Physics-Inspired Extrapolation for efficient error mitigation and hardware certification

TL;DR

The paper addresses the challenge of extracting useful quantum observables from noisy hardware by introducing Physics-Inspired Extrapolation (PIE), an EMRE-based, linear-runtime error-mitigation protocol. PIE leverages a fitting form $f_{PIE}(\lambda) = \langle \mathcal{O} \rangle_{ideal} s^{-{\lambda}}$, with $\lambda$ set by circuit folding and $s$ linked to the max-relative entropy between the ideal and noisy circuits, yielding an interpretable, hardware-certifying extrapolation that requires only linear data processing. It demonstrates competitive accuracy and low variance relative to standard zero-noise extrapolation methods on Ising-model dynamics and quantum-chemistry benchmarks, including large $N=84$ Ising simulations on IBMQ hardware and molecules like $\mathrm{H_2}$ and $\mathrm{LiH}$. The work also analyzes alternative approaches (inverse EMRE) and realistic noise models, outlining a path toward practical error mitigation and hardware certification suitable for the EFT and early fault-tolerant quantum computing era. Overall, PIE provides a scalable, interpretable, and hardware-friendly route to improved quantum utility with built-in diagnostic capability via the max-relative entropy slope.

Abstract

Quantum error mitigation is essential for the noisy intermediate-scale quantum era, and will remain relevant for early fault-tolerant quantum computers, where logical error rates are still significant. However, most QEM methods incur an exponential sampling overhead to achieve unbiased estimates, limiting their practical applicability. Recently, error mitigation by restricted evolution was shown to estimate expectation values with constant sampling overhead, albeit with a small bias that grows with circuit size and noise level. Building upon the EMRE framework, here, we propose physics-inspired extrapolation, a linear circuit runtime protocol that achieves enhanced accuracy without incurring substantial overhead. Unlike traditional zero-noise extrapolation methods, PIE provides an operational interpretation of its fitting parameters and converges to unbiased estimates as noise decreases. Distinctively, the slope of the extrapolation fit corresponds to the max-relative entropy between the ideal and noisy circuits, enabling quantitative hardware certification alongside error mitigation, with no additional computational overhead. We also demonstrate the efficacy of this method on IBMQ hardware and apply it to simulate 84-qubit quantum dynamics efficiently. Our results show that PIE yields accurate, low-variance error mitigated estimates, establishing it as a practical and scalable strategy for both error mitigation and hardware certification for near-term and early fault-tolerant quantum computers.

Figures (14)

• Figure 1: Workflow and Performance Comparison of the PIE Method. (a) Expectation values $\langle \mathcal{O} \rangle$ are measured from a quantum circuit executed on a noisy quantum processor. Measurements are repeated at different effective noise levels, achieved by inserting identity-equivalent operations (e.g., gate folding) into the circuit. (b) Extrapolation to the zero-noise limit using the collected data. The yellow curve shows the PIE fit, defined as $f_{\text{PIE}}(\lambda) = \langle \mathcal{O} \rangle_{\text{ideal}} s^{-\lambda}$, where $s$ is the max relative entropy between the noisy and ideal operations. This is compared to Exponential Zero-Noise Extrapolation (Exp. ZNE) (blue region). PIE exhibits three key advantages: (i) Exponentially reduced data requirements to achieve accurate extrapolation, (ii) Lower variance in the inferred zero-noise observable, as evidenced by the narrower confidence region, (iii) Physically meaningful fit parameters, including the parameter $s$, interpreted as the maximum relative entropy, which can then be used to certify quantum circuit performance (see the main text.)
• Figure 2: (Color online) Illustration of estimating expectation values using PIE on different quantum hardware. The slope of each curve reflects the hardware's sensitivity to noise (yellow (least steep), orange, and red (steepest)), enabling resource-efficient certification via max-relative entropy without prior knowledge of the noise or ideal circuit.
• Figure 3: (Color online) Experimental error-mitigated global magnetization $M_z$ on the two Trotter steps simulation of the one-dimensional Ising chain with $N=84$ spins. (a) PIE extrapolation from the noise amplified expectation values for $t=1.5$. (b) Comparison of extrapolation methods. (c) Topology of the IBM Eagle superconducting processor ibm kyiv with the ring of qubits selected for the experiments in dark blue. All experiments were performed using four extrapolation points and 4096 shots.
• Figure 4: (Color online) PIE performance as we increase the number of Trotter steps and hence the depth of the quantum circuit and the noise in the computation of the global magnetization $M_z$. We report results from experiments on $N=84$ qubits of the IBM Eagle superconducting processor ibm kyiv (the selected qubits are displayed in Figure \ref{['fig_experimentalresults']}). We also report the mitigated results obtained by other common extrapolation techniques. We observe that PIE is the most accurate protocol for low noise levels, while the results deviate from the noise-free and the exponential ZNE result for large circuit depths.
• Figure 5: (Color online) Numerical results of the global magnetization $M_z$ on the three Trotter steps simulation of the one-dimensional Ising chain at time $t=1.5$ with N = 8 spins using a depolarizing noise model. We display the error mitigated results (dots) from PIE, exponential ZNE, and polynomial quadratic ZNE techniques, together with the standard deviation of the results (bars). We also show the unmitigated or noisy result and the noise-free simulation, with the corresponding standard deviation (bars and shaded area, respectively). All experiments were performed using four extrapolation points and 4096 shots.