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arXiv · 2112.01666

Computing the quantum guesswork: a quadratic assignment problem

Michele Dall'Arno·Francesco Buscemi·Takeshi Koshiba
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Abstract

The quantum guesswork quantifies the minimum number of queries needed to guess the state of a quantum ensemble if one is allowed to query only one state at a time. Previous approaches to the computation of the guesswork were based on standard semi-definite programming techniques and therefore lead to approximated results. In contrast, we show that computing the quantum guesswork of qubit ensembles with uniform probability distribution corresponds to solving a quadratic assignment problem and we provide an algorithm that, upon the input of any qubit ensemble over a discrete ring, after finitely many steps outputs the exact closed-form expression of its guesswork. While in general the complexity of our guesswork-computing algorithm is factorial in the number of states, our main result consists of showing a more-than-quadratic speedup for symmetric ensembles, a scenario corresponding to the three-dimensional analog of the maximization version of the turbine-balancing problem. To find such symmetries, we provide an algorithm that, upon the input of any point set over a discrete ring, after finitely many steps outputs its exact symmetries. The complexity of our symmetries-finding algorithm is polynomial in the number of points. As examples, we compute the guesswork of regular and quasi-regular sets of qubit states.

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Michele Dall'Arno, Francesco Buscemi, Takeshi Koshiba. 2021-12-03. Computing the quantum guesswork: a quadratic assignment problem. https://doi.org/10.26421/qic23.9-10-1

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