# Techniques: Error-mitigation aware benchmarking strategy for quantum optimization problems

## What the paper contributes (one paragraph)
The paper develops a benchmarking framework for ground-state energy estimation that decides whether a noisy quantum device can plausibly achieve quantum advantage on an optimization task, while accounting for two effects earlier entropy-based benchmarks ignored: finite-shot statistics and the sampling overhead of quantum error mitigation (QEM). Quantum advantage is defined operationally as the confidence (success probability) that the estimated energy lands within an interval bounded by the best-known classical lower and upper bounds on the ground-state energy. Using probabilistic error cancellation (PEC) as the representative QEM method, the authors derive a closed-form success-probability expression that depends on noise level, circuit depth, and shot budget through the PEC negativity. Applied to an 8x8 Fermi-Hubbard instance under global depolarizing noise, the framework partitions the noise-versus-shots plane into regimes where PEC helps, where raw sampling suffices, and where neither reaches advantage, giving a deploy-or-not decision criterion.

## Techniques offered
- **error-mitigation-aware quantum-advantage benchmarking (finite-shot success-probability framework)** - An application-level, hardware-agnostic benchmark that scores potential quantum advantage as the confidence that an estimated ground-state energy falls between classically certified upper and lower bounds, given a shot budget and noise level.
  - guarantee: statistical confidence bound (success probability that the energy estimate lies within the classical bound interval, compared against a fixed threshold); no interactive soundness or hardware certificate.
  - quote: "We address this limitation by developing a benchmarking framework that explicitly incorporates finite-shot statistics and the resource overhead induced by QEM."
- **probabilistic error cancellation (PEC) success-probability model** - A closed-form expression for the confidence in quantum advantage under PEC, linking the negativity (sampling overhead) and shot count to the variance of the unbiased energy estimator via the error function.
  - guarantee: statistical bound on the unbiased PEC estimator (Gaussian success probability as a function of negativity, depth, and shots) under perfect noise characterisation and a normality assumption; no residual-bias guarantee.
  - quote: "To illustrate the framework, we consider ground-state energy estimation for the two-dimensional Fermi-Hubbard model and adopt probabilistic error cancellation (PEC)~\cite{temme_error_2017, endo_practical_2018} as a representative QEM strategy."
- **task-level QEM deployment criterion (break-even / winning-strategy phase diagram)** - A decision rule over the noise-versus-shot-budget plane that flags, for a target task, whether PEC should be used, whether raw sampling suffices, or whether neither reaches the advantage threshold.
  - guarantee: a threshold-based decision criterion derived from the compared PEC and raw success probabilities; not a proof, but an operational classification at a chosen confidence threshold.
  - quote: "From the QEM perspective, our framework also provides a concrete, task-level criterion to identify the regimes in which QEM can be operationally beneficial."

## Where it could apply
- Target primitive(s): VQE-style variational ground-state energy estimation; Trotterised or ansatz-based state preparation (Hamiltonian Variational Ansatz); expectation-value estimation under QEM.
- Target application group(s): chemistry_materials (lattice Hamiltonians, Fermi-Hubbard and strongly correlated models); platforms_tooling (benchmarking and QEM resource assessment); optimization (ground-state energy minimisation framed as quantum optimization).
- Code family / hardware assumptions: noisy near-term hardware without fault tolerance; global (or layerwise) depolarizing noise model; gate-level calibration data (two-qubit gate fidelities); assumes classically computable lower and upper energy bounds exist for the target Hamiltonian; not tied to a specific code or qubit modality.

## Caveats
The framework supplies a benchmarking and QEM-deployment criterion rather than an algorithmic speed-up, and its quantitative outputs rest on strong assumptions: a noiseless ansatz that exactly prepares the target state (expressivity and optimisation set aside), perfect noise characterisation and inversion, a bias-free Gaussian PEC estimator, and global depolarizing noise for the closed-form results. It requires pre-existing classical upper and lower bounds on the ground-state energy, and the demonstration is a proof-of-principle numerical study on a single 8x8 Fermi-Hubbard instance, with residual PEC bias and broader noise or QEM models left to future work. The advantage threshold is set somewhat arbitrarily. No paywall or access limitation; the LaTeX source and referenced code are available.