本篇文章主要探討一個源自量子通道搜尋，並且具有正交約束的最佳化問題。給定一對具有單位跡且對稱正定的量子態，我們的目標是尋找一個正交轉換，使得給定的輸入態能夠映射至對應的輸出態。此問題可被建構為在Stiefel 流形上求解的最小化問題。為求解此問題，我們採用並比較三種黎曼最佳化方法：黎曼梯度流、黎曼牛頓法，以及擬瞬態連續法。數值實驗顯示，三種方法皆能有效將黎曼梯度範數下降到零，表明其收斂至一個臨界點。其中，擬瞬態連續法在穩定性、準確性與計算效率之間展現出良好的平衡，為此類問題提供了一個實用的求解策略。

In this paper, we study an orthogonality-constrained optimization problem motivated by quantum channel search. Given a pair of symmetric positive definite quantum states with unit trace, the objective is to determine an orthogonal transformation that maps the input state to the output state. This task is formulated as minimizing the Frobenius norm of their difference, subject to orthogonality constraints on the Stiefel manifold. To solve this problem, we employ and compare three Riemannian optimization methods: Riemannian gradient flow, Riemannian Newton's method, and pseudo-transient continuation. Numerical experiments demonstrate that all methods successfully drive the Riemannian gradient norm to near zero, indicating convergence to stationary points. Among these approaches, the pseudo-transient continuation method achieves a favorable balance between stability, accuracy, and computational efficiency.

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