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Fast state tomography with optimal error bounds
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Projected least squares (PLS) is an intuitive and numerically cheap technique for quantum state tomography. The method first computes the least-squares estimator (or a linear inversion estimator) and then projects the initial estimate onto the space of states. The main result of this paper equips this point estimator with a rigorous, non-asymptotic confidence region expressed in terms of the trace distance. The analysis holds for a variety of measurements, including 2-designs and Pauli measurements. The sample complexity of the estimator is comparable to the strongest convergence guarantees available in the literature and -- in the case of measuring the uniform POVM -- saturates fundamental lower bounds.The results are derived by reinterpreting the least-squares estimator as a sum of random matrices and applying a matrix-valued concentration inequality. The theory is supported by numerical simulations for mutually unbiased bases, Pauli observables, and Pauli basis measurements.
Forward citations
Cited by 2 Pith papers
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Predicting Features of Quantum Systems from Very Few Measurements
Random Clifford measurements produce a classical shadow of a quantum state that predicts M linear features using only O(log M) measurements, independent of system size, with a matching lower bound.
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On estimating operator norm distance, with optimal trace distance estimation when one state is pure
Rank-independent quantum estimators achieve Θ(1/ε) queries for operator-norm (and trace) distance when one state is pure, and Õ(1/ε^{3/2}) queries for general states, proving BQP-completeness.
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