Noise-Robust Estimation of Quantum Observables in Noisy Hardware

Abstract

Error mitigation is essential for extracting reliable results from quantum computations performed on noisy intermediate-scale quantum hardware. Here we introduce Noise-Robust Estimation (NRE), a noise-agnostic framework that suppresses estimation bias through a two-stage post-processing protocol. The method combines measurement data from a target circuit and a corresponding noise-canceling companion circuit to construct a baseline estimator with reduced sensitivity to noise. We show that the residual bias of this estimator is governed by the variation of an auxiliary quantity across amplified noise realizations, motivating the use of a measurable diagnostic quantity: the normalized dispersion of this auxiliary estimator. When the dispersion approaches zero, contributions arising from imperfect noise amplification vanish and the remaining bias terms are expected to diminish for smooth stationary noise profiles. Leveraging this relationship, NRE performs a final extrapolation to the zero-dispersion limit using bootstrapped measurement data. We experimentally validate the method on a 20-qubit IQM superconducting quantum processor using circuits containing up to 480 entangling CZ gates. Across a variety of circuits and noise levels, NRE consistently achieves substantially reduced bias compared to existing mitigation techniques while maintaining moderate sampling overhead. These results establish NRE as a practical and broadly applicable error-mitigation strategy for quantum computations on noisy hardware.

Figures (8)

• Figure 1: (a) Workflow of NRE, with each step detailed in Sec. \ref{['sec:NRE_workflow']}. (b) Schematic illustration of expectation values obtained from executing the target circuit $\langle \tilde{O} \rangle_t$, the noise-canceling circuit $\langle \tilde{O} \rangle_{ncc}$, and the auxiliary quantity $\mathcal{A}$ as a function of the noise scale factor $\lambda_i$. Expectation values and $\mathcal{A}$ are shown for two different bootstrap indices ($s$). The mean absolute deviation (MAD) of the sets $\{\langle \tilde{O}^s_t(\lambda_i)\rangle\}$ and $\{\mathcal{A}^s(\lambda_i)\}$ is schematically represented. As explained in Sec. \ref{['sec:NRE_workflow']}, the first post-processing step ensures that $\mathcal{A}(\lambda_1)$ serves as a baseline estimator for the ideal expectation value. (c) Experimentally observed correlation between the normalized dispersion $\mathcal{D}$ and the residual bias of the baseline estimator, $\mathcal{B}_{\mathrm{b\text{-}NRE}}$. The light blue cloud represents the distribution of baseline estimations obtained from different bootstrap realizations. Blue and red points correspond to the mean and standard deviation of the baseline estimator (b-NRE) and the final estimator (NRE), respectively. The residual bias of an error-mitigated estimator is defined as $\mathcal{B}_{\mathrm{EM}} = \langle O\rangle_{t}-\langle O\rangle_{\mathrm{EM}}$, where $\langle O\rangle_{t}$ denotes the ideal expectation value of the target circuit and $\langle O\rangle_{\mathrm{EM}}$ denotes the expectation value obtained from an error-mitigation estimator. While the bias can take both positive and negative values, the magnitude of the baseline-estimator bias, $|\mathcal{B}_{\mathrm{b\text{-}NRE}}|$, is expected to correlate with the normalized dispersion $\mathcal{D}$ under mild smoothness assumptions on the noise response, enabling extrapolation toward the $\mathcal{D}\to 0$ limit. A detailed explanation of the experiment is provided in Sec. \ref{['sec:TFIM_implementation']}.
• Figure 2: (a) Schematic of the IQM Garnet, a 20-qubit superconducting quantum processor with square-grid connectivity. (b) Measured ground-state energy of the transverse-field Ising model as a function of the noise scale factor $\lambda$, normalized to the noiseless expectation value. Results are shown for both the target circuit (t) and the noise-canceling circuit (ncc).
• Figure 3: (a–e) Noise-Robust Estimation of the TFIM ground-state energy for different sets of noise scale factors. Dark blue shaded regions show $2\times10^{4}$ baseline estimations obtained by resampling the first bootstrap set ($s{=}1$) of expectation values using the extended NRE procedure (see Appendix \ref{['subsec:extended_NRE']}). Light blue shaded regions aggregate the corresponding baseline estimations across all 500 bootstrap sets. Red points indicate the mean and standard deviation of the final NRE estimations, while blue points show the mean and standard deviation of the baseline estimations; in both cases error bars are derived from 500 bootstrap samples. Panel (c) corresponds to the dataset shown in Fig. \ref{['fig:NRE_Schematics']}(c). (f–j) Relative bias magnitude $|\mathcal{B}_\mathrm{EM}/\langle O\rangle_t|$ after applying different error-mitigation strategies, using the same noise scale factors as in (a–e). All methods are compared under a fixed budget of $2.4\times10^{5}$ measurement shots. Because CDR does not require noise amplification (unlike vnCDR), its predictions remain constant across panels (f–i). In panel (j), where $\lambda_{1}=2$, the CDR ansatz is trained using noisy measurements of the training circuits at $\lambda_1=2$.
• Figure 4: (a) Geometry of the $\mathrm{H}_4$ molecule in the STO-3G basis set. The target circuits are executed on the subset of 8 qubits highlighted in blue on the processor. (b) Measured expectation values of the $\mathrm{H}_4$ energy observable as a function of the noise scale factor $\lambda_i$, normalized to the corresponding noiseless values, $\langle \tilde{O} \rangle_{\mathrm{cir}}(\lambda_i) / \langle O \rangle_{\mathrm{cir}}$, for both the target circuit (t) and the corresponding noise-canceling circuit (ncc).
• Figure 5: (a–e) Noise-Robust Estimation of the ground-state energy of the H$_4$ molecule for different choices of noise scale factors. As in Fig. \ref{['fig:NRE_on_Garnet']}, dark blue shaded regions show $2\times10^{4}$ baseline estimations obtained by resampling the first bootstrap set ($s{=}1$), while light blue shaded regions aggregate the corresponding baseline estimations across all 500 bootstrap sets. (f–j) Relative bias magnitude after applying different error-mitigation strategies, including ZNE, the method of Ref. Urbanek_2021, and NRE. The shaded area marks the threshold for chemical precision.