Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Reducing Circuit Depth in Lindblad Simulation via Step-Size Extrapolation

Pegah Mohammadipour, Xiantao Li

TL;DR

The paper addresses the challenge of simulating open quantum systems governed by the Lindblad equation on quantum hardware with limited circuit depth. It shows that Richardson-type extrapolation over a sequence of step sizes can suppress leading discretization errors for two first-order Lindblad primitives (Kraus-form and dilated-Hamiltonian) by exploiting a backward-error expansion, yielding Gevrey-smooth observable maps. The main results prove that the required circuit depth drops from $\mathcal{O}((lT)^2/\varepsilon)$ to $\mathcal{O}((lT)^2 \log l \log^2(1/\varepsilon))$ while maintaining the standard $1/\varepsilon^2$ sampling cost, i.e., an exponential improvement in $1/\varepsilon$. This extends extrapolation-based error mitigation from Hamiltonian to Lindblad dynamics, enabling more feasible open-system simulations on NISQ devices with provable bias–variance control and practical numerical validation.

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 $n=Ω(\log(1/\varepsilon))$-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.

Reducing Circuit Depth in Lindblad Simulation via Step-Size Extrapolation

TL;DR

The paper addresses the challenge of simulating open quantum systems governed by the Lindblad equation on quantum hardware with limited circuit depth. It shows that Richardson-type extrapolation over a sequence of step sizes can suppress leading discretization errors for two first-order Lindblad primitives (Kraus-form and dilated-Hamiltonian) by exploiting a backward-error expansion, yielding Gevrey-smooth observable maps. The main results prove that the required circuit depth drops from to while maintaining the standard sampling cost, i.e., an exponential improvement in . This extends extrapolation-based error mitigation from Hamiltonian to Lindblad dynamics, enabling more feasible open-system simulations on NISQ devices with provable bias–variance control and practical numerical validation.

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 , our main theorem shows that an -point extrapolator reduces the maximum circuit depth needed for accuracy from polynomial to polylogarithmic scaling, an exponential improvement in~, while keeping sampling complexity to the standard level, thus extending such results for Hamiltonian simulations to Lindblad simulations. Several numerical experiments illustrate the practical viability of the method.

Paper Structure

This paper contains 20 sections, 18 theorems, 137 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Related work.
  4. Outline and Notation.
  5. Background
  6. First-order Approximation by a Kraus Form
  7. First-order Approximation by Hamiltonian Dilation
  8. Numerical Experiments
  9. Kraus Form approximation
  10. Dilated Hamiltonian Approximations
  11. Summary and further discussions
  12. Extensions.
  13. Outlook.
  14. Supplemental Material
  15. Detailed derivation of the error expansion
  16. ...and 5 more sections

Key Result

Lemma 1

Assume that the approximation method has an expansion in eq:kappa_expansion. Let $\rho_\tau (t)$ denote the corresponding approximate solution at time $t$ with stepsize $\tau$. There exists a sequence of smooth functions $\Gamma_k(t)$ such that where $\rho(t)$ is the solution to the exact evolution eq:lindblad. For a discrete solution eq:kappa_expansion, these coefficient matrices satisfy the ini

Figures (6)

  • Figure 1: Richardson extrapolation of $\langle O \rangle(\tau)$ using first-order Kraus evolution. The curves represent a degree-8 polynomial interpolating noisy data at $n=9$ points and then extrapolating to $\tau = 0$. $N_\mathrm{shots}=2 \times 10^7$. Left: Equidistant time steps. Right: perturbed Chebyshev time steps.
  • Figure 2: Least-square based extrapolation of $\langle O \rangle(\tau)$ with data generated by first-order Kraus-form approximation. $N_\mathrm{shots}=2 \times 10^7$. Left: Equidistant time steps. Right: Chebyshev time steps. The continuous curves represent degree-7 polynomials fit to $9$ noisy data points, then extrapolated to $\tau = 0$.
  • Figure 3: Richardson extrapolation of $\langle M_x\rangle$ via Kraus-form approximation. $N_\mathrm{shots}=2 \times 10^3$. Left: Equidistant time steps. Right: Chebyshev time steps. Both are computed using a degree‑8 polynomial interpolating 9 noisy data points. Chebyshev nodes reduce bias and variance, enhancing agreement with the true expectation value.
  • Figure 4: Least-square based extrapolation of $\langle M_x\rangle$ via Kraus-form approximation. $N_\mathrm{shots}=2 \times 10^3$. Left: Equidistant time steps. Right: Chebyshev time steps. A degree‑7 polynomial fit is used on 9 noisy data points and is shown as a curve in each plot.
  • Figure 5: Richardson extrapolation of $\langle M_x\rangle$ via dilated-Hamiltonian Lindblad evolution. $N_\mathrm{shots}=2 \times 10^3$. Left: Equidistant time steps. Right: Chebyshev time steps. Both are computed using a degree‑8 polynomial interpolating 9 noisy data points. Chebyshev grids reduce bias and variance, enhancing agreement with the true expectation value.
  • ...and 1 more figures

Theorems & Definitions (31)

  • Lemma 1
  • Definition 2: Generating sequence for Kraus Form
  • Lemma 3
  • Lemma 4
  • Corollary 5
  • Theorem 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Theorem 10
  • ...and 21 more