AI 中文总结
研究将量子算法双边界过程视为相干量子薛定谔桥，利用阿哈罗诺夫双态矢量和庞特里亚金原理得出最优哈密顿量，应用此框架重构多种算法，将不同算法范式统一为选择边界条件和实现最优流的几何原理。
AI 中文摘要
量子算法本质上是双边界过程：准备输入态并选择输出态或子空间作为计算答案。我们将此视为相干量子薛定谔桥（QSB），它是薛定谔桥理论的纯态哈密顿量对应物，端点约束施加于态矢量，传输成本为二次控制作用。在此设置下，阿哈罗诺夫的双态矢量成为自然的最优控制对。庞特里亚金原理产生一个通用的最优哈密顿量，其弱值在测地规范中为纯虚数。将此框架应用于无结构搜索、周期性查找和矩阵运算，重构了格罗弗算法、肖尔算法背后的量子傅里叶变换以及量子奇异值变换（QSVT）。常见电路组件如预言机、扩散反射、受控相位和信号处理旋转，作为最优弱值漂移的李代数合成出现。此观点将不同算法范式统一为单一几何原理：算法设计是选择计算边界条件并实现相应最优流的问题。
英文摘要
Quantum algorithms are intrinsically two-boundary processes: an input state is prepared, and an output state or subspace is selected as the computational answer. We formulate this observation as a coherent Quantum Schrödinger Bridge (QSB), a pure-state Hamiltonian counterpart of Schrödinger bridge theory in which the endpoint constraint is imposed on state vectors and the transport cost is the quadratic control action. In this setting Aharonov's two-state vector becomes the natural optimal-control pair: a forward state from the input and a backward state from the target. Pontryagin's principle then yields a universal optimal Hamiltonian whose weak value is purely imaginary in the geodesic gauge. Thus weak values are not an auxiliary interpretation; they are the local response functions that quantify the drift of the pre-selected state toward the post-selected boundary. Applying this framework to unstructured search, periodicity finding, and matrix arithmetic, we reconstruct Grover's algorithm, the quantum Fourier transform underlying Shor's algorithm, and quantum singular value transformation (QSVT). The usual circuit components -- oracles, diffusion reflections, controlled phases, and signal-processing rotations -- emerge as Lie-algebraic syntheses of the optimal weak-value drift. This perspective unifies distinct algorithmic paradigms into a single geometric principle: algorithm design is the problem of choosing computational boundary conditions and realizing the corresponding optimal flow.
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