Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2607.11273

Fixed-Protocol Amortized MPS Tomography with Conformalized Predictive Uncertainty

Jian Xu, Delu Zeng, John Paisley, Qibin Zhao

quant-phcs.LGSubmitted Jul 13, 2026• Announced Jul 14, 2026• 0 citations
arXivHTMLPDF

Abstract

Quantum state tomography is sample-starved, and the states one prepares live on a narrow, learnable manifold. A $k{=}0$ prior-only control shows that on concentrated families a prior estimate is already near-optimal, so ``high fidelity at few measurements'' can be family memorization rather than tomography; genuine measurement-efficiency needs a model that conditions on the measurements and demonstrably uses them. On a shared matrix-product-state (MPS) core parameterization we study two routes. Approach~A learns a generative prior over MPS cores with measurement-guided posterior inference (gold-standard-validated, but whose few-measurement accuracy the control shows is largely the prior). Approach~B, our main proposal, is a \emph{fixed-protocol amortized} MPS estimator trained once with a gauge-invariant fidelity loss; we deliberately do not rest it on a permutation-invariant set encoder (a plain MLP matches it). The decisive lever is the measurement design: motivated by the fact that local reduced density matrices determine a $χ$-MPS, conditioning on an \emph{informative local} Pauli set rather than random strings turns a modest, memorization-prone estimator into a high-fidelity one ($\approx\!0.95$, up to $+0.59$ over prior-only, decisively passing a shuffled-measurement control). A dropout ensemble, conformally recalibrated, gives $\approx\!90\%$-coverage intervals -- including for observables never measured, where a shot-based interval does not exist. Quality holds as the system grows (fidelity $0.90$ at $n{=}10$, gain \emph{growing} in $n$; $0.88$ at bond dimension $χ{=}4$), the parameterization is polynomial (native contraction to $20$ qubits), and we close the loop on IBM hardware ($5$ states at $0.97$ from hardware-measured Paulis).

Related papers

Verifying random matrix product states with autoregressive local measurements

Hyunho Cha, Subin Kim, Jungwoo Lee

Matrix product states (MPS) are a central language for one-dimensional quantum matter and a practical target for near-term quantum simulators and variational algorithms. Yet, while substantial effort has focused on preparing MPS with shallow circuits, scalable methods to \emph{verify} that a many-body device has actually produced the intended state remain underdeveloped. Direct fidelity estimation (DFE) relies only on local Pauli measurements, but in many-body settings it suffers an exponential classical overhead from the preprocessing needed to sample Pauli strings. We eliminate this obstacle by introducing an \emph{autoregressive} importance sampler that draws Pauli strings sequentially from efficiently computable conditional distributions, reducing the per-shot classical overhead to linear scaling in the number of qubits. We further develop a grouped extension that constructs qubit-wise commuting measurement settings via a \emph{sorting string} and simultaneously estimates the entire commuting group from a single setting, significantly reducing estimator variance while preserving efficient postprocessing. Our approach extends naturally to matrix product operators (MPO), enabling scalable verification of tensor-network states and observables in long one-dimensional quantum systems. We utilize random MPS as a natural benchmark for generic 1D entangled states.

quant-pharXiv:2604.16578•Apr 20, 2026•0 citations

Optimal quantum state tomography with local informationally complete measurements

Casey Jameson, Zhen Qin, Alireza Goldar +3 more

Quantum state tomography (QST) remains the gold standard for benchmarking and verification of near-term quantum devices. While QST for a generic quantum many-body state requires an exponentially large amount of resources, most physical quantum states are structured and can often be represented by a much smaller number of parameters, making efficient QST potentially possible. A prominent example is a matrix product state (MPS) or a matrix product density operator (MPDO), which is believed to represent most physical states generated by one-dimensional (1D) quantum devices. We study whether a general MPS/MPDO state can be recovered with bounded errors using only a number of state copies polynomial in the number of qubits, which is necessary for efficient QST. To make this question practically interesting, we assume only local measurements of qubits directly on the target state. By using a local symmetric informationally complete positive operator-valued measurement (SIC-POVM), we provide a positive answer to the above question for a variety of common many-body quantum states, including typical short-range entangled states, random MPS/MPDO states, and thermal states of one-dimensional Hamiltonians. In addition, we also provide an affirmative no answer for certain long-range entangled states such as a family of generalized GHZ states, but with the exception of target states that are known to have real-valued wavefunctions. Our answers are supported by a near-perfect agreement between an efficient calculation of the Cramer-Rao bound that rigorously bounds the sample complexity and numerical optimization results using a machine learning assisted maximal likelihood estimation (MLE) algorithm. This agreement also leads to an optimal QST protocol using local SIC-POVM that can be practically implemented on current quantum hardware and is highly efficient for most 1D physical states.

arXiv:2408.07115•Aug 13, 2024•1 citation

Efficient Fidelity Estimation with Few Local Pauli Measurements

Sun, Mingyu, Gabriel Waite, Michael J. Bremner +1 more

As quantum devices scale, quantifying how close an experimental state aligns with a target becomes both vital and challenging. Fidelity is the standard metric, but existing estimators either require full tomography or apply only to restricted state/measurement families. Huang, Preskill, and Soleimanifar (Nature Physics, 2025) introduced an efficient certification protocol for Haar-random states using only a polynomial number of non-adaptive, single-copy, local Pauli measurements. Here, we adopt the same data collection routine but recast it as a fidelity estimation protocol with rigorous performance guarantees and broaden its applicability. We analyze the bias in this estimator, linking its performance to the mixing time $τ$ of a Markov chain induced by the target state, and resolve the three open questions posed by Huang, Preskill, and Soleimanifar (Nature Physics, 2025). Our analysis extends beyond Haar-random states to state $t$-designs, states prepared by low-depth random circuits, physically relevant states and families of mixed states. We introduce a $k$-generalized local escape property that identifies when the fidelity estimation protocol is both efficient and accurate, and design a practical empirical test to verify its applicability for arbitrary states. This work enables scalable benchmarking, error characterization, and tomography assistance, supports adaptive quantum algorithms in high dimensions, and clarifies fundamental limits of learning from local measurements.

arXiv:2510.08155•Oct 9, 2025•0 citations

Scalable bayesian shadow tomography for quantum property estimation with set transformers

Cha, Hyunho, Kim, Wonjung, Jungwoo Lee

A scalable Bayesian machine learning framework is introduced for estimating scalar properties of an unknown quantum state from measurement data, which bypasses full density matrix reconstruction. This work is the first to integrate the classical shadows protocol with a permutation-invariant set transformer architecture, enabling the approach to predict and correct bias in existing estimators to approximate the true Bayesian posterior mean. Measurement outcomes are encoded as fixed-dimensional feature vectors, and the network outputs a residual correction to a baseline estimator. Scalability to large quantum systems is ensured by the polynomial dependence of input size on system size and number of measurements. On Greenberger-Horne-Zeilinger state fidelity and second-order Rényi entropy estimation tasks -- using random Pauli and random Clifford measurements -- this Bayesian estimator always achieves lower mean squared error than classical shadows alone, with more than a 99\% reduction in the few copy regime.

arXiv:2509.18674•Sep 23, 2025•0 citations

Uncertainty Quantification for Quantum Fidelity Estimation

Kuchibhotla Aditi, Stephen Becker

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.

quant-pharXiv:2610.10885•Oct 9, 2026•0 citations

Uncertainty-Aware Quantum State Tomography via Latent Posterior Factors: A Multi-Evidence Aggregation Approach

Aliyu Agboola Alege

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

•Mar 18, 2026•0 citations
Quantum state tomography is sample-starved, and the states one prepares live on a narrow, learnable manifold. A k=0k{=}0k=0 prior-only control shows that on concentrated families a prior estimate is already near-optimal, so ``high fidelity at few measurements'' can be family memorization rather than tomography; genuine measurement-efficiency needs a model that conditions on the measurements and demonstrably uses them. On a shared matrix-product-state (MPS) core parameterization we study two routes. ApproachA learns a generative prior over MPS cores with measurement-guided posterior inference (gold-standard-validated, but whose few-measurement accuracy the control shows is largely the prior). ApproachB, our main proposal, is a \emph{fixed-protocol amortized} MPS estimator trained once with a gauge-invariant fidelity loss; we deliberately do not rest it on a permutation-invariant set encoder (a plain MLP matches it). The decisive lever is the measurement design: motivated by the fact that local reduced density matrices determine a χχχ-MPS, conditioning on an \emph{informative local} Pauli set rather than random strings turns a modest, memorization-prone estimator into a high-fidelity one (≈ ⁣0.95\approx\!0.95≈0.95, up to +0.59+0.59+0.59 over prior-only, decisively passing a shuffled-measurement control). A dropout ensemble, conformally recalibrated, gives ≈ ⁣90%\approx\!90\%≈90%-coverage intervals -- including for observables never measured, where a shot-based interval does not exist. Quality holds as the system grows (fidelity 0.900.900.90 at n=10n{=}10n=10, gain \emph{growing} in nnn; 0.880.880.88 at bond dimension χ=4χ{=}4χ=4), the parameterization is polynomial (native contraction to 202020 qubits), and we close the loop on IBM hardware (555 states at 0.970.970.97 from hardware-measured Paulis).
Matrix product states (MPS) are a central language for one-dimensional quantum matter and a practical target for near-term quantum simulators and variational algorithms. Yet, while substantial effort has focused on preparing MPS with shallow circuits, scalable methods to \emph{verify} that a many-body device has actually produced the intended state remain underdeveloped. Direct fidelity estimation (DFE) relies only on local Pauli measurements, but in many-body settings it suffers an exponential classical overhead from the preprocessing needed to sample Pauli strings. We eliminate this obstacle by introducing an \emph{autoregressive} importance sampler that draws Pauli strings sequentially from efficiently computable conditional distributions, reducing the per-shot classical overhead to linear scaling in the number of qubits. We further develop a grouped extension that constructs qubit-wise commuting measurement settings via a \emph{sorting string} and simultaneously estimates the entire commuting group from a single setting, significantly reducing estimator variance while preserving efficient postprocessing. Our approach extends naturally to matrix product operators (MPO), enabling scalable verification of tensor-network states and observables in long one-dimensional quantum systems. We utilize random MPS as a natural benchmark for generic 1D entangled states.
Quantum state tomography (QST) remains the gold standard for benchmarking and verification of near-term quantum devices. While QST for a generic quantum many-body state requires an exponentially large amount of resources, most physical quantum states are structured and can often be represented by a much smaller number of parameters, making efficient QST potentially possible. A prominent example is a matrix product state (MPS) or a matrix product density operator (MPDO), which is believed to represent most physical states generated by one-dimensional (1D) quantum devices. We study whether a general MPS/MPDO state can be recovered with bounded errors using only a number of state copies polynomial in the number of qubits, which is necessary for efficient QST. To make this question practically interesting, we assume only local measurements of qubits directly on the target state. By using a local symmetric informationally complete positive operator-valued measurement (SIC-POVM), we provide a positive answer to the above question for a variety of common many-body quantum states, including typical short-range entangled states, random MPS/MPDO states, and thermal states of one-dimensional Hamiltonians. In addition, we also provide an affirmative no answer for certain long-range entangled states such as a family of generalized GHZ states, but with the exception of target states that are known to have real-valued wavefunctions. Our answers are supported by a near-perfect agreement between an efficient calculation of the Cramer-Rao bound that rigorously bounds the sample complexity and numerical optimization results using a machine learning assisted maximal likelihood estimation (MLE) algorithm. This agreement also leads to an optimal QST protocol using local SIC-POVM that can be practically implemented on current quantum hardware and is highly efficient for most 1D physical states.
As quantum devices scale, quantifying how close an experimental state aligns with a target becomes both vital and challenging. Fidelity is the standard metric, but existing estimators either require full tomography or apply only to restricted state/measurement families. Huang, Preskill, and Soleimanifar (Nature Physics, 2025) introduced an efficient certification protocol for Haar-random states using only a polynomial number of non-adaptive, single-copy, local Pauli measurements. Here, we adopt the same data collection routine but recast it as a fidelity estimation protocol with rigorous performance guarantees and broaden its applicability. We analyze the bias in this estimator, linking its performance to the mixing time τττ of a Markov chain induced by the target state, and resolve the three open questions posed by Huang, Preskill, and Soleimanifar (Nature Physics, 2025). Our analysis extends beyond Haar-random states to state ttt-designs, states prepared by low-depth random circuits, physically relevant states and families of mixed states. We introduce a kkk-generalized local escape property that identifies when the fidelity estimation protocol is both efficient and accurate, and design a practical empirical test to verify its applicability for arbitrary states. This work enables scalable benchmarking, error characterization, and tomography assistance, supports adaptive quantum algorithms in high dimensions, and clarifies fundamental limits of learning from local measurements.
A scalable Bayesian machine learning framework is introduced for estimating scalar properties of an unknown quantum state from measurement data, which bypasses full density matrix reconstruction. This work is the first to integrate the classical shadows protocol with a permutation-invariant set transformer architecture, enabling the approach to predict and correct bias in existing estimators to approximate the true Bayesian posterior mean. Measurement outcomes are encoded as fixed-dimensional feature vectors, and the network outputs a residual correction to a baseline estimator. Scalability to large quantum systems is ensured by the polynomial dependence of input size on system size and number of measurements. On Greenberger-Horne-Zeilinger state fidelity and second-order Rényi entropy estimation tasks -- using random Pauli and random Clifford measurements -- this Bayesian estimator always achieves lower mean squared error than classical shadows alone, with more than a 99\% reduction in the few copy regime.
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.
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