Orders of magnitude runtime reduction in quantum error mitigation

TL;DR

This work tackles the high sampling overhead of quantum error mitigation (QEM), particularly for agnostic noise amplification (ANA) methods. It introduces virtual noise scaling (VNS) to shift the effective noise spectrum into a favorable plateau and couples it with a layer-based mitigation approach (Layered-KIK) to suppress higher-order errors with minimal additional overhead. The authors derive analytical bounds on infidelity and runtime cost, propose a practical procedure to determine the VNS scale from measured data, and demonstrate exponential reductions in runtime overhead in strong-noise regimes, especially when combining VNS with multiple layers. The results indicate that, while QEM cannot replace full quantum error correction, VNS-based strategies can render ANA-based QEM far more practical for near-term devices and dynamic circuits, including mid-circuit measurements and SPAM mitigation.

Abstract

Quantum error mitigation (QEM) infers noiseless expectation values by combining outcomes from intentionally modified, noisy variants of a target quantum circuit. Unlike quantum error correction, QEM requires no additional hardware resources and is therefore routinely employed in experiments on contemporary quantum processors. A central limitation of QEM is its substantial sampling overhead, which necessitates long execution times where device noise may drift, potentially compromising the reliability of standard mitigation protocols. QEM strategies based on agnostic noise amplification (ANA) are intrinsically resilient to such noise variations, but their sampling cost remains a major practical bottleneck. Here we introduce a mitigation framework that combines virtual noise scaling with a layered mitigation architecture, yielding orders of magnitude reduction in runtime overhead compared to conventional zero-noise extrapolation post-processing. The proposed approach is compatible with dynamic circuits and can be seamlessly integrated with error detection and quantum error correction schemes. In addition, it naturally extends to ANA-based mitigation of mid-circuit measurements and preparation errors. We validate our post-processing approach by applying it to previously reported experimental data, where we observe a substantial improvement in mitigation efficiency and accuracy.

Figures (6)

• Figure 1: Plots of the mitigation function $G(m,s)$ for odd (a) and even (b) orders. As the order increases, the plateaus become wider, and a broader range of noise eigenvalues is mapped to 1, corresponding to successful mitigation. The left histogram in (a) represents the noise histogram of the circuit. By applying our VNS method, the histogram is shifted to the center of the plateau (right histogram), where the mitigation performance is significantly improved. The histograms in (b) illustrate that when the scaled noise interval $[gs_{\min},g]$ is narrower than the width of the plateau, small changes in $g$ do not affect the value of the mitigated observable. We exploit this principle to determine $g$ from the noise-amplified expectation value.
• Figure 2: (a) The red curve shows the runtime overhead $\mathcal{R}$ of standard Taylor mitigation as a function of the mitigated worst-case infidelity with the markers indicating the mitigation order $m$. The black curve, shows that the virtual noise scaling (VNS) introduced in the present work, leads to a substantial reduction of the runtime overhead. The initial infidelity is $0.6$ which corresponds to strong noise. To achieve an even larger reduction in $\mathcal{R}$, we combine VNS with a two-layer mitigation protocol (blue curve). The corresponding runtime overhead reduction factors for several target infidelities are indicated below the vertical dotted lines. This orders-of-magnitude reduction in $\mathcal{R}$ can make previously unrealistic mitigation performance feasible. For reference, the orange curve shows two-layer mitigation without VNS. (b) The solid purple funnel shows the mitigated infidelity as a function of the noise eigenvalue for mitigation order seven ($m=7$). The smallest eigenvalue, $s_{\min}=0.4$, determines the worst-case infidelity. After applying VNS, the original noise interval (red) is mapped to the black interval. Although this interval is wider, the new extreme point $g_{eq}s_{\min}$ corresponds to a smaller infidelity. For comparison, the dashed purple funnel shows the infidelity for mitigation order $m=14$, which is the order required to achieve the same infidelity as the $m=7$ mitigation with VNS.
• Figure 3: (a) The value of g can be extracted from the mitigated expectation value as a function of the VNS parameter g. As the mitigation order m increases, a broader plateau develops. The value of $g$ is then chosen as the extremum or the inflection point within this plateau. The numerical simulation is based on a four-qubit Ising Trotterized evolution; see main text for simulation details. In (a), the observable is $\sigma_{z}$ of the left qubit, whereas in (b) it is $\sigma_{x}$ of the same qubit. While in (a) the optimal value is $g\simeq1.1$, in (b) it is $g\simeq1.28$. This difference reflects the fact that the $\sigma_{z}$ observable is less affected by noise (and therefore requires a smaller value of $g$) and can consequently be mitigated with a lower runtime overhead.
• Figure 4: The red curve shows the fidelity of a GHZ state creation experiment reported in Parity2025REM. The GHZ state is created using a dynamic circuit. The mid-circuit measurements, mid-circuit resets, and final measurements are simultaneously mitigated using the parity mitigation method, which is resilient to noise drifts. The black curve shows an improvement over the Taylor post-processing when applying VNS to the same experimental data.
• Figure 5: (a) Slopes of the $\mathcal{R}$-infidelity curves for various mitigation schemes as a function of the smallest eigenvalue of the noise operator, $s_{\min}$. Lower absolute values correspond to slower growth in mitigation overheads as infidelity decreases. Without VNS, two-layer Taylor mitigation outperforms single-layer mitigation when the noise is above a critical value ($s_{\min}$ is below a critical value). The VNS slopes are always less steep (than the Taylor slopes, yet VNS also shows a crossover: beyond a critical noise value, two-layer VNS becomes more efficient than single-layer VNS in terms of runtime overhead. For both Taylor and VNS, the crossover value is $\sim0.62$ for $m\to\infty$, while for finite $m$ we find the crossover to be $\sim0.65$. (b) Same plot as Fig. \ref{['Fig2: loglog']}(a) but at the crossover point $s_{min,2L}^{\text{tot}}=0.65$. While VNS still outperforms Taylor mitigation at this point, two-layer mitigation should be used only when the noise is stronger i.e., $s_{min}^{\text{tot}}<s_{min,2L}^{\text{tot}}$.