Computing the quantum guesswork: a quadratic assignment problem

TL;DR

This paper introduces an exact algorithm for computing quantum guesswork of qubit ensembles by formulating it as a quadratic assignment problem, with a focus on symmetric ensembles where the method is significantly faster.

Contribution

It establishes a novel connection between quantum guesswork and quadratic assignment problems, providing an exact, efficient algorithm for symmetric ensembles and symmetry detection.

Findings

Exact guesswork computation for qubit ensembles

Quadratic speedup for symmetric ensembles

Algorithm for symmetry detection in point sets

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.