Quantum state tomography is fundamental to quantum computing verification and quantum cryptography validation, yet existing methods provide point estimates without quantifying when additional measurements are needed. We introduce Latent Posterior Factors (LPF) \citep{aliyu2026lpf,aliyu2026theory}, the first multi-evidence aggregation system with exact uncertainty decomposition for quantum measurements. LPF achieves 99.8% classification accuracy and 0.979 ± 0.051 fidelity on single-qubit tomography, with 50% fewer errors than maximum likelihood estimation (MLE) (0.21% vs. 0.43% error rate) despite MLE's asymptotic optimality. Critically, LPF provides perfectly calibrated uncertainties (Expected Calibration Error 0.17%) and exact epistemic-aleatoric decomposition: 46% of uncertainty is reducible through additional measurements, while 54% represents irreducible quantum randomness. When LPF reports 95% confidence, it is correct 95.1% of the time—enabling experimentalists to make trustworthy decisions about measurement sufficiency. The framework's learned aggregation captures Pauli measurement correlations that violate MLE's independence assumptions, explaining LPF's superior error rate despite lower continuous fidelity metrics. Cross-seed analysis across 11 random initializations demonstrates exceptional robustness (fidelity σ = 0.0003), validating theoretical soundness. Epistemic uncertainty reliably indicates measurement insufficiency, enabling adaptive protocols that achieve 10% measurement reduction in controlled experiments while maintaining accuracy—with production deployments estimated to realize greater savings when accounting for inter-measurement overhead. Keywords:Quantum state tomography, quantum computing verification, uncertainty quantification, epistemic uncertainty, aleatoric uncertainty, uncertainty decomposition, probabilistic inference, multi-evidence aggregation, Latent Posterior Factors (LPF), measurement optimization, maximum likelihood estimation (MLE), calibration (ECE), quantum measurement, adaptive protocols, interpretable uncertainty, quantum information processing, experimental physics AI
Quantum state tomography is fundamental to quantum computing verification and quantum cryptography validation, yet existing methods provide point estimates without quantifying when additional measurements are needed. We introduce Latent Posterior Factors (LPF) \citep{aliyu2026lpf,aliyu2026theory}, the first multi-evidence aggregation system with exact uncertainty decomposition for quantum measurements. LPF achieves 99.8% classification accuracy and 0.979 ± 0.051 fidelity on single-qubit tomography, with 50% fewer errors than maximum likelihood estimation (MLE) (0.21% vs. 0.43% error rate) despite MLE's asymptotic optimality. Critically, LPF provides perfectly calibrated uncertainties (Expected Calibration Error 0.17%) and exact epistemic-aleatoric decomposition: 46% of uncertainty is reducible through additional measurements, while 54% represents irreducible quantum randomness. When LPF reports 95% confidence, it is correct 95.1% of the time—enabling experimentalists to make trustworthy decisions about measurement sufficiency. The framework's learned aggregation captures Pauli measurement correlations that violate MLE's independence assumptions, explaining LPF's superior error rate despite lower continuous fidelity metrics. Cross-seed analysis across 11 random initializations demonstrates exceptional robustness (fidelity σ = 0.0003), validating theoretical soundness. Epistemic uncertainty reliably indicates measurement insufficiency, enabling adaptive protocols that achieve 10% measurement reduction in controlled experiments while maintaining accuracy—with production deployments estimated to realize greater savings when accounting for inter-measurement overhead. Keywords:Quantum state tomography, quantum computing verification, uncertainty quantification, epistemic uncertainty, aleatoric uncertainty, uncertainty decomposition, probabilistic inference, multi-evidence aggregation, Latent Posterior Factors (LPF), measurement optimization, maximum likelihood estimation (MLE), calibration (ECE), quantum measurement, adaptive protocols, interpretable uncertainty, quantum information processing, experimental physics AI
Quantum state tomography is fundamental to quantum computing verification and quantum cryptography validation, yet existing methods provide point estimates without quantifying when additional measurements are needed. We introduce Latent Posterior Factors (LPF) \citep{aliyu2026lpf,aliyu2026theory}, the first multi-evidence aggregation system with exact uncertainty decomposition for quantum measurements. LPF achieves 99.8% classification accuracy and 0.979 ± 0.051 fidelity on single-qubit tomography, with 50% fewer errors than maximum likelihood estimation (MLE) (0.21% vs. 0.43% error rate) despite MLE's asymptotic optimality. Critically, LPF provides perfectly calibrated uncertainties (Expected Calibration Error 0.17%) and exact epistemic-aleatoric decomposition: 46% of uncertainty is reducible through additional measurements, while 54% represents irreducible quantum randomness. When LPF reports 95% confidence, it is correct 95.1% of the time—enabling experimentalists to make trustworthy decisions about measurement sufficiency. The framework's learned aggregation captures Pauli measurement correlations that violate MLE's independence assumptions, explaining LPF's superior error rate despite lower continuous fidelity metrics. Cross-seed analysis across 11 random initializations demonstrates exceptional robustness (fidelity σ = 0.0003), validating theoretical soundness. Epistemic uncertainty reliably indicates measurement insufficiency, enabling adaptive protocols that achieve 10% measurement reduction in controlled experiments while maintaining accuracy—with production deployments estimated to realize greater savings when accounting for inter-measurement overhead. Keywords:Quantum state tomography, quantum computing verification, uncertainty quantification, epistemic uncertainty, aleatoric uncertainty, uncertainty decomposition, probabilistic inference, multi-evidence aggregation, Latent Posterior Factors (LPF), measurement optimization, maximum likelihood estimation (MLE), calibration (ECE), quantum measurement, adaptive protocols, interpretable uncertainty, quantum information processing, experimental physics AI
Quantum state tomography (QST) estimates an unknown quantum state from a measurement process typically performed on identical copies of the target state and plays a core role in characterizing quantum information systems. However, because quantum measurement outcomes are probabilistic, QST methods often require many shots per basis to achieve high-fidelity estimates, which hinders scalability to many-qubit applications and slows the transmission of quantum information. A more efficient procedure is therefore required for modern-day quantum operations in quantum computing, communications, and sensing. We address this need with an adaptive QST approach that measures a single-qubit state in a wide set of bases. We include bases beyond the Pauli triad {X,Y,Z} by leveraging the infinite-dimensional Hilbert space of quantum mechanics, increasing the likelihood of producing highly polarized measurement outcomes and finding an optimally informative basis. With this basis, the state can be identified with fewer measurements by focusing on directions that most reduce uncertainty of the state. In addition to theoretical innovation in [9], we experimentally verify in a simplified photonic realization using 13 bases that having many more available measurement bases than the Pauli triad benefits QST. This engineering implementation demonstrates that adaptive QST is both efficient in performance and practical to implement.
Quantum fidelity estimation is a fundamental task in quantum information and computing to benchmark the quality of quantum states, and hence of quantum processes, on noisy quantum hardware. Point estimates of fidelity by existing methods generally lack the rigorous uncertainty quantification required for reliable preparation and verification of these tasks. While Bayesian and bootstrap frameworks are widely used for uncertainty quantification, they are sensitive to model misspecification or lack formal coverage guarantees in non-asymptotic regimes. This paper introduces a hybrid "Bayesian-informed" conformal prediction-based framework to provide distribution-free, rigorous prediction intervals for quantum state fidelity estimation which can be used, with care, to approximate confidence intervals with respect to the quantum randomness of Born's rule. The framework requires a point estimator, and we consider both classical point estimators as well as data-driven neural net estimators. We evaluate the robustness of the methodology across a diverse range of experimental challenges, including finite-sample noise, tomographic incompleteness, physical model error, and out-of-distribution data. Our results demonstrate that conformalized estimators maintain statistical validity where traditional methods fail, offering a scalable and reliable approach for the verification of modern quantum systems. The best performing variant of our methodology is based on a class-conditional conformal prediction, as this most closely approximates the conditional coverage desired for a true confidence interval.
Recent experimental progress in the preparation and control of quantum systems has brought to light the importance of Quantum State Tomography (QST) in validating the results. In this thesis we investigate several aspects of QST, whose central problem is to devise estimation schemes for the recovery of an unknown state, given an ensemble of n independent identically prepared systems. The key issues in tackling QST for large dimensional systems is the construction of physically relevant low dimensional state models, and the design of appropriate measurements. Inspired by compressed sensing tomography, in chapters 4, 5 we consider the statistical problem of estimating low rank states (r ≪ d) in the set-up of Multiple Ions Tomography (MIT), where r and d are the rank and the dimension of the state respectively. We investigate how the estimation error behaves with a reduction in the number of measurement settings, compared to ‘full’ QST in two setups - Pauli and random bases measurement designs. We study the estimation errors in this ‘incomplete’ measurement setup in terms of a concentration of the Fisher information matrix. For the random bases design we demonstrate that O(r logd) settings suffice for the mean square error w.r.t the Frobenius norm to achieve the optimal O(1/n) rate of estimation. When the error functions are locally quadratic, like the Frobenius norm, then the expected error (or risk) of standard procedures achieves this optimal rate. However, for fidelity based errors such as the Bures distance we show that no ‘compressive’ recovery exists for states close to the boundary, and it is known that even with conventional ‘full’ tomography schemes the risk scales as O(1/√n) for such states and error functions. For qubit states this boundary is the surface of the Bloch sphere. Several estimators have been proposed to improve this scaling with ‘adaptive’ tomography. In chapter 6 we analyse this problem from the perspective of the maximum Bures risk over all qubit states. We propose two adaptive estimation strategies, one based on local measurements and another based on collective measurements utilising the results of quantum local asymptotic normality. We demonstrate a scaling of O(1/n) for the maximum Bures risk with both estimation strategies, and also discuss the construction of a minimax optimal estimator. In chapter 7 we return to the MIT setup and systematically compare several tomographic estimators in an extensive simulation study. We present and analyse results from this study, investigating the performance of the estimators across various states, measurement designs and error functions. Along with commonly used estimators like maximum likelihood, we propose and evaluate a few new ones. We finally introduce two web-based applications designed as tools for performing QST simulations online.
Quantum state vector reconstruction is a fundamental problem in quantum information theory, crucial for tasks like quantum state tomography and quantum error correction. Traditional methods often struggle with noisy measurements and lack a principled framework for uncertainty quantification. This paper introduces a novel algorithm, Quantum State Vector Reconstruction via Bayesian Inference, that leverages Bayesian inference to reconstruct a quantum state vector from a set of measurements, enhancing accuracy and offering a more robust approach compared to existing methods. We detail the core mechanism, emphasizing the integration of Bayesian estimation for uncertainty quantification and refinement of the reconstruction process. The algorithm incorporates a probabilistic approach to estimate the state vector distribution, improving reconstruction by incorporating uncertainty estimates directly into the reconstruction process. This results in a more reliable and accurate state vector reconstruction, particularly when dealing with noisy measurements. This work addresses the limitations of conventional methods by providing a principled Bayesian framework for state vector reconstruction, offering a significant advancement in this field.
Drawing inspiration from gradient-descent methods developed for data processing in quantum state tomography [\href{https://iopscience.iop.org/article/10.1088/2058-9565/ae0baa}{Quantum Sci.~Technol.~\textbf{10} 045055 (2025)}] and quantum process tomography [\href{https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.130.150402}{Phys.~Rev.~Lett.~\textbf{130}, 150402 (2023)}], we introduce stochastic gradient descent (SGD) algorithms for fast quantum measurement tomography (QMT), applicable to both discrete- and continuous-variable quantum systems -- thus completing the tomography trio. A measurement device or detector in a quantum experiment is characterized by a set of positive operator-valued measure (POVM) elements; the goal of QMT is to estimate these operators from experimental data. To ensure physically valid (positive and complete) POVM reconstructions, we propose two distinct parameterization schemes within the SGD framework: one leveraging optimization on a Stiefel manifold and one based on Hermitian operator normalization via eigenvalue scaling. Within the SGD-QMT framework, we further investigate two loss functions: mean squared error, equivalent to L2 or Euclidean norm, and average negative log-likelihood, inspired by maximum likelihood estimation. We benchmark performance against state-of-the-art constrained convex optimization methods. Numerical simulations demonstrate that, compared to standard methods, our SGD-QMT algorithms offer significantly lower computational cost, superior reconstruction fidelity, and enhanced robustness to noise. We make a Python implementation of the SGD-QMT algorithms publicly available at \href{https://github.com/agtomo/SGD-QMT}{github.com/agtomo/SGD-QMT}.