发表机构
University of Delaware; University of California, Riverside（特拉华大学; 加州大学河滨分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对树结构上的对称约化QAOA，提出一种多项式时间经典算法，可高效估计约化DLA，实现表达能力诊断与可控子系统证明，为变分量子算法的量子动力学分析提供经典方法。
AI 中文摘要
优化问题的经典等价形式未必能得到等价的量子算法。对于MaxCut问题，固定单个顶点的值可消除简单的全局对称性，且不改变底层优化问题，但会显著改变量子近似优化算法（Quantum Approximate Optimization Algorithm，QAOA）产生的量子动力学。这些动力学由电路的动力学李代数（dynamical Lie algebra，DLA）刻画，而其直接构造可能呈指数级昂贵。本文表明，对于树结构上的对称约化QAOA，可仅从图高效获取约化DLA的大量信息。我们提出一种多项式时间经典算法，该算法利用最短路径结构和度奇偶性递归区分顶点。当所有顶点被单独区分时，该方法可确定完整的约化DLA，并在对应图条件下，证明约化QAOA ansatz具有最大表达能力；即使未实现完全区分，该算法也能识别嵌入子代数、给出DLA维数的严格下界，并证明可控子系统的存在。我们的理论与实验结果表明，可利用经典图结构在量子硬件上运行变分算法前，诊断甚至潜在指导其量子动力学。
英文摘要
Classically equivalent formulations of an optimization problem need not lead to equivalent quantum algorithms. For MaxCut, fixing the value of a single vertex removes a simple global symmetry without changing the underlying optimization problem, yet it can substantially alter the quantum dynamics generated by the Quantum Approximate Optimization Algorithm (QAOA). These dynamics are captured by the circuit's dynamical Lie algebra (DLA), whose direct construction can become exponentially expensive. Here we show that, for symmetry-reduced QAOA on trees, substantial information about the reduced DLA can instead be obtained efficiently from the graph alone. We introduce a polynomial classical algorithm that recursively distinguishes vertices using shortest path structure and degree parity. When all vertices are resolved individually, the method determines the complete reduced DLA and, under the corresponding graph conditions, certifies maximal expressibility of the reduced QAOA ansatz. Even when full resolution is not achieved, the algorithm identifies embedded subalgebras, provides rigorous lower bounds on DLA dimension, and certifies controllable subsystems. Our theoretical and experimental results show how classical graph structure can be used to diagnose and potentially guide the quantum dynamics of variational algorithms before running them on quantum hardware.
Comments23 pages, 5 figures, 2 tables
