发表机构
Yale University; The Ohio State University; Texas Tech University; Brown University（耶鲁大学; 俄亥俄州立大学; 德克萨斯理工大学; 布朗大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对应用于最大独立集问题的浅层量子近似优化算法，发现动力李代数理论的预测失效，训练经验硬度模型可高保真恢复地形缩放类别，凸显需开发更具经验依据的变分量子算法损失地形模型。
AI 中文摘要
变分量子算法（VQA）的动力李代数（DLA）理论预测，对于足够深的参数化电路，损失和梯度方差会呈指数级消失。本研究表明，当将量子近似优化算法（QAOA）应用于最大独立集（MIS）问题时，这些预测在浅层电路（尤其是常数深度）的 regime 中会出现严重偏差。在针对约23000个问题实例的大规模数值研究中，我们发现贫瘠高原极为罕见，而方差随系统大小呈多项式增长的地形（我们称之为“崎岖地形”）在各类图族中普遍存在。这种整体多项式增长既出现在通用低对称性随机图中，也出现在高度对称的顶点传递图中，表明基于DLA的方差预测无法描述该 regime 下的地形缩放规律。作为该理论的临时替代方案，我们训练经验硬度模型以预测QAOA-MIS的实例级硬度指标。尽管这些模型的泛化能力较差，但它们仍能以高保真度恢复正确的地形缩放类别（贫瘠高原vs.崎岖地形）。综上，我们的结果确定了针对MIS的浅层QAOA是一个典型场景，其中以渐近酉设计为中心的预测可能根本不足以更广泛地描述浅层变分量子算法，强调了开发更具经验依据的VQA损失地形模型的必要性。
英文摘要
The dynamical Lie algebraic (DLA) theory of variational quantum algorithms (VQAs) predicts commonplace exponentially vanishing loss and gradient variances for sufficiently deep parametrized circuits. In this work, we show that these predictions fail dramatically in the shallow-circuit (and particularly constant-depth) regime for the Quantum Approximate Optimization Algorithm (QAOA) applied to the maximum independent set (MIS) problem. In a large-scale numerical study across $\sim$23,000 problem instances, we find that barren plateaus are rare, while landscapes whose variances polynomially increase with system size---which we term "cragged terrains"---are common across graph families. This aggregate polynomial growth persists both for generic, low-symmetry random graphs and for highly symmetric vertex-transitive graphs, indicating that DLA-based variance predictions do not describe landscape scaling in this regime. As a stopgap alternative to the theory, we train empirical hardness models to predict instance-wise hardness metrics for QAOA-MIS. While these models generalize poorly, they nonetheless recover the correct landscape scaling class (barren plateau vs. cragged terrain) with high fidelity. Taken together, our results identify shallow QAOA for MIS as a prototypical setting in which asymptotic, unitary-design-centric predictions may be fundamentally insufficient to describe shallow variational quantum algorithms more broadly, emphasizing the need for more empirically-informed models of VQA loss landscapes.
Comments18 pages, 9 figures
