Hamiltonian Simulation in the Interaction Picture Using the Magnus Expansion

TL;DR

This paper introduces an efficient quantum simulation algorithm for local Hamiltonians perturbed by small local terms, leveraging the Magnus expansion in the interaction picture to improve performance and reduce resource requirements.

Contribution

The paper presents a novel algorithm that uses the Magnus expansion in the interaction frame, achieving optimal scaling and avoiding ancillary qubits for local Hamiltonian simulation.

Findings

Achieves optimal scaling in certain regimes

Outperforms existing algorithms in efficiency

Provides a framework for error analysis of Magnus truncation

Abstract

We propose an algorithm for simulating the dynamics of a geometrically local Hamiltonian $A$ under a small geometrically local perturbation $\alpha B$. In certain regimes, the algorithm achieves the optimal scaling and outperforms the state-of-the-art algorithms. By moving into the interaction frame of $A$ and classically computing the Magnus expansion of the interaction-picture Hamiltonian, our algorithm bypasses the need for ancillary qubits. In analyzing its performance, we develop a framework to capture the quasi-locality of the Magnus operators, leading to a tightened bound for the error of the Magnus truncation. The Lieb-Robinson bound also guarantees the efficiency of computing the Magnus operators and of their subsequent decomposition into elementary quantum gates. These features make our algorithm appealing for near-term and early-fault-tolerant simulations.