Papers › Maximum Likelihood, Minimum Effort

Maximum Likelihood, Minimum Effort

27 Jun 2011arXiv:1106.5458links table onlyarchive 2025-07-28

John A. Smolin, Jay M. Gambetta, Graeme Smith

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We provide an efficient method for computing the maximum likelihood mixed quantum state (with density matrix ρ) 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 μ which may have nonphysical (negative) eigenvalues, and then finding the nearest physical state under the 2-norm. Our algorithm takes at worst O(d⁴) for the basis change plus O(d³) for finding ρ 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(d³) 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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