Quantum measurements are the means by which we recover messages encoded into quantum states. They are at the forefront of quantum hypothesis testing, wherein the goal is to perform an optimal measurement for arriving at a correct conclusion. Mathematically, a measurement operator is Hermitian with eigenvalues in [0,1]. By noticing that this constraint on each eigenvalue is the same as that imposed on fermions by the Pauli exclusion principle, we interpret every eigenmode of a measurement operator as an independent effective fermionic mode. Under this perspective, various objective functions in quantum hypothesis testing can be viewed as the total expected energy associated with these fermionic occupation numbers. By instead fixing a temperature and minimizing the total expected fermionic free energy, we find that optimal measurements for these modified objective functions are Fermi-Dirac thermal measurements, wherein their eigenvalues are specified by Fermi-Dirac distributions. In the low-temperature limit, their performance closely approximates that of optimal measurements for quantum hypothesis testing, and we show that their parameters can be learned by classical or hybrid quantum-classical optimization algorithms. This leads to a new quantum machine-learning model, termed Fermi-Dirac machines, consisting of parameterized Fermi-Dirac thermal measurements-an alternative to quantum Boltzmann machines based on thermal states. Beyond hypothesis testing, we show how general semidefinite optimization problems can be solved using this approach, leading to a novel paradigm for semidefinite optimization on quantum computers, in which the goal is to implement thermal measurements rather than prepare thermal states. Finally, we propose quantum algorithms for implementing Fermi-Dirac thermal measurements, and we also propose second-order hybrid quantum-classical optimization algorithms.

翻译：量子测量是我们从量子态中恢复编码信息的手段。在量子假设检验中，其核心目标是执行最优测量以获得正确结论。从数学上看，测量算符是厄米算符，其特征值位于[0,1]区间内。通过注意到每个特征值的约束条件与泡利不相容原理对费米子的限制相同，我们将测量算符的每个本征模式解释为独立的等效费米子模式。在此视角下，量子假设检验中的各类目标函数可视为与这些费米子占据数相关的总期望能量。通过固定温度并最小化总期望费米子自由能，我们发现针对这些修正目标函数的最优测量是费米-狄拉克热测量，其特征值由费米-狄拉克分布确定。在低温极限下，其性能可紧密逼近量子假设检验的最优测量，并且我们证明其参数可通过经典或混合量子-经典优化算法学习。这催生了一种新的量子机器学习模型——费米-狄拉克机，它由参数化的费米-狄拉克热测量构成，成为基于热态的量子玻尔兹曼机的替代方案。除假设检验外，我们展示了如何利用该方法求解一般半定优化问题，从而为量子计算机上的半定优化开辟了新范式：其目标在于实现热测量而非制备热态。最后，我们提出了实现费米-狄拉克热测量的量子算法，以及二阶混合量子-经典优化算法。