Quantum Uncertainty Quantification

• Quantum Uncertainty Quantification is a rigorous framework that defines and measures both intrinsic and extrinsic uncertainties in quantum systems using mathematical and estimation-theoretic principles.
• It integrates operator theory, statistical inference, and probabilistic models to separate genuine quantum effects from classical noise and enhance model validation.
• Applications span quantum chemistry, many-body simulations, and quantum machine learning, enabling robust prediction errors and actionable insights.

Quantum Uncertainty Quantification refers to a set of mathematically rigorous methodologies developed to characterize, measure, and propagate the intrinsic and extrinsic uncertainties present in quantum systems, models, algorithms, or data-driven quantum predictions. It encompasses both foundational theories of quantum indeterminacy and practical probabilistic frameworks for quantifying uncertainty in quantum simulations, measurements, and machine learning models. The topic draws from operator theory, statistical inference, entropic and geometric frameworks, and active learning, and connects directly to how non-classicality and probabilistic uncertainty permeate physical observables and computational predictions.

1. Foundational Principles: Axiomatic and Operational Views

Rigorous formulations of quantum uncertainty arise from both axiomatic and operational perspectives. Budiyono and Rohrlich formalize quantum uncertainty as a uniquely determined estimation-theoretic trade-off, derived from three postulates: (i) a nonseparable global ontic random variable $\lambda$ with variance $\langle \lambda^2\rangle = \hbar^2$, (ii) position-probabilities irreducibly parameterized by an underlying momentum field, and (iii) a principle of estimation independence, enforcing that independent system estimates remain independent. Under these assumptions, the best estimator of momentum at $q$ is $\,\nabla S(q)$, and the single-shot estimation error

$$\Delta (\mathbf{p}|q) = \frac{\lambda}{2}\nabla \ln P_{p}(q)$$

with mean-square error $E^2_j = (\hbar^2/4) I_j$ where $I_j$ is the Fisher information, yields exactly the Heisenberg–Kennard bound $\Delta x_j \Delta p_j \ge \hbar/2$. Thus, the uncertainty relation is realized as a Cramér–Rao trade-off in a classical estimation problem augmented by the epistemic restriction—Planck's constant appears intrinsically as the epistemic noise strength (Budiyono, 2020).

Measure-independent frameworks extend this via operational axioms—monotonicity under information-reducing maps, symmetry invariance, and the independence of joint-uncertainty from the measurement procedure—enabling the broad formalization and generalization of uncertainty and joint uncertainty beyond any single measure (variance, entropy, etc.) (1505.02223).

2. Mathematical Structures and Quantitative Formalisms

Standard quantum uncertainty is quantified by the variances (or quadratic moments) of non-commuting operators, with the canonical example being position $x$ and momentum $p$. More generally, the sl(2,R) algebra formed by $$\{\widehat{x^2},\, \widehat{p^2},\, \widehat{xp}+\widehat{px}\}$$ underpins a hierarchy of uncertainty relations, where the Casimir operator

$$\mathcal{C} = \frac{1}{2} (\widehat{x^2}\widehat{p^2} + \widehat{p^2}\widehat{x^2}) - (\frac{1}{2} (\widehat{x}\widehat{p} + \widehat{p}\widehat{x}))^2$$

characterizes quantized uncertainty "levels" in second-quantized field theory (Livine, 2023).

More broadly, uncertainty can be geometrized as a region (rather than a bound) in the space of variances of observables. For instance, the set

$$R_p = \{(\Delta O_1, ..., \Delta O_n): |\psi\rangle \in \mathcal{H}\}$$

is semialgebraic, describable by polynomial constraints—meaning all possible fluctuations for given observables can be algorithmically characterized, enabling tight, state-independent error quantification in theory and experiment (Xie et al., 2019).

The quantum–classical decomposition formalism further separates total measurement uncertainty into "genuine quantum" and "irreducible classical" parts, via generalized entropies and the structure of Kirkwood–Dirac quasiprobabilities (Budiyono, 13 Dec 2024). This split admits

$$U_\text{total}(\rho; \{M_x\}) = U_\text{quantum}(\rho; \{M_x\}) + U_\text{classical}(\rho; \{M_x\})$$

where the quantum contribution is operationally linked to contextuality and measurement disturbance. The minimal measurement uncertainty is fully classical and set by the impurity of the quantum state.

3. Measurement, Information, and Contextuality

Quantum uncertainty is rigorously distinguished from classical ignorance. Operational guessing-game protocols demonstrate that part of the entropic uncertainty relation can be attributed to inaccessible side information, but, especially in higher dimensions ($d\geq 3$), a strictly intrinsic quantum component remains. Experimental implementations confirm that for $d=2$, quantum side information can eliminate uncertainty entirely, while for $d\geq3$ intrinsic gaps are experimentally observable (Zhao et al., 2021). This aligns with the quantum-classical decomposition: nonzero genuine quantum uncertainty occurs exactly when weak values (or KD quasiprobabilities) show non-reality or negativity, corresponding to contextuality (Budiyono, 13 Dec 2024).

4. Practical Uncertainty Quantification in Quantum Chemistry and Simulation

In computational sciences, quantum uncertainty quantification is critical for model validation and prediction. In electronic structure prediction, system-specific Bayesian UQ frameworks infer errors and confidence intervals for quantum chemical models via a Gaussian process prior on residuals, integrating high-level reference calculations as benchmarks (Reiher, 2021). The LoUQAL strategy leverages cheap low-fidelity calculations to bias active learning towards regions where the model is least certain, with the UQ metric $\phi_\text{LoUQ}^f(x) = |\hat{g}^f(x) - g^f(x)|$ providing efficient and empirically nearly optimal sample selection (Vinod et al., 21 Aug 2025).

Bound-to-Bound Data Collaboration (B2BDC) interval-based UQ rigorously tracks the propagation of experimental and model uncertainties to quantum-chemical predictions, certifying the existence of feasible parameter sets and quantifying the growth of prediction errors outside chemically interpolating regimes (Oreluk et al., 2018).

In modeling the dynamics of quantum many-body systems, such as heavy-ion fusion, Monte Carlo sampling of parameter posteriors in time-dependent mean-field models propagates uncertainty in static nuclear properties to physical observables like fusion cross sections. Largest sources are traced to poorly constrained static properties (e.g., neutron skin thickness), and dynamical effects may amplify or reduce the static uncertainty impact (Godbey et al., 2022).

Quantum algorithms for uncertainty quantification in PDEs exploit superpositions to lift uncertainty into higher-dimensional initial data, yielding quantum costs independent of the number of samples $M$ and providing regimes of exponential quantum advantage in physical dimension and precision (Golse et al., 2022).

5. Uncertainty Quantification in Quantum Machine Learning and Quantum Algorithms

Emerging work on quantum-enhanced machine learning has directly imported classical UQ methodologies—Bayesian neural networks, Monte Carlo dropout, deep ensembles, and Gaussian processes—into variational quantum circuit settings, with modifications to account for parameter encoding and quantum-specific priors. Bayesian approaches and Gaussian-parameter noise yield the best-calibrated UQ in quantum settings (Wendlinger et al., 20 Jul 2025).

Measurements from quantum models are stochastic due to both quantum indeterminacy and device noise. Quantum conformal prediction (QCP) offers finite-sample, distribution-free predictive sets for quantum models, leveraging the randomness of measurement shots and gate/measurement noise. Nonconformity scores derived from quantum shot samples are used in quantile-based set construction, preserving calibration guarantees even in the presence of temporal noise drifts and nonclassical stochasticity (Park et al., 2023).

Quantum Approximate Bayesian Optimization (QABOA) integrates UQ at a circuit level: a quantum Matérn kernel injects empirical kurtosis of the measured state distribution as a diagonal uncertainty term in the GP surrogate, self-consistently connecting the quantum circuit's amplitude concentration to the exploration–exploitation balance. Regions of high kurtosis (amplified solutions) are treated as reliable, and the acquisition function adjusts accordingly, significantly improving search efficiency and solution consistency (Kim et al., 2023).

Quantum digital twins introduce ensemble-based UQ for hybrid quantum–classical computing, where digital replicas of noisy quantum devices with sampled calibration parameters are used in parallelized hybrid models. The ensemble mean and variance over digital twins provide robust estimates of predictive uncertainty, concurrently capturing both algorithmic and hardware-induced stochasticity (Otgonbaatar et al., 29 Oct 2024).

The Schrödinger Neural Network (SNN) paradigm reframes uncertainty quantification in conditional modeling as the computation of moments, credible intervals, and calibration diagnostics with respect to wave-function amplitudes, exploiting the Born rule and leveraging quantum-inspired regularizers directly linked to uncertainty relations (Hammad, 27 Oct 2025).

6. Formalizations, Generalizations, and Unified Frameworks

The "two-component" framework treats quantum uncertainty as the composition of (i) a state–observable derived quantum probability distribution and (ii) a classical uncertainty function—variance, entropy, geometric, or sine-defined—subject to axioms of vanishing for certainty, maximizing for uniformity, symmetry, and concavity. This architecture unifies informational, metrological, and purity-based quantifiers, ensuring all such $U(\rho;M) = U(Q(\rho,M))$ are operational and bounded (Gudder, 7 Aug 2024).

Measurement uncertainty relations are optimally quantified using semidefinite programming given arbitrary cost functions, operationally interpreting error through optimal transport distances between the marginals of joint measurements and ideal observables for every input state. This yields valid state-independent uncertainty regions, higher-order joint-uncertainty measures, and their respective tradeoffs (Schwonnek et al., 2016).

7. Applications and Implications

Quantum uncertainty quantification enables:

• Rigorous confidence intervals and error bars for quantum chemical energies, observables, and reaction barriers, crucial for actionable computational predictions (Reiher, 2021, Vinod et al., 21 Aug 2025)
• Model validation and interpretability in quantum simulations, including evidence-based assessment of model consistency and guidelines for reliable extrapolation (Oreluk et al., 2018, Godbey et al., 2022)
• Set-based probabilistic prediction with provable coverage in quantum machine learning, integrating hardware noise and shot stochasticity (Park et al., 2023, Wendlinger et al., 20 Jul 2025)
• Calibration of digital quantum simulations and distributed quantum models under realistic noise profiles, with robust propagation of uncertainty from hardware to application layer (Otgonbaatar et al., 29 Oct 2024)
• Operational certification of cryptographically secure randomness, via entropic uncertainty relations and min/max entropy-based bounds (Vallone et al., 2014)
• Estimation and separation of quantum versus classical (statistical) origins of experimental uncertainty, enabling improved understanding of contextuality and disturbance (Budiyono, 13 Dec 2024, Zhao et al., 2021)

A plausible implication is that future developments in quantum uncertainty quantification will continue to blend foundational statistical principles, information-theoretic constraints, and hardware-aware probabilistic modeling, underpinning the next generation of quantum computational assurance, verification, and robust inference.