发表机构
Center for Advanced Systems Understanding (CASUS); Helmholtz-Zentrum Dresden-Rossendorf (HZDR); Leiden University; Covestro Deutschland AG; Dahlem Center for Complex Quantum Systems, Freie Universität Berlin; Walrus Computing; Institute of Artificial Intelligence, Technische Universität Dresden; QuSoft, HIMS & IvI, University of Amsterdam; Center for Scalable Data Analytics and Artificial Intelligence (ScaDS.AI) Dresden/Leipzig; Technical University Dresden（高级系统理解中心; 德累斯顿-罗斯多夫亥姆霍兹中心; 莱顿大学; 科思创德国股份公司; 柏林自由大学达勒姆复杂量子系统中心; 鲸鱼计算公司; 德累斯顿工业大学人工智能研究所; 阿姆斯特丹大学QuSoft、HIMS及IvI; 德累斯顿/莱比锡可扩展数据分析与人工智能中心; 德累斯顿工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究非厄米转相关哈密顿量的量子算法，通过可解模型和二次量子化分析，揭示TC关联子可改善基组收敛（从N^-1/2到N^-3/2）并降低查询复杂度，为精度-成本权衡提供理论框架。
AI 中文摘要
电子-电子库仑奇异性迫使电子波函数呈现尖点结构。这种尖点结构限制了波函数的正则性，并减缓了在光滑基组中展开的收敛速度，需要较大的基组才能达到目标精度。转相关（TC）方法将这种短程行为纳入哈密顿量，从而改善变换后波函数的正则性，并加速向完全基组极限的收敛。由此产生的哈密顿量是非厄米且非正规的，这限制了标准量子算法的直接适用性。对于此类非正规算符，近期若干量子算法的复杂度显式依赖于与本征向量条件数和本征值敏感性相关的量。理解这些量对于评估算法复杂度和量子资源需求至关重要。在本工作中，我们分两个阶段解决这些问题。我们首先研究一个可解模型，该模型提供了一个受控环境，用于隔离转相关对基组收敛和本征向量条件数的影响。在此环境中，我们研究两种不同的单参数Jastrow关联子。我们推导了渐近能量误差标度：对于裸投影为$N^{-1/2}$，而对于两种TC关联子在一般固定参数下均为$N^{-3/2}$，其中$N$表示基组大小。此外，我们表明这种更快的收敛可以在渐近查询复杂度上界中产生多项式改进。然后，我们将分析扩展到二次量子化的电子结构哈密顿量，并使用粒子数扇区诊断来区分目标扇区与辅助扇区的条件数。我们的工作为理解TC哈密顿量中核心的精度-成本权衡提供了一个框架。
英文摘要
The electron--electron Coulomb singularity forces cusp structure in electronic wavefunctions. This cusp structure limits wavefunction regularity and slows the convergence of expansions in smooth bases, requiring large basis sets to reach a target accuracy. The transcorrelated (TC) method incorporates this short-range behavior into the Hamiltonian, thereby improving the regularity of the transformed wavefunction and accelerating convergence toward the complete-basis-set limit. The resulting Hamiltonian is non-Hermitian and non-normal, which limits the direct applicability of standard quantum algorithms. For such non-normal operators, the complexity of several recent quantum algorithms depends explicitly on quantities related to eigenvector conditioning and eigenvalue sensitivity. Understanding these quantities is essential for assessing both algorithmic complexity and quantum resource requirements. In this work, we address these issues in two stages. We first study a solvable model, which provides a controlled setting for isolating how transcorrelation affects basis convergence and eigenvector conditioning. Within this setting, we investigate two distinct one-parameter Jastrow correlators. We derive the asymptotic energy-error scaling: $N^{-1/2}$ for the bare projection and $N^{-3/2}$ for both TC correlators at generic fixed parameters, with $N$denoting the basis size. Furthermore, we show that this faster convergence can yield a polynomial improvement in the asymptotic query-complexity upper bound. We then extend the analysis to second-quantized electronic-structure Hamiltonians and use particle-number-sector diagnostics to distinguish target-sector from auxiliary-sector conditioning. Our work provides a framework for understanding the central accuracy--cost trade-off in TC Hamiltonians.
Comments35 + 29 pages, 15 figures, 1 table
