Abstract
Projected least squares is an intuitive and numerically cheap technique for quantum state tomography: compute the least-squares estimator and project it onto the space of states. The main result of this paper equips this point estimator with rigorous, non-asymptotic convergence guarantees expressed in terms of the trace distance. The estimator’s sample complexity is comparable to the strongest convergence guarantees available in the literature and—in the case of the uniform POVM—saturates fundamental lower bounds. Numerical simulations support these competitive features.
Highlights
Quantum state tomography is the task of reconstructing a quantum state from experimental measurement data
In this work we present an in depth analysis of an alternative method, the projected least squares (PLS) estimator, and show that it improves on the status quo in several significant directions
We focus on explaining the novel results and the key techniques used in establishing them
Summary
Quantum state tomography is the task of reconstructing a quantum state from experimental measurement data. Our main theoretical results show that PLS achieves fundamental lower bounds [20] for tomography with separate measurements: to reconstruct an arbitrary d dimensional state ρ of rank-r with accuracy in trace distance it suffices to measure r2d −2log d independent samples with a two-design measurement, or r2d −2 samples with a covariant measurement. This sampling rate improves upon existing results [21] and is competitive with the most powerful techniques in the literature [13, 22].
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