Nearly tight Trotterization of interacting electrons (QIP 2021)

Abstract

We consider simulating quantum systems on digital quantum computers. We show that the performance of quantum simulation can be improved by simultaneously exploiting the commutativity of Hamiltonian, the sparsity of interactions, and the prior knowledge of initial state. We achieve this using Trotterization for a class of interacting electrons that encompasses various physical systems, including the plane-wave-basis electronic structure and the Fermi-Hubbard model. We estimate the simulation error by taking the transition amplitude of nested commutators of Hamiltonian terms within the $\eta$-electron manifold. We develop multiple techniques for bounding the transition amplitude and expectation of general fermionic operators, which may be of independent interest. We show that it suffices to use $\mathcal{O}\left(\frac{n^{5/3}}{\eta^{2/3}}+n^{4/3}\eta^{2/3}\right)$ gates to simulate electronic structure in the plane-wave basis with $n$ spin orbitals and $\eta$ electrons up to a negligible factor, improving the best previous result in second quantization while outperforming the first-quantized simulation when $n=\mathcal{O}\left(\eta^2\right)$. We also obtain an improvement for simulating the Fermi-Hubbard model. We construct concrete examples for which our bounds are almost saturated, giving a nearly tight Trotterization of interacting electrons.