Time-Dependent Hamiltonian Simulation in the Low-Energy Subspace

TL;DR

This paper advances the understanding of simulating time-dependent Hamiltonians in the low-energy subspace, providing improved algorithms and theoretical bounds relevant for quantum physics and computation.

Contribution

It introduces improved Trotter formulas for low-energy subspace simulation and establishes lower bounds on query complexity for such Hamiltonian simulations.

Findings

Enhanced Trotter number bounds for low-energy Hamiltonian simulation

Derived low-energy simulation error with commutator scaling

Proved lower bounds on query complexity for general time-dependent Hamiltonians

Abstract

Hamiltonian simulations are key subroutines in adiabatic quantum computation, quantum control, and quantum many-body physics, where quantum dynamics often happen in the low-energy sector. In contrast to time-independent Hamiltonian simulations, a comprehensive understanding of quantum simulation algorithms for time-dependent Hamiltonians under the low-energy assumption remains limited hitherto. In this paper, we investigate how much we can improve upon the standard performance guarantee assuming the initial state is supported on a low-energy subspace. In particular, we compute the Trotter number of digital quantum simulation based on product formulas for time-dependent spin Hamiltonians under the low-energy assumption that the initial state is supported on a small number of low-energy eigenstates, and show improvements over the standard cost for simulating full unitary simulations. Technically, we derive the low-energy simulation error with commutator scaling for product formulas by leveraging adiabatic perturbation theory to analyze the time-variant energy spectrum of the underlying Hamiltonian. We further discuss the applications to simulations of non-equilibrium quantum many-body dynamics and adiabatic state preparation. Finally, we prove a lower bound of query complexity for generic time-dependent Hamiltonian simulations.