Uncertainty Quantification for Quantum Computing

Ryan Bennink; Olena Burkovska; Konstantin Pieper; Jorge Ramirez; Elaine Wong

quant-phMar 26, 2026

Uncertainty Quantification for Quantum Computing

Authors: Ryan Bennink, Olena Burkovska, Konstantin Pieper, Jorge Ramirez, Elaine Wong

Organizations: Quantum Information Sciences Section, Oak Ridge National Laboratory · Mathematics in Computation Section, Oak Ridge National Laboratory · Universidad Nacional de Colombia, Sede Medellin, Colombia · Oak Ridge National Laboratory

Abstract

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.

Figures & tables

Domain	Role of UQ in Decision Making	Example / Reference
Earth Systems Modeling	Quantifies the spread of predictions from different models and initial conditions, helping policymakers assess the risks of various climate scenarios.	Assessing future sea level rise projections to inform coastal infrastructure planning [ 36 ] .
Aerospace Design	Enables efficient design by accounting for uncertainties in aerodynamics, material properties, and operating environments. Used in reliability-based optimization of structures and flight trajectories.	Design of reusable launch vehicles under uncertain aerodynamic loads [ 59 ] .
Biomedical Imaging	Reduces diagnostic errors and enables better treatment decisions by quantifying noise and model-based uncertainty in image reconstruction (e.g., computed tomography, magnetic resonance imaging)	Bayesian UQ for positron emission tomography (PET) image reconstruction improves tumor detection confidence [ 37 ] .

Table 1: The role of uncertainty quantification in decision making across scientific domains.

Operator type	Defining property	Physical / computational role
Unitary $U$	$U^{\dagger}U=UU^{\dagger}=I$	Describes reversible quantum evolution and quantum gates .
Observable $O$	$O^{\dagger}=O$	Represents a measurable quantity . Has real eigenvalues and orthogonal eigenvectors; the eigenvalues correspond to possible measurement outcomes.
Positive $M$	$\braket{\psi\|M\|\psi}\geq 0$ for all $\ket{\psi}\in\mathcal{H}_{n}$	Special case of observable. Represents a possible measurement outcome in a POVM. Also called effect operator.
Projection $P$	$P^{2}=P=P^{\dagger}$	Special case of positive operator. Describes an ideal measurement outcome or event . Form Projective Measurements (PVMs).
Density $\rho$	$\rho=\rho^{\dagger},\;\rho\geq 0,\;\mathrm{Tr}(\rho)=1$	Represents a general (possibly mixed) quantum state . Encodes statistical mixtures and allows computation of expectations via $\langle A\rangle=\mathrm{Tr}(A\rho)$ .

Table 2: Main types of operators in quantum computing: definitions and roles.

Figure 1: Example of a quantum circuit on three qubits

Figure 2: Example of a variational quantum circuit on four qubits.

Panel of Figure 3 — Figure 3: Schematic representation of the open and closed open quantum systems.

Method	Typical variance scaling	UQ handles
Zero-noise extrapolation	Mild–moderate increase from fit uncertainty; grows with extrapolation factor	Bootstrap/Bayesian fits to extrapolant; drift-aware weighting of noise-scaled points.
Probabilistic error cancellation	Variance multiplies by quasi-probability 1-norm squared ( $\gamma^{2}$ ), often exponential in depth	Importance sampling bounds, stratified sampling, control variates; report $\gamma$ and effective sample size.
Symmetry verification / subspace checks	Mild shot loss from discarding symmetry-violating shots	Binomial/hypothesis tests on discard rate; propagate to confidence intervals on retained-sample estimators.
Learning-assisted mitigation (e.g., ML regressors, drift models)	Depends on model bias plus prediction variance	Cross-validation and posterior predictive checks; interval estimates that combine model and shot noise.
Virtual distillation	Variance grows with number of copies/shots and overlap estimation error	Delta-method or bootstrap error bars on overlap estimates; report depth/shot overhead explicitly.

Table 3: Comparative view of common estimator-based error-mitigation tools, emphasizing how variance inflation enters and which UQ techniques keep reported error bars honest.

Uncertainty Quantification

Uncertainty quantification estimates how reliable predictions and scientific inferences are, and how uncertainty propagates into decisions.

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On the QUEST for Uncertainty Quantification via Highest Density Regions

Sam Goring, Tom Kuipers, Nicola Paoletti, David S. Watson

King’s College London · Northeastern University London

Uncertainty Quantification Aleatoric Uncertainty Epistemic Uncertainty

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Uncertainty quantification (UQ) is essential for reliable decision-making in safety-critical applications in probabilistic machine learning. For regression problems, dominant scalar UQ approaches - notably, those based on proper scoring rules - measure uncertainty via pointwise predictive risk. This can lead to counterintuitive results when the target statistic is not the conditional expectation. We propose an alternative framework, in which uncertainty is characterised by the volume of the most probable subset of a distribution's support. QUEST (Quantifying Uncertainty via highest dEnSiTy regions) is a novel approach to UQ based on the concentration of Lebesgue measure at a distribution's peak(s), evaluated at one or more values of a robustness parameter

α

. We establish connections between our measures and classical statistics from information theory and economics. We show that, unlike popular alternatives based on proper scoring rules, QUEST measures of epistemic and aleatoric uncertainty satisfy a set of axioms adapted from the UQ literature, including monotonicity under distributional spread and invariance to location shifts. Selective prediction benchmarks confirm that QUEST performs favourably against standard measures such as variance and differential entropy.

Abstract

Uncertainty Quantification

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