AI 中文总结
针对经典优化问题全局最优解计数的#P困难问题，开发基于CTPQ态的CTPQsd#量子算法，利用探针退相干与简并度的关系，通过小型探针测量实现高效计数，模拟验证了算法有效性。
AI 中文摘要
计数经典优化问题的全局最优解是#P困难任务。我们开发了基于正则热纯量子（CTPQ）态的简并计数（CTPQsd#）算法，该算法仅通过测量小型探针S即可确定经典优化问题P的全局最优解数量，无需找到单个最小值。该方法利用当S和P共同处于CTPQ态时，S的退相干度量与P的简并度之间的微扰关系。我们提供了首个数值演示，证明该关系可用于计数全局最小值，将其应用于对角随机能量哈密顿量编码的问题，作为经典二元优化问题的最大非结构化测试平台。对最多包含20个问题量子比特的经典模拟，量化了算法对CTPQ态温度、哈密顿量能量范围、问题规模和简并度变化的敏感性。我们确定了用于确定精确简并度的温度阈值，并识别出第二个更低的阈值，该阈值提供了一个温度窗口，可在用户定义的能量容差内计数近简并最小值。通过将测量限制在S上，该协议将对指数级大的问题希尔伯特空间的层析成像，替换为对仅由四个量子比特表示的小型探针的层析成像。
英文摘要
Counting the global optima of a classical optimization problem is a #P-hard task. We develop the canonical thermal pure quantum (CTPQ) state-based degeneracy counting (CTPQsd#) algorithm that determines the number of global optima of a classical optimization problem P by measuring only a small probe S, without finding individual minima. The method exploits a perturbative relation between the decoherence measure of S and the degeneracy of P when S and P are together in a CTPQ state. We provide the first numerical demonstration that this relation can be used to count the global minima, applying it to problems encoded by diagonal random-energy Hamiltonians as a maximally unstructured testbed for classical binary optimization problems. Classical simulations of up to 20 problem qubits quantify the algorithm's sensitivity to variations in the temperature of the CTPQ state, the Hamiltonian energy range, the problem size, and degeneracy. We establish the temperature threshold for determining the exact degeneracy and identify a second, lower threshold that provides a temperature window to count near-degenerate minima within a user-defined energy tolerance. By confining measurement to S, the protocol replaces tomography over the exponentially large problem Hilbert space with tomography over a small probe represented by only four qubits.
