Error-mitigation aware benchmarking strategy for quantum optimization problems

TL;DR

This work addresses the challenge of evaluating quantum advantage for optimization tasks on noisy, near-term devices by incorporating finite-shot statistics and quantum error mitigation (QEM) into a benchmarking framework. It defines quantum advantage as the confidence that a PEC-corrected energy estimate lies within classical energy bounds $E_-$ and $E_+$, with success probabilities that depend on the shot budget, circuit depth, and QEM overhead. Through a numerical study of the 8×8 two-dimensional Fermi-Hubbard model, the authors identify regimes where probabilistic error cancellation (PEC) improves the likelihood of beating classical bounds and regimes where it does not, revealing a practical Goldilocks zone for QEM feasibility. The framework provides a hardware- and algorithm-agnostic, statistically grounded tool for end-users to assess potential practical quantum advantage and to guide QEM deployment on near-term quantum hardware, given a fixed shot budget and device noise level.

Abstract

Assessing whether a noisy quantum device can potentially exhibit quantum advantage is essential for selecting practical quantum utility tasks that are not efficiently verifiable by classical means. For optimization, a prominent candidate for quantum advantage, entropy benchmarking provides insights based concomitantly on the specifics of the application and its implementation, as well as hardware noise. However, such an approach still does not account for finite-shot effects or for quantum error mitigation (QEM), a key near-term error suppression strategy that reduces estimation bias at the cost of increased sampling overhead. We address this limitation by developing a benchmarking framework that explicitly incorporates finite-shot statistics and the resource overhead induced by QEM. Our framework quantifies quantum advantage through the confidence that an estimated energy lies within an interval defined by the best-known classical upper and lower bounds. Using a proof-of-principle numerical study of the two-dimensional Fermi-Hubbard model at size $8\times8$, we demonstrate that the framework effectively identifies noise and shot-budget regimes in which the probabilistic error cancellation (PEC), a representative QEM method, is operationally advantageous, and potential quantum advantage is not hindered by finite-shot effects. Overall, our approach equips end-users with a framework based on lightweight numerics for assessing potential practical quantum advantage in optimization on near-future quantum hardware, in light of the allocated shot budget.

Figures (6)

• Figure 1: Illustration of the method. For a given number of shots, compared with the raw results in red, probabilistic error cancellation (PEC) yields an unbiased energy distribution with larger variance. We consider the error-mitigated experiment to have succeeded if we obtain an estimated energy within the interval $[E_{-}, E_{+}]$, where $E _ {-}$ and $E _ {+}$ are the tightest lower and upper bounds to the target energy achieved by classical algorithms. The success probability $\mathbb{P}_\text{success}$ is then depicted as the area of the green-shaded region between these bounds. The success probability for unmitigated distribution (red-shaded area) can be used to assess whether applying QEM is advantageous.
• Figure 2: Success probability of ground state energy estimation under global depolarizing noise. We consider a 2D Fermi-Hubbard Hamiltonian with parameters $(U, t, \mu) = (8, 1, 3.75)$ with $L=64$ sites on a square lattice. We assume the candidate ground state was obtained with a layered circuit with width $n=2L=128$ and depth $D=L=64$ under global depolarizing noise with layerwise global depolarizing probability $P$. We take values of the lower and upper bounds to the ground state energy for this specific energy minimization problem from valenti_rigorous_1991. Figures \ref{['fig:success_prob_globalDP']}(a) and \ref{['fig:success_prob_globalDP']}(b) show the success probability in the presence \ref{['eq:P_tilde']} and absence \ref{['eq:P_success_raw']} of PEC respectively, as a function of the noise level $P$ and the available number of shots $N_\text{shots}$.
• Figure 3: Winning strategy as a function of the layerwise global depolarizing probability and the shot count. We consider a 2D Fermi-Hubbard Hamiltonian with parameters $(U, t, \mu) = (8, 1, 3.75)$ with $L=64$ sites. We assume the quantum solution for the ground state energy was obtained with a layered circuit with width $n=2L=128$ and depth $D=L=64$ under global depolarizing noise with layerwise global depolarizing probability $P$. We take lower bounds and upper bounds to the ground state energy for this specific energy minimization problem from valenti_rigorous_1991. We show three regimes as a function of the noise level $P$ and the available number of shots $N_\text{shots}$ and for a fixed threshold value of $0.95$: 'PEC' means PEC should be preferred over the raw distribution and ensures that the success probability is above the threshold, 'raw' means the raw distribution should be preferred over PEC and gives a success probability above the threshold, and 'none' corresponds to the regime where the success probability of both strategies is below the threshold.
• Figure 4: Schematic illustration of the PEC process.
• Figure 5: Illustration of the study of the accuracy of the proxy $\widetilde{\mathbb{P}}_{\mathrm{success}}$ to the success probability. We compare $\widetilde{\mathbb{P}}_{\mathrm{success}}$ (red-shaded area) obtained by assuming a Gaussian distribution of standard deviation $\sigma$ and mean 0 with the (actual) success probability $\mathbb{P}_{\mathrm{success}}$ (green-shaded area) obtained for a Gaussian distribution of standard deviation $\sigma$ and mean $E_0$. We vary the relative shift $E_0/(\Delta/2)$ and the relative width of the distributions, $\sigma/\Delta$.