Quantum computing offers speedups, but NISQ processors face hardware-induced errors that degrade fidelity and reproducibility. We introduce an uncertainty-aware benchmarking framework that combines uncertainty quantification with global sensitivity analysis to evaluate not only peak fidelity but also its reliability over time. Using Bayesian Optimization with SGLD refinement under fixed budgets, seeds, bounds, and trust-region rules, and repeating runs across days, we capture calibration drift and quantify efficiency, stability, landscape complexity, and maintenance cost. A noise-aware Gaussian process surrogate provides scalable sensitivity estimates without error mitigation. Applied to VQAMET and VQC on three IBMQ backends, the framework delivers actionable guidance for co-tuning and backend selection, complementing and often outperforming quantum volume style metrics.
Quantum computing offers speedups, but NISQ processors face hardware-induced errors that degrade fidelity and reproducibility. We introduce an uncertainty-aware benchmarking framework that combines uncertainty quantification with global sensitivity analysis to evaluate not only peak fidelity but also its reliability over time. Using Bayesian Optimization with SGLD refinement under fixed budgets, seeds, bounds, and trust-region rules, and repeating runs across days, we capture calibration drift and quantify efficiency, stability, landscape complexity, and maintenance cost. A noise-aware Gaussian process surrogate provides scalable sensitivity estimates without error mitigation. Applied to VQAMET and VQC on three IBMQ backends, the framework delivers actionable guidance for co-tuning and backend selection, complementing and often outperforming quantum volume style metrics.
We present an uncertainty quantification (UQ) framework for application level benchmarking and characterization of noisy quantum backends. The framework compares two workload classes under one statistical pipeline: noisy intermediate scale quantum (NISQ) variational quantum algorithms (VQAs) and Quantum Singular Value Transformation (QSVT) based Green's function reconstruction. For the VQA branch, we evaluate ten benchmark families spanning chemistry, optimization, simulation, compiling, linear solving, partial differential equations, metrology, error correction, tomography, and channel fidelity estimation. For the QSVT branch, we reconstruct orbital resolved Green's functions and spectral peaks from a block encoded real time propagator. The workflow combines Bayesian optimization, posterior distribution refinement, sensitivity analysis, robust parameter density estimation, backend ranking, noise correlation, and resource estimation analysis. Instead of reporting only one best parameter vector, the framework identifies robust parameter regions, residual gaps to ideal behavior, backend specific failure modes, and calibration sensitive uncertainty. The result is a common benchmark for variational and non-variational workloads that measures how reliably each backend reaches useful task level behavior.
This review is designed to introduce mathematicians and computational scientists to quantum computing (QC) through the lens of uncertainty quantification (UQ) by presenting a mathematically rigorous and accessible narrative for understanding how noise and intrinsic randomness shape quantum computational outcomes in the language of mathematics. By grounding quantum computation in statistical inference, we highlight how mathematical tools such as probabilistic modeling, stochastic analysis, Bayesian inference, and sensitivity analysis, can directly address error propagation and reliability challenges in today's quantum devices. We also connect these methods to key scientific priorities in the field, including scalable uncertainty-aware algorithms and characterization of correlated errors. The purpose is to narrow the conceptual divide between applied mathematics, scientific computing and quantum information sciences, demonstrating how mathematically rooted UQ methodologies can guide validation, error mitigation, and principled algorithm design for emerging quantum technologies, in order to address challenges and opportunities present in modern-day quantum high performance and fault-tolerant quantum computing paradigms.
Benchmarking noisy quantum computers for variational workloads requires more than a single best energy number: two backends may reach similar energies while differing sharply in convergence speed, stability, and parameter sensitivity. We present a compact uncertainty quantification driven benchmarking study for the Variational Quantum Eigensolver (VQE), combining Bayesian optimization with Metropolis-Hastings refinement to search a backend conditioned VQE landscape and characterizing each backend by its best observed energy, evaluations to target, objective dispersion, and Sobol sensitivity fingerprint. On a 92 parameters LiH VQE across seven IBM fake backends, Brisbane wins overall (best energy 1.0207, top normalized quality), Kyoto matches its energy but converges five times more slowly, and Osaka trades a modest energy penalty for near optimal speed and low dispersion. The Sobol fingerprints further show that the dominant VQE parameters differ across backends, indicating backend specific robust regions rather than a universal calibration. Uncertainty quantification therefore provides a practical application level view of backend quality for near term variational workloads.
Quantum machine learning (QML) models introduce unique sources of uncertainty arising from finite measurement shots, hardware noise, and variational optimization instability. Despite rapid progress in quantum algorithms, systematic uncertainty quantification (UQ) in QML remains largely underexplored and lacks standardized tooling. We present Quan-tumUQ, an open-source Python library for uncertainty quantification in quantum machine learning with native support for PennyLane and Qiskit. QuantumUQ provides shot-based uncertainty estimation, ensemble-based epistemic uncertainty, measurement-budget stability profiling, and calibration metrics for classification and regression tasks. We formalize the uncertainty decomposition in QML, describe the architectural design of the library, and demonstrate its capabilities through reproducible experiments. Our results show that measurement-induced uncertainty exhibits predictable scaling behavior and that calibration analysis reveals systematic overconfidence in variational quantum classifiers. QuantumUQ aims to make robustness, reliability, and trustworthiness first-class citizens in quantum ML research.
Variational Quantum Algorithms (VQAs) are promising methods for solving combinatorial optimization problems on noisy intermediate-scale quantum (NISQ) devices. However, benchmarking VQAs is difficult due to their stochastic behavior and the lack of standardized performance criteria. This work introduces a general framework for evaluating VQAs applied to Quadratic Unconstrained Binary Optimization (QUBO) problems. The framework uses three complementary metrics: feasibility, quality, and reproducibility. It also introduces a quality diagram that visualizes trade-offs between success probability and computational resources. Reproducibility is formalized using Shannon entropy, and a decision rule is defined for selecting algorithms under resource constraints. As a demonstration, the framework is applied to several VQAs using Conditional Value at Risk (CVaR) cost functions and different shot counts on a 16-qubit QUBO instance. The results show how the framework supports systematic benchmarking and provides a foundation for adaptive algorithm selection in hybrid quantum-classical workflows.
Quantum Machine Learning (QML) unites quantum computing with modern AI to achieve potential gains in expressivity, runtime, and sample efficiency. This paper synthesizes advances in quantum feature maps, variational quantum circuits (VQCs), and hybrid quantum-classical workflows deployable on noisy intermediate-scale quantum (NISQ) hardware. We outline design principles for data encoding, ansatz selection, and trainability, and summarize error-aware practices—error mitigation, shot-frugal estimation, and compiler co-design. Using synthetic but instructive experiments, we illustrate scaling behavior, noise sensitivity, end-to-end inference latency across backends, and optimizer convergence in variational training. We conclude with an agenda emphasizing domain-aligned encodings, rigorous benchmarking, and reproducible open science to validate advantage beyond asymptotic claims.