量子 Arimoto-Blahut 算法中全局最优性的条件

中文摘要 AI 辅助

广义 Arimoto--Blahut (AB) 算法在信息论和量子优化中被广泛使用，但单调的目标函数下降和数值稳定化并不能保证全局最优性。我们针对线性约束下凸可微目标函数，建立了满秩 AB 不动点全局最优性的充分必要条件。一个 AB 不动点是全局最优的，当且仅当在该点处，AB 更新方向与目标函数梯度之差属于约束法空间中的一个元素。相同的相容性条件刻画了单个 AB 更新与镜像下降 (MD) 更新之间的一致性，而路径等价性则要求该条件沿整个共同轨迹成立。因此，AB 算法可能遵循与 MD 不同的轨迹，但仍能达到全局最优。我们还基于可行方向导数开发了后验最优性证书，包括仅需目标函数评估的目标间隙的有限差分上界。对于去相位信道和去极化信道之间的信道相对熵，我们解析地确定了全局最小化器是唯一的满秩 AB 不动点，尽管路径等价性不成立。数值实验说明了 AB 和 MD 沿不同轨迹的收敛性，并验证了这些证书。相比之下，一个振幅阻尼示例表明，单调的 AB 迭代可能稳定在一个次优不动点，而有限差分证书可以检测到其非最优性。这些结果为评估量子 AB 算法中的全局最优性提供了结构性和可计算的判据。

英文摘要

Generalized Arimoto--Blahut (AB) algorithms are widely used in information theory and quantum optimization, but monotonic objective decrease and numerical stabilization do not guarantee global optimality. We establish necessary and sufficient conditions for the global optimality of full-rank AB fixed points for convex differentiable objectives under linear constraints. An AB fixed point is globally optimal if and only if the AB update direction and the objective gradient differ by an element of the constraint normal space at that point. The same compatibility condition characterizes agreement between individual AB and mirror-descent (MD) updates, whereas pathwise equivalence requires it along the entire common trajectory. Thus, an AB algorithm may follow a trajectory different from MD and still reach the global optimum. We also develop a posteriori optimality certificates based on feasible directional derivatives, including finite-difference upper bounds on the objective gap that require only objective evaluations. For channel relative entropy between dephasing and depolarizing channels, we analytically identify the global minimizer as the unique full-rank AB fixed point, although pathwise equivalence fails. Numerical experiments illustrate convergence of AB and MD along different trajectories and validate the certificates. By contrast, an amplitude-damping example shows that a monotone AB iteration can stabilize at a suboptimal fixed point, whose nonoptimality is detected by the finite-difference certificate. These results provide structural and computable criteria for assessing global optimality in quantum AB algorithms.

发表机构

• The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））
• International Quantum Academy (SIQA)（国际量子科学院）
• Nagoya University（名古屋大学）

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