费米-狄拉克热测量：量子假设检验和半定规划的框架
Fermi-Dirac thermal measurements: A framework for quantum hypothesis testing and semidefinite optimization
AI总结：
该研究提出费米-狄拉克热测量作为量子假设检验和半定规划的新框架，通过热力学方法优化量子测量，构建了费米-狄拉克机模型。
摘要：
量子测量是恢复编码在量子态中的信息的手段。它们在量子假设检验中处于前沿位置，其目标是进行最优测量以得出正确结论。数学上，测量算子是厄米特的，其本征值在[0,1]区间内。通过注意到这一对每个本征值的约束与泡利不相容原理对费米子施加的约束相同，我们将每个测量算子的本征模式视为独立的有效费米子模式。在此视角下，量子假设检验中的各种目标函数可视为这些费米子占据数的总期望能量。通过固定温度并最小化总期望费米子自由能，我们发现这些修改目标函数的最优测量是费米-狄拉克热测量，其本征值由费米-狄拉克分布指定。在低温极限下，其性能接近于量子假设检验的最优测量，我们展示了其参数可通过经典或混合量子-经典优化算法学习。这导致了一种新的量子机器学习模型，称为费米-狄拉克机，由参数化的费米-狄拉克热测量组成，这是基于热态的量子玻尔兹曼机的替代方案。除了假设检验外，我们展示了如何利用这种方法解决一般的半定规划问题，从而在量子计算机上提出了一种新的半定规划范式，其目标是实现热测量而非准备热态。最后，我们提出了实现费米-狄拉克热测量的量子算法，并提出了二阶混合量子-经典优化算法。
英文摘要：
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.
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