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Non-Markovian Noise Mitigation: Practical Implementation, Error Analysis, and the Role of Environment Spectral Properties

Ke Wang, Xiantao Li

TL;DR

This paper addresses mitigating non-Markovian noise in near-term quantum devices by extending probabilistic error cancellation to NMNM. It develops a time-local quantum master equation whose memory term is governed by bath correlation functions, linking error mitigation performance and sampling overhead to the environment's spectral properties. A fixed operator-basis PEC mapping enables practical one-step and multi-step NMNM along with explicit error and resource bounds that depend on environmental parameters. Numerical experiments on spin-boson models demonstrate effective suppression of non-Markovian errors in both single- and two-qubit settings, while highlighting how stronger environmental coupling or richer spectra increase sampling overhead. The findings offer design guidance for choosing or engineering environments to facilitate efficient error mitigation on NISQ devices.

Abstract

Quantum error mitigation(QEM), an error suppression strategy without the need for additional ancilla qubits for noisy intermediate-scale quantum~(NISQ) devices, presents a promising avenue for realizing quantum speedups of quantum computing algorithms on current quantum devices. However, prior investigations have predominantly been focused on Markovian noise. In this paper, we propose a non-Markovian Noise Mitigation(NMNM) method by extending the probabilistic error cancellation (PEC) method in the QEM framework to treat non-Markovian noise. We present the derivation of a time-local quantum master equation where the decoherence coefficients are directly obtained from bath correlation functions(BCFs), key properties of a non-Markovian environment that will make the error mitigation algorithms environment-aware. We further establish a direct connection between the overall approximation error and sampling overhead of QEM and the spectral property of the environment. Numerical simulations performed on a spin-boson model further validate the efficacy of our approach.

Non-Markovian Noise Mitigation: Practical Implementation, Error Analysis, and the Role of Environment Spectral Properties

TL;DR

This paper addresses mitigating non-Markovian noise in near-term quantum devices by extending probabilistic error cancellation to NMNM. It develops a time-local quantum master equation whose memory term is governed by bath correlation functions, linking error mitigation performance and sampling overhead to the environment's spectral properties. A fixed operator-basis PEC mapping enables practical one-step and multi-step NMNM along with explicit error and resource bounds that depend on environmental parameters. Numerical experiments on spin-boson models demonstrate effective suppression of non-Markovian errors in both single- and two-qubit settings, while highlighting how stronger environmental coupling or richer spectra increase sampling overhead. The findings offer design guidance for choosing or engineering environments to facilitate efficient error mitigation on NISQ devices.

Abstract

Quantum error mitigation(QEM), an error suppression strategy without the need for additional ancilla qubits for noisy intermediate-scale quantum~(NISQ) devices, presents a promising avenue for realizing quantum speedups of quantum computing algorithms on current quantum devices. However, prior investigations have predominantly been focused on Markovian noise. In this paper, we propose a non-Markovian Noise Mitigation(NMNM) method by extending the probabilistic error cancellation (PEC) method in the QEM framework to treat non-Markovian noise. We present the derivation of a time-local quantum master equation where the decoherence coefficients are directly obtained from bath correlation functions(BCFs), key properties of a non-Markovian environment that will make the error mitigation algorithms environment-aware. We further establish a direct connection between the overall approximation error and sampling overhead of QEM and the spectral property of the environment. Numerical simulations performed on a spin-boson model further validate the efficacy of our approach.
Paper Structure (12 sections, 5 theorems, 73 equations, 7 figures, 1 table, 1 algorithm)

This paper contains 12 sections, 5 theorems, 73 equations, 7 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. A Brief Introduction to Probabilistic Error Cancellation(PEC)
  3. Non-Markovian Noise Mitigation
  4. Time-Local Master Equation for Non-Markovian Noise
  5. One-step Noise Mitigation Algorithm
  6. Error and Complexity Analysis
  7. Numerical Experiments
  8. Single qubit coupled to a Ohmic bath
  9. Two qubits coupled to a Ohmic bath
  10. Discussion
  11. Acknowledgements
  12. Fourth-Order Representation of the non-Markovian noise

Key Result

Theorem 1

For a quantum circuit coupled to an environment with bath correlation functions meier1999nonritschel2014analytic$C_{j,k}(t) = \sum_{\mu}g^*_{j,\mu}g_{k,\mu}e^{i\omega_\mu t}, t>0, \omega_\mu\in\mathbb{C}$, the Non-Markovian Noise Mitigation(NMNM) method with $N_r$ number of samples and discrete time where the bound ${\color{black} G_{\mathrm{env}}}$ are related to the spectral properties of the en

Figures (7)

  • Figure 1: Schematic representation of the multi-step NMNM method in \ref{['eq:nstepQEM']}. (a) Interlacing structure of the multi-step NMNM method; (b) A sample circuit used in the Monte Carlo sampling implementation.
  • Figure 2: Error mitigated time evolution of spin-boson model with 1 qubit. The expectation of the density operator with respect to Pauli matrices observables $O_x, O_y, O_z$. The coupling strength $\lambda^2 = 0.81$ and the splitting energy $\Delta =8$. Monte-Carlo method is used to sample the measurement of the noisy quantum state $\rho_N(t)$, $\rho_Q(t)$ with number of samples $N_r = 10^4$. The blue and red shaded area represents the standard deviation of the population for the error mitigated trajectory $\mathcal{E}_Q\mathcal{E}_N$ and the noisy trajectories $\mathcal{E}_N$. The step size is set to $\delta t = 0.025$.
  • Figure 3: Error mitigated time evolution of spin-boson model with 1 qubit. The coupling strength $\lambda^2 = 0.01$ and the splitting energy $\Delta =2$. Monte-Carlo method is used to sample the measurement of the noisy quantum state $\rho_N(t)$ and $\rho_Q(t)$ with $N_r = 10^4$. The blue and red shaded area represents the standard deviation of the population for the error mitigated trajectory $\mathcal{E}_Q\mathcal{E}_N$ and the noisy trajectories $\mathcal{E}_N$. The step size is set to $\delta t = 0.1$.
  • Figure 4: Influence of ${\color{black} G_{\mathrm{env}}}$ on the normalization coefficient and sampling overhead. The coupling strength $\lambda^2 = 0.81$ and the splitting energy $\Delta =8$. \ref{['fig:Genv1']}: The normalization factor $\gamma_{\textbf{tot}}$ is computed for environments with different exponential cutoff coefficients $\omega_c = 1, 1.5, 2, 2.5, 3$. \ref{['fig:Genv1spinOzWeakCoupling']}: Use the larger ${\color{black} G_{\mathrm{env}}}$ settings from \ref{['fig:Genv1']} with $\omega_c = 2$, we reapply the noise mitigation algorithm with $N_r = 10^4$. The blue and red shaded regions indicate the standard deviation of the population for the error mitigated trajectory $\mathcal{E}_Q\mathcal{E}_N$ and the noisy trajectory $\mathcal{E}_N$. The step size is set to $\delta t = 0.025$.
  • Figure 5: The eigenvalues of the coefficients $(\Gamma_{\alpha\beta})_{\alpha,\beta}$ in the incoherent noise operator $\mathcal{L}_D(t)$ in \ref{['eq:LDLC']} for each time step $k\delta t,$ as well as the resource overhead $\gamma(t,\delta t)$ defined in \ref{['eq:normalizationconstant']}. The coupling parameter $\lambda^2 = 0.81$, the splitting energy $\Delta = 8$ and the time step $\delta t = 0.025$.
  • ...and 2 more figures

Theorems & Definitions (9)

  • Theorem : Informal version of \ref{['thm:bias', 'thm:gamma', 'thm:complexity']}
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3: Error Mitigation for the incoherent part of the non-Markovian noise
  • proof
  • Theorem 4: Sampling Complexity
  • proof