Tighter Error Bounds for the qDRIFT Algorithm

TL;DR

This paper improves the error bounds for the qDRIFT quantum simulation algorithm, reducing the number of steps needed for accurate results across various quantum system types.

Contribution

It refines the existing error bounds for qDRIFT by incorporating Jensen's inequality, leading to more efficient quantum simulation with fewer steps.

Findings

Significantly reduces the number of steps for fixed accuracy

Applicable to both closed and open quantum systems

Demonstrates practical improvements in quantum chemistry and machine learning

Abstract

Randomized algorithms such as qDRIFT provide an efficient framework for quantum simulation by sampling terms from a decomposition of the system's generator. However, existing error bounds for qDRIFT scale quadratically with the norm of the generator, limiting their efficiency for large-scale closed or open quantum system simulation. In this work, we refine the qDRIFT error bound by incorporating Jensen's inequality and a careful treatment of the integral form of the error. This yields an improved scaling that significantly reduces the number of steps required to reach a fixed simulation accuracy. Our result applies to both closed and open quantum systems, and we explicitly recover the improved bound in the Hamiltonian case. To demonstrate the practical impact of this refinement, we apply it to three settings: quantum chemistry simulations, dissipative transverse field Ising models, and Hamiltonian encoding of classical data for quantum machine learning. In each case, our bound leads to a substantial reduction in gate counts, highlighting its broad utility in enhancing randomized simulation techniques.