Hamiltonian Simulation in the Interaction Picture

TL;DR

This paper introduces a low-space overhead quantum simulation algorithm using the truncated Dyson series, achieving exponential and quadratic improvements in gate and query complexities for various Hamiltonian classes.

Contribution

It presents a novel simulation method leveraging the interaction picture, significantly improving complexity bounds for time-dependent and certain time-independent Hamiltonians.

Findings

Exponential improvement in gate complexity for diagonally dominant Hamiltonians

Reduced gate complexity for simulating N-site Hubbard models with long-range interactions

Quadratic improvement in query complexity for sparse time-dependent Hamiltonians

Abstract

We present a low-space overhead simulation algorithm based on the truncated Dyson series for time-dependent quantum dynamics. This algorithm is applied to simulating time-independent Hamiltonians by transitioning to the interaction picture, where some portions are made time-dependent. This can provide a favorable complexity trade-off as the algorithm scales exponentially better with derivatives of the time-dependent component than the original Hamiltonian. We show that this leads to an exponential improvement in gate complexity for simulating some classes of diagonally dominant Hamiltonian. Additionally we show that this can reduce the gate-complexity scaling for simulating $N$-site Hubbard models for time $t$ with arbitrary long-range interactions as well as reduce the cost of quantum chemistry simulations within a similar-sized plane-wave basis to $\widetilde{\mathcal{O}}(N^2t)$ from $\widetilde{\mathcal{O}}(N^{11/3}t)$. We also show a quadratic improvement in query complexity for simulating sparse time-dependent Hamiltonians, which may be of independent interest.