Faster Algorithmic Quantum and Classical Simulations by Corrected Product Formulas

TL;DR

This paper introduces corrected product formulas (CPFs) that enhance the accuracy of Hamiltonian simulations on quantum computers by injecting correctors, especially effective for lattice and perturbed systems, with minimal additional cost.

Contribution

The paper proposes corrected product formulas that significantly improve simulation accuracy for certain Hamiltonians while maintaining low additional computational cost.

Findings

CPFs outperform standard product formulas in accuracy for lattice Hamiltonians.

Numerical simulations confirm theoretical error bounds of CPFs.

CPFs are effective on actual quantum hardware and simulators.

Abstract

Hamiltonian simulation using product formulas is arguably the most straightforward and practical approach for algorithmic simulation of a quantum system's dynamics on a quantum computer. Here we present corrected product formulas (CPFs), a variation of product formulas achieved by injecting auxiliary terms called correctors into standard product formulas. We establish several correctors that greatly improve the accuracy of standard product formulas for simulating Hamiltonians comprised of two partitions that can be exactly simulated, a common feature of lattice Hamiltonians, while only adding a small additive or multiplicative factor to the simulation cost. We show that correctors are particularly advantageous for perturbed systems, where one partition has a relatively small norm compared to the other, as they allow the small norm to be utilized as an additional parameter for controlling the simulation error. We demonstrate the performance of CPFs by numerical simulations for several lattice Hamiltonians. Numerical results show our theoretical error bound for CPFs matches or exceeds the empirical error of standard product formulas for these systems. We also demonstrate improvements offered by CPFs by implementation on actual quantum hardware as well as noisy and noiseless quantum simulators. CPFs could be a valuable algorithmic tool for early fault-tolerant quantum computers with limited computing resources. As for standard product formulas, CPFs could also be used for simulations on a classical computer.