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arXiv · 1106.5458

Maximum Likelihood, Minimum Effort

John A. Smolin·Jay M. Gambetta·Graeme Smith
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Abstract

We provide an efficient method for computing the maximum likelihood mixed quantum state (with density matrix $\rho$) given a set of measurement outcome in a complete orthonormal operator basis subject to Gaussian noise. Our method works by first changing basis yielding a candidate density matrix $\mu$ which may have nonphysical (negative) eigenvalues, and then finding the nearest physical state under the 2-norm. Our algorithm takes at worst $O(d^4)$ for the basis change plus $O(d^3)$ for finding $\rho$ where $d$ is the dimension of the quantum state. In the special case where the measurement basis is strings of Pauli operators, the basis change takes only $O(d^3)$ as well. The workhorse of the algorithm is a new linear-time method for finding the closest probability distribution (in Euclidean distance) to a set of real numbers summing to one.

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John A. Smolin, Jay M. Gambetta, Graeme Smith. 2011-06-27. Maximum Likelihood, Minimum Effort. https://doi.org/10.1103/physrevlett.108.070502

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