaches is therefore particularly attractive. Moreover, heuristic variational forms can be employed to overcome the limitation of the hard decomposition of Trotterized first quantized Hamiltonians into quantum gates. We apply this quantum algorithm to the dynamics of several systems in one and two dimensions. Our simulations exhibit the previously observed numerical instabilities of variational time propagation approaches. We show how they can be significantly attenuated through subspace diagonalization at a cost of an additional $\mathcal{O}(MN^2)$ 2-qubit gates where $M$ is the number of dimensions and $N^M$ is the total number of grid points.