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Stability of digital and analog quantum simulations under noise

Jayant Rao, Jens Eisert, Tommaso Guaita

TL;DR

This work analyzes the robustness of analog and digital quantum simulators to perturbations by deriving rigorous worst-case and average-case bounds for the estimation of local observables under perturbations of magnitude $\delta$. It develops a unified framework that spans time-dependent Gaussian and white-noise perturbations, as well as Brownian and Lindbladian limits, and employs Lieb-Robinson bounds and Suzuki–Trotter product formulas to compare stability across paradigms. The results show that worst-case scaling is similar for both analog and digital approaches, while average-case behavior exhibits enhanced cancellation and regime-dependent advantages for digital simulations with suitably tuned parameters. These insights inform practical strategies for noise-aware quantum simulation, including parameter choices and potential error-mitigation directions, with implications for near-term quantum devices.

Abstract

Quantum simulation is a central application of near-term quantum devices, pursued in both analog and digital architectures. A key challenge for both paradigms is the effect of imperfections and noise on predictive power. In this work, we present a rigorous and physically transparent comparison of the stability of digital and analog quantum simulators under a variety of perturbative noise models. We provide rigorous worst- and average-case error bounds for noisy quantum simulation of local observables. We find that the two paradigms show comparable scaling in the worst case, while exhibiting different forms of enhanced error cancellation on average. We further analyze Gaussian and Brownian noise processes, deriving concentration bounds that capture typical deviations beyond worst-case guarantees. These results provide a unified framework for quantifying the robustness of noisy quantum simulations and identify regimes where digital methods have intrinsic advantages and when we can see similar behavior.

Stability of digital and analog quantum simulations under noise

TL;DR

This work analyzes the robustness of analog and digital quantum simulators to perturbations by deriving rigorous worst-case and average-case bounds for the estimation of local observables under perturbations of magnitude . It develops a unified framework that spans time-dependent Gaussian and white-noise perturbations, as well as Brownian and Lindbladian limits, and employs Lieb-Robinson bounds and Suzuki–Trotter product formulas to compare stability across paradigms. The results show that worst-case scaling is similar for both analog and digital approaches, while average-case behavior exhibits enhanced cancellation and regime-dependent advantages for digital simulations with suitably tuned parameters. These insights inform practical strategies for noise-aware quantum simulation, including parameter choices and potential error-mitigation directions, with implications for near-term quantum devices.

Abstract

Quantum simulation is a central application of near-term quantum devices, pursued in both analog and digital architectures. A key challenge for both paradigms is the effect of imperfections and noise on predictive power. In this work, we present a rigorous and physically transparent comparison of the stability of digital and analog quantum simulators under a variety of perturbative noise models. We provide rigorous worst- and average-case error bounds for noisy quantum simulation of local observables. We find that the two paradigms show comparable scaling in the worst case, while exhibiting different forms of enhanced error cancellation on average. We further analyze Gaussian and Brownian noise processes, deriving concentration bounds that capture typical deviations beyond worst-case guarantees. These results provide a unified framework for quantifying the robustness of noisy quantum simulations and identify regimes where digital methods have intrinsic advantages and when we can see similar behavior.

Paper Structure

This paper contains 42 sections, 40 theorems, 244 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Analog and digital quantum simulation
  4. Meaningful error models
  5. Stability
  6. Results for worst case errors
  7. Analog quantum simulation
  8. Digital quantum simulation
  9. Comparison of digital vs. analog simulation
  10. Results for average errors
  11. Analog quantum simulation
  12. Digital quantum simulation
  13. Comparison of digital vs. analog simulation
  14. Lindbladians and Brownian random walks
  15. Outlook and conclusion
  16. ...and 27 more sections

Key Result

Theorem 1

Consider a perturbed analog time evolution $V(t)$ defined by the time-dependent local Hamiltonian where $\norm{H'_\gamma(s) - H_\gamma} \leq \delta$ for all $s<t$ and all $\gamma\in\Gamma$. Then, the error on the time-evolution of a local observable $O$ is at most

Figures (1)

  • Figure 1: In this work, (a) analog quantum simulation provided by precisely controlled quantum systems naturally evolving in time $t>0$ is comprehensively and rigorously compared to digital quantum simulation run on non-quantum error corrected quantum circuits with respect to the 1. worst case and 2. average case impact of natural errors.

Theorems & Definitions (75)

  • Definition 1: Product unitary
  • Definition 2: Truncated Hamiltonian
  • Theorem 1: Upper bound for worst case errors in analog simulators
  • Theorem 2: Upper bound for worst case errors in digital simulators with gate-dependent perturbations
  • Theorem 3: Upper bound for worst case errors in digital simulators with constant gate perturbations
  • Theorem 4: Average case bounds for errors in analog simulators with Gaussian perturbations
  • Theorem 5: Average case bounds for errors in analog simulators with white noise perturbations
  • Theorem 6: Upper bound for average case errors in analog simulators
  • Theorem 7: Average case errors in digital simulators with gate-dependent perturbations
  • Theorem 8: Average case errors in digital simulators with constant gate perturbations
  • ...and 65 more