Simulating Quantum Dynamics On A Quantum Computer

TL;DR

This paper introduces efficient quantum algorithms for simulating time-dependent Hamiltonian dynamics, achieving time complexities comparable to the best methods for time-independent cases under certain smoothness conditions.

Contribution

It presents the first quantum algorithms with optimal time complexity for simulating time-dependent Hamiltonians and provides new error bounds for Lie-Trotter-Suzuki approximations.

Findings

Algorithms with constant and adaptive time steps are developed.

Cost analysis includes discretization errors in time and Hamiltonian representation.

New upper bounds for Lie-Trotter-Suzuki approximation errors are established.

Abstract

We present efficient quantum algorithms for simulating time-dependent Hamiltonian evolution of general input states using an oracular model of a quantum computer. Our algorithms use either constant or adaptively chosen time steps and are significant because they are the first to have time-complexities that are comparable to the best known methods for simulating time-independent Hamiltonian evolution, given appropriate smoothness criteria on the Hamiltonian are satisfied. We provide a thorough cost analysis of these algorithms that considers discretizion errors in both the time and the representation of the Hamiltonian. In addition, we provide the first upper bounds for the error in Lie-Trotter-Suzuki approximations to unitary evolution operators, that use adaptively chosen time steps.