Optimal, reliable estimation of quantum states

By Robin Blume-Kohout. 2006

384 citations recorded by the source corpus.

Accurately inferring the state of a quantum device from the results of measurements is a crucial task in building quantum information processing hardware. The predominant state estimation procedure, maximum likelihood estimation (MLE), generally reports an estimate with zero eigenvalues. These cannot be justified. Furthermore, the MLE estimate is incompatible with error bars, so conclusions drawn from it are suspect. I propose an alternative procedure, Bayesian mean estimation (BME). BME never yields zero eigenvalues, its eigenvalues provide a bound on their own uncertainties, and under certain circumstances it is provably the most accurate procedure possible. I show how to implement BME numerically, and how to obtain natural error bars that are compatible with the estimate. Finally, I briefly discuss the differences between Bayesian and frequentist estimation techniques.

Subfields: Artificial Intelligence

View Paper: https://www.semanticscholar.org/paper/27231f749bb4ee8fa7b94c059be1febb41dbe84b

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In Artificial Intelligence

Citations in the corpus

Quantum tomography via compressed sensing: error bounds, sample complexity and efficient estimators (2012, methodology): “Blume-Kohout’s hedged MLE [53] and Bayesian mean estimation [52] are good examples of this type of estimator, as are the minimax estimator of Ref.”
Quantum metrology with nonclassical states of atomic ensembles (2016, methodology): “Different techniques have been developed for full state tomography (Blume-Kohout, 2010; Paris and Řeháček, 2004; Schmied and Treutlein, 2011), estimating all spin moments up to order N .”
Hamiltonian Simulation by Qubitization (2016, methodology): “applications of Table.2. As Hermitian operators are also quantum observables, our methodology also oers a new approach for implementing arbitrary functions of measurement operators on quantum states [1,13,24,38]. We conjecture that this application would be optimal, which is supported by our optimal”
Predicting many properties of a quantum system from very few measurements (2020, neutral): “[5, 7, 34, 39] and references therein.”
Quantum certification and benchmarking (2019, neutral): “More recently, techniques to obtain more rigorous region estimates have appeared including Bayesian credibility regions [11, 33] and conﬁdence regions [12, 23, 99] where the former are usually smaller but depend strongly upon the Bayesian prior.”
Quantum computing with trapped ions (2008, background): “Even though maximum likelihood estimation has been criticized (Blume-Kohout, 2006) for being less accurate than Bayesian estimation techniques (Paris and Rehacek, 2004), it has the practical merit of being easily implemented in experiments.”