用于变分量子算法的Krylov - 李代数：对表达能力和可训练性的几何、深度感知洞察
Krylov-Lie Algebras for Variational Quantum Algorithms: Geometric, Depth-Aware Insights into Expressivity and Trainability
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中文总结 AI 辅助
研究变分量子算法损失景观问题，引入Krylov代数为其景观理论框架，用Krylov - 李代数和群近似可达流形，推导方差公式，分析哈罗收敛情况，表明非哈罗贡献或可缓解贫瘠高原。
中文摘要 AI 辅助
变分量子算法是近期量子计算的主要方法，但其效用受损失景观中贫瘠高原等问题限制。现有景观理论有局限，本文引入Krylov代数作为变分量子算法景观理论框架，展示了相关近似方法，推导方差公式，分析哈罗收敛情况，表明非哈罗贡献或可缓解贫瘠高原。
英文摘要
Variational quantum algorithms (VQAs) are a leading approach to near-term quantum computation, but their utility is limited by barren plateaus and other pathologies in their loss landscapes. Existing landscape theories based on dynamical Lie algebras, Jordan-algebraic Wishart systems, approximate t-designs, and Haar-random circuits are foundational, but they often neglect the finite-depth geometry of realistic ansätze and are therefore ill-suited to the shallow-depth regime, where VQAs are poor approximators of 2-designs and trainability is most feasible. This work introduces Krylov algebras, algebraic structures induced by the Krylov span of a finite generator set acting on one or more seed vectors, as a framework for VQA landscape theory. We show that VQA reachable manifolds can be approximated in a numerically robust, geometrically faithful fashion by Krylov-Lie algebras and groups, and that these structures induce canonical invariant measures for computing expectation values and variances under general sampling measures. In particular, we derive weighted non-Haar variance formulas that recover the usual Lie-algebraic Haar formulas as a special case while isolating non-Haar effects into explicit correction terms. We also show that the common heuristic that sufficiently deep circuit ensembles must converge to Haar fails in general without additional hypotheses, identify concrete obstructions to naive Haar convergence, and recover convergence under natural necessary and sufficient ergodic conditions. Lastly, our formulas further imply that non-Haar contributions to landscape statistics may mitigate barren plateaus by reweighting the visible sectors of the loss landscape, suggesting that VQAs may be more trainable than recent literature has posited.
发表机构
- Mathematics and Statistics（数学与统计学）
机构由 AI 辅助整理，请以论文原文为准。
