Error-mitigation aware benchmarking strategy for quantum optimization problems

Technical Summary: Error-Mitigation Aware Benchmarking Strategy for Quantum Optimization Problems

Problem Statement
Assessing whether noisy intermediate-scale quantum (NISQ) devices can demonstrate quantum advantage for optimization tasks is critical, particularly for problems where classical verification is intractable. While entropy benchmarking frameworks have emerged as a rigorous tool to link physical entropy to performance degradation, they face significant limitations in realistic experimental settings. Specifically, existing methods do not account for finite-shot statistics or Quantum Error Mitigation (QEM). QEM techniques, such as Probabilistic Error Cancellation (PEC), reduce estimation bias by trading it for increased sampling overhead (negativity). Consequently, the physical state entropy no longer accurately reflects the quality of the error-mitigated estimate, rendering deterministic entropy-energy curves insufficient for characterizing the end-to-end performance of statistical, error-mitigated computations. Furthermore, there is a lack of quantitative, task-level criteria to determine whether applying QEM actually enhances or undermines quantum advantage under a fixed shot budget.

Methodology
The authors propose a benchmarking framework that explicitly incorporates finite-shot statistics and the resource overhead induced by QEM. The core of the methodology involves redefining "quantum advantage" not as a deterministic energy value, but as the confidence (success probability) that an estimated energy lies within a classically certified interval $[E_-, E_+]$, where $E_-$ and $E_+$ are the tightest known lower and upper bounds for the true ground-state energy $E_0$.

Key methodological components include:

1. Statistical Modeling: The framework assumes that the energy distribution yielded by PEC follows a normal distribution $N(E_0, \sigma^2)$ centered on the true ground state. The variance $\sigma^2$ is linked to the sampling overhead (negativity $\gamma$) and the number of shots ($N_{shots}$) via the relation $N_{shots} \propto \gamma^2 / \sigma^2$.
2. Success Probability Formulation: The success probability $P_{success}$ is calculated as the probability mass of the estimated energy distribution falling within the interval $[E_-, E_+]$. For PEC, this is derived using the error function based on the total sampling overhead $\gamma_{tot}$ and the shot budget.
3. Approximation for Unknown $E_0$: Since $E_0$ is unknown in the target application, the authors introduce a proxy success probability $\tilde{P}_{success}$ by assuming the mean of the error-mitigated distribution is the midpoint of the classical bounds, $(E_- + E_+)/2$.
4. Comparison with Raw Results: The framework also calculates the success probability for unmitigated (raw) measurements. In this case, the distribution is biased ($E_{noisy} \neq E_0$) but has lower variance. The success probability for raw data depends on both the variance and the shift in the mean caused by noise.
5. Case Study: The framework is applied to the ground-state energy estimation of a 2D Fermi-Hubbard model on an $8 \times 8$ lattice ($L=64$ sites, $n=128$ qubits) using a Hamiltonian Variational Ansatz (HVA) circuit of depth $D=64$ under global depolarizing noise.

Key Results
Through numerical proof-of-principle studies, the authors demonstrate that the framework effectively partitions the parameter space of noise level ($P$) and shot count ($N_{shots}$) into three distinct regimes:

• Raw Regime: In low-noise environments with sufficient shots, the raw (unmitigated) strategy yields a higher success probability than PEC. Here, the sampling overhead of PEC is unnecessary, and the bias in raw data is negligible.
• PEC Regime: As noise increases, the bias in raw data shifts the mean outside the success interval, reducing its success probability. Conversely, PEC maintains the mean at $E_0$ but increases variance. In an intermediate noise regime, PEC provides a higher success probability than raw sampling, making it the advantageous strategy.
• None Regime: For high noise levels or extremely low shot budgets, neither strategy achieves the defined success threshold (set at 0.95 in the study). In these regions, quantum advantage is not guaranteed regardless of the mitigation strategy.

The study reveals that the transition between these regimes is non-trivial. For instance, in the raw strategy, increasing the shot count beyond a certain noise threshold can paradoxically decrease the success probability because the distribution becomes more peaked around a biased mean that lies outside the target interval.

Significance and Claims
The paper claims to provide a task-level criterion for assessing potential practical quantum advantage in optimization on near-future hardware. Its primary significance lies in:

• Moving beyond deterministic metrics: It shifts the benchmarking paradigm from deterministic entropy-energy mappings to a statistical assessment of confidence intervals, which is essential for finite-shot, error-mitigated experiments.
• Operational decision-making: The framework offers end-users a lightweight numerical tool to determine when to deploy QEM. It identifies specific "Goldilocks zones" where hardware noise and available shot budgets align such that PEC meaningfully enhances the prospects of achieving quantum advantage.
• Realistic constraints: By explicitly modeling the trade-off between bias reduction and sampling overhead, the framework addresses the practical limitations of current quantum hardware, contrasting with previous negative results regarding the scaling of PEC.

The authors maintain modesty regarding their claims, noting that their analysis relies on specific assumptions (e.g., Gaussian distributions, perfect noise characterization) and a specific Hamiltonian instance. They suggest that while the framework is demonstrated on the Fermi-Hubbard model, it is applicable to other lattice Hamiltonians with available classical bounds and can be extended to other QEM protocols.