Sample-efficient quantum error mitigation via classical learning surrogates

TL;DR

The paper tackles the substantial measurement overhead of zero-noise extrapolation (ZNE) in quantum error mitigation for families of parametrized circuits. It introduces surrogate-enabled ZNE (S-ZNE), which trains classical surrogates to predict noisy expectation values across noise levels and performs the zero-noise extrapolation entirely on the classical side. The authors provide theoretical bounds showing comparable scaling to conventional ZNE and demonstrate through numerical experiments on up to 100-qubit ground-state energy estimation and GHZ-based quantum metrology that S-ZNE achieves similar mitigation accuracy with much reduced quantum sampling. They also propose a hybrid S-ZNE variant that blends direct measurements with surrogates to balance accuracy and resource use, and discuss extensions to other QEM strategies, making S-ZNE a practical template for scalable error mitigation on near-term devices.

Abstract

The pursuit of practical quantum utility on near-term quantum processors is critically challenged by their inherent noise. Quantum error mitigation (QEM) techniques are leading solutions to improve computation fidelity with relatively low qubit-overhead, while full-scale quantum error correction remains a distant goal. However, QEM techniques incur substantial measurement overheads, especially when applied to families of quantum circuits parameterized by classical inputs. Focusing on zero-noise extrapolation (ZNE), a widely adopted QEM technique, here we devise the surrogate-enabled ZNE (S-ZNE), which leverages classical learning surrogates to perform ZNE entirely on the classical side. Unlike conventional ZNE, whose measurement cost scales linearly with the number of circuits, S-ZNE requires only constant measurement overhead for an entire family of quantum circuits, offering superior scalability. Theoretical analysis indicates that S-ZNE achieves accuracy comparable to conventional ZNE in many practical scenarios, and numerical experiments on up to 100-qubit ground-state energy and quantum metrology tasks confirm its effectiveness. Our approach provides a template that can be effectively extended to other quantum error mitigation protocols, opening a promising path toward scalable error mitigation.

Figures (8)

• Figure 1: The scheme of surrogate-enabled zero noise extrapolation (S-ZNE). The S-ZNE framework towards a class of parametrized quantum circuits comprises three key stages. a. Data Collection. Noise scaling (e.g., via unitary folding) is applied to generate the training dataset. At each noise level, different classical inputs are fed into the quantum circuit, followed by quantum sampling. b. Surrogate Modeling. The constructed dataset trains classical learning surrogates to estimate expectation values under target noise conditions purely on the classical side. c. Surrogate-Enabled Extrapolation: Given any new input, the optimized classical learning surrogates at all noise factors $\lambda$ can be used to predict the corresponding expectation values. In the main text, we use surrogate prediction for all noise levels, while a hybrid approach combining surrogate predictions and direct measurements from nosiy quantum processor (hybrid S-ZNE) is discussed in SI E. These results extrapolate to $\lambda \to 0$, yielding an error-mitigated estimate with reduced quantum overhead.
• Figure 2: Performance of S-ZNE in ground state energy estimation for quantum many-body systems.a. Surrogate model prediction accuracy under varying training set sizes ($n_j$) and noise levels ($\lambda_j$). Prediction error, quantified by $\log(1+\text{MSE})$, is evaluated on 1000 test samples for both transverse field Ising model (TFIM) and Heisenberg model (HM) under depolarizing (DP) and combined depolarizing+coherent (DP+CO) noise channels. b. Distribution of residuals ($\mathcal{R}$) of unmitigated, S-ZNE, and conventional ZNE protocols, evaluated by the kernel density estimation over 1000 test configurations. c. Sampling overhead analysis during the optimization. The main panel shows cumulative measurement costs for S-ZNE (constant overhead) versus ZNE (linear scaling with iteration). Inset displays estimated ground-state energies at epochs 1 and 1500 for TFIM (exact value: $-50.50$), demonstrating comparable convergence between S-ZNE and ZNE despite significantly reduced quantum resource requirements for S-ZNE.
• Figure 3: Application of S-ZNE to GHZ-state quantum metrology.a. Training dynamics of learning surrogates under varied noise levels $\lambda_i$. b. Reconstructed phase estimation signals after quantum error mitigation. S-ZNE predictions and ZNE estimates both recover the ideal $2\pi/100$-periodic response. c. Residual analysis of error-mitigated expectation values. Both S-ZNE and ZNE maintain residuals close to the ideal $\cos(100x)$ signal, with an inset comparing overall MSE performance. d. Sampling overhead of ZNE and S-ZNE. The y-axis refers to the cumulative number of measurements consumed by S-ZNE/ZNE for error mitigation.
• Figure 4: Ansatz circuits used in numerical experiments.a. Trotterized Hamiltonian variational ansatz for the 1D Transverse Field Ising Model (TFIM). b. Trotterized Hamiltonian variational ansatz for the 1D Heisenberg Model (HM). c. Circuit implementation for GHZ-state-based quantum metrology.
• Figure 5: Residual distributions of conventional ZNE and S-ZNE for different extrapolation functions. Probability density of residuals (error-mitigate estimation minus ideal value) for unmitigated, ZNE, and S-ZNE results under depolarizing (DP) and DP+coherent (CO) noise. Three extrapolation functions are compared: Linear, Quadratic, and Richardson. Results are aggregated over 1000 test instances.

Theorems & Definitions (5)

• Theorem 1: Informal
• Lemma 1: Liao2025surrogate
• Lemma 2: Liao2025surrogate
• Theorem : Formal statement of Theorem 1
• proof : Proof of Theorem 1