A Unified Framework for Sample Complexity of Structured Quantum State Tomography under Noisy Observations

Abstract:Quantum state tomography (QST) has attracted considerable attention due to its fundamental role in quantum information processing. In this paper, we develop a unified theoretical framework for analyzing the sample complexity of structured QST under noisy observations arising from state preparation noise, measurement noise, and finite-shot statistical noise. The proposed framework applies to a broad family of structured quantum-state classes, including general mixed states, sparse states, low-rank states, matrix product states (MPSs), matrix product operators (MPOs), projected entangled-pair states (PEPSs), and projected entangled-pair operators (PEPOs), while further introducing physically consistent structured models---including low-rank and sparse states, low-rank MPOs (LR-MPOs), and low-rank PEPOs (LR-PEPOs)---that simultaneously exploit low-dimensional structures and preserve the physical constraint. Within this framework, we derive unified non-asymptotic sample complexity guarantees for two constrained least-squares estimators under noisy observations: a noise-aware estimator that incorporates the calibrated noise model and a noise-unaware estimator based on the ideal Born measurement model. For the noise-aware estimator, we derive unified trace-norm recovery guarantees that explicitly characterize the dependence of the sample complexity on three fundamental quantities: the complexity of the underlying structured state class, the complexity of the measurement ensemble, and the state preparation and measurement noise levels. For the noise-unaware estimator, we establish a unified non-asymptotic recovery guarantee consisting of a statistical error term and an additional deterministic bias term arising from the mismatch between the assumed reconstruction model and the noisy observation process.

Quantum Physics (quant-ph); Signal Processing (eess.SP); Optimization and Control (math.OC)

arXiv:2608.05526 [quant-ph]

(or arXiv:2608.05526v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2608.05526 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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