Hamiltonian simulation in the low-energy subspace

TL;DR

This paper analyzes low-energy Hamiltonian simulation, providing improved error bounds for product formulas and demonstrating exponential decay of leakage to high-energy states, advancing quantum simulation methods.

Contribution

It introduces new error bounds based on low-energy norms and proves exponential decay of leakage, improving simulation efficiency for low-energy states.

Findings

Error bounds depend on a low-energy norm of H

Exponential decay bounds on leakage to high-energy subspaces

Improvements are significant for long evolution times

Abstract

We study the problem of simulating the dynamics of spin systems when the initial state is supported on a subspace of low energy of a Hamiltonian $H$. This is a central problem in physics with vast applications in many-body systems and beyond, where the interesting physics takes place in the low-energy sector. We analyze error bounds induced by product formulas that approximate the evolution operator and show that these bounds depend on an effective low-energy norm of $H$. We find improvements over the best previous complexities of product formulas that apply to the general case, and these improvements are more significant for long evolution times that scale with the system size and/or small approximation errors. To obtain these improvements, we prove exponentially decaying upper bounds on the leakage to high-energy subspaces due to the product formula. Our results provide a path to a systematic study of Hamiltonian simulation at low energies, which will be required to push quantum simulation closer to reality.