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.

翻译：量子猜测工作量量化了在每次仅允许查询一个状态的情况下，猜测一个量子系综状态所需的最小查询次数。以往计算猜测工作量的方法基于标准半定规划技术，因此只能得到近似结果。相比之下，我们证明在均匀概率分布下计算量子比特系综的猜测工作量对应于求解一个二次分配问题，并提出一种算法：对于任何定义在离散环上的量子比特系综，该算法在经过有限步后能输出其猜测工作量的精确闭式表达式。尽管我们猜测工作量计算算法的一般复杂度随状态数量呈阶乘增长，但我们的主要贡献在于展示了对对称系综而言亚二次的超加速效应——这一场景对应于涡轮平衡问题最大化版本的三维类比。为发现此类对称性，我们提供了一种算法：对于任何定义在离散环上的点集，该算法在经过有限步后能输出其精确对称性。该对称性发现算法的复杂度随点数量呈多项式增长。作为示例，我们计算了正则和拟正则量子比特态集的猜测工作量。