AI 中文总结
研究将N个相同热量子比特中选出的M个冷却到最低共同局部温度的问题，提出由被动重排和复哈达玛变换组成的最优协议，推导出相关标准并分类可行岛，还得出冷却曲线等结论，证明辅助量子比特必要性及联合冷却优势。
AI 中文摘要
我们解决了将从N个相同热量子比特中选出的M个量子比特冷却到幺正性允许的最低共同局部温度的封闭系统问题。最优协议由两个概念上不同的步骤组成。首先，被动重排将初始状态的最大本征值分配到最低汉明重量的目标扇区，从而使总目标能量最小化。其次，在每个固定汉明重量子空间内进行仅针对目标的复哈达玛变换，使单量子比特目标边缘分布均匀化，而不改变任何目标扇区概率或总能量。因此，施加共同局部温度既不消耗冷却深度也不消耗额外功：对于每个N>M和每个初始温度，受约束的最优值与无约束的被动最小值一致。然而，复哈达玛校正在电路层面可能成本高昂。我们因此推导出一个精确的算术标准，用于确定何时仅通过与温度无关的计算基排列就能达到相同的最优值，并对M+2≤N≤128时产生的有限尺寸可行岛进行了详尽分类。在孤立温度下，通过不同热本征值壳之间的数值抵消可能会出现进一步的最优排列。这些替代实现方式可能会降低实现复杂度，但无法改善通用协议已经达到的冷却曲线。我们还推导出了精确的冷却曲线，证明至少两个辅助量子比特对于非平凡冷却既必要又充分，并表明联合多目标冷却可以严格优于并行单目标策略。
英文摘要
We solve the closed-system problem of cooling $M$ qubits, selected from $N$ identical thermal qubits, to the lowest common local temperature allowed by unitarity. The optimal protocol consists of two conceptually distinct steps. First, a passive rearrangement assigns the largest eigenvalues of the initial state to target sectors of lowest Hamming weight, thereby minimizing the total target energy. Second, a target-only complex-Hadamard transformation within each fixed-Hamming-weight subspace equalizes the one-qubit target marginals without changing any target-sector probability or the total energy. Consequently, imposing a common local temperature costs neither cooling depth nor additional work: the constrained optimum coincides with the unconstrained passive minimum for every $N>M$ and every initial temperature. The complex-Hadamard correction may nevertheless be costly at the circuit level. We therefore derive an exact arithmetic criterion for when the same optimum can be attained by a temperature-independent computational-basis permutation alone, and exhaustively classify the resulting finite-size islands of feasibility for $M+2\leq N\leq 128$. At isolated temperatures, further optimal permutations can arise through numerical cancellations between different thermal eigenvalue shells. These alternative realizations may reduce implementation complexity, but they cannot improve the cooling curve already attained by the universal protocol. We also derive the exact cooling curve, prove that at least two ancillary qubits are necessary and sufficient for nontrivial cooling, and show that joint many-target cooling can strictly outperform parallel single-target strategies.
Comments12 + 8 pages, 14 figures
