Reducing Circuit Depth in Lindblad Simulation via Step-Size Extrapolation

TL;DR

This paper introduces a step-size extrapolation method to significantly reduce the circuit depth required for accurate Lindblad quantum simulations, achieving exponential improvements in error scaling while maintaining manageable sampling complexity.

Contribution

It extends Richardson extrapolation techniques to Lindblad dynamics, providing a theoretical framework and bounds for reducing circuit depth in open quantum system simulations.

Findings

Exponential reduction in circuit depth for Lindblad simulation accuracy

Bounded the bias and variance in the extrapolation process

Numerical experiments demonstrate practical effectiveness

Abstract

We study algorithmic error mitigation via Richardson-style extrapolation for quantum simulations of open quantum systems modelled by the Lindblad equation. Focusing on two specific first-order quantum algorithms, we perform a backward-error analysis to obtain a step-size expansion of the density operator with explicit coefficient bounds. These bounds supply the necessary smoothness for analyzing Richardson extrapolation, allowing us to bound both the deterministic bias and the shot-noise variance that arise in post-processing. For a Lindblad dynamics with generator bounded by $l$, our main theorem shows that an $n=\Omega (\log(1/\varepsilon))$-point extrapolator reduces the maximum circuit depth needed for accuracy $\varepsilon$ from polynomial $\mathcal{O} ((lT)^{2}/\varepsilon)$ to polylogarithmic $\mathcal{O} ((lT)^{2} \log l \log^2(1/\varepsilon))$ scaling, an exponential improvement in~$1/\varepsilon$, while keeping sampling complexity to the standard $1/\varepsilon^2$ level, thus extending such results for Hamiltonian simulations to Lindblad simulations. Several numerical experiments illustrate the practical viability of the method.