A Trotter-Suzuki approximation for Lie groups with applications to Hamiltonian simulation

TL;DR

This paper introduces a new product formula based on Trotter-Suzuki approximation for efficiently simulating Lie group exponentials, especially in quantum systems with large or complex generators.

Contribution

The paper develops a novel product formula that reduces the number of terms needed for Hamiltonian simulation, particularly for systems with large norms or non-quadratic potentials.

Findings

Number of terms can be significantly reduced under certain conditions.

Applicable to continuous-variable and bosonic quantum systems.

Potential for sublinear or subpolynomial complexity in local Hilbert space dimension.

Abstract

We present a product formula to approximate the exponential of a skew-Hermitian operator that is a sum of generators of a Lie algebra. The number of terms in the product depends on the structure factors. When the generators have large norm with respect to the dimension of the Lie algebra, or when the norm of the effective operator resulting from nested commutators is less than the product of the norms, the number of terms in the product is significantly less than that obtained from well-known results. We apply our results to construct product formulas useful for the quantum simulation of some continuous-variable and bosonic physical systems, including systems whose potential is not quadratic. For many of these systems, we show that the number of terms in the product can be sublinear or subpolynomial in the dimension of the relevant local Hilbert spaces, where such a dimension is usually determined by an energy scale of the problem. Our results emphasize the power of quantum computers for the simulation of various quantum systems.