发表机构
The University of Osaka; RIKEN; Kyoto University（大阪大学; 理化学研究所; 京都大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明量子化贝里相位判定在逆多项式谱间隙下为BQP完全，并给出恒定间隙下的经典多项式算法，揭示谱间隙而非精度是经典可处理性的关键。
AI 中文摘要
贝里相位是表征量子多体系统几何与拓扑性质的基本几何量。此前的研究表明，在给定近似初始基态的试探态的情况下，以逆多项式精度估计贝里相位具有超多项式量子优势。然而，对于由对称性量子化、可在恒定精度下区分的贝里相位，这种困难性是否仍然成立此前尚不清楚。在本工作中，我们刻画了量子化贝里相位估计的计算复杂性。首先，我们证明当谱间隙为逆多项式小时，判定贝里相位是精确为$0$还是$\pi$是$\mathsf{BQP}$完全的。这一结果意味着，在假设$\mathsf{BPP}\neq \mathsf{BQP}$的前提下，区分量子化贝里相位仍具有超多项式量子优势。为证明该结果，我们引入了一种编码，将任何量子计算的输出映射到精确为$0$或$\pi$的贝里相位上。其次，我们将这种困难性扩展到物理上受启发的二维方格上的哈密顿量，包括海森堡和XY相互作用。第三，我们为固定维晶格上具有恒定谱间隙的几何局域哈密顿量提供了一种经典多项式时间算法。因此，我们识别出谱间隙是经典可处理性的关键资源，而非精度。这些结果确立了计算在对称性和间隙保持形变下不变的量子化拓扑不变量的超多项式量子优势，并为量子物理中的实际量子优势提供了新的步骤。
英文摘要
The Berry phase is a fundamental geometric quantity for characterizing the geometry and topology of quantum many-body systems. Previous work established a super-polynomial quantum advantage in Berry phase estimation at inverse-polynomial precision, given an ansatz state approximating the initial ground state. However, whether this hardness persists for Berry phases quantized by symmetry, which can be distinguished at constant precision, remained open. In this work, we characterize the computational complexity of quantized Berry phase estimation. First, we prove that deciding whether the Berry phase is exactly $0$ or $π$ is $\mathsf{BQP}$-complete when the spectral gap is inverse-polynomially small. This result implies that distinguishing the quantized Berry phase still has a super-polynomial quantum advantage, assuming $\mathsf{BPP}\neq \mathsf{BQP}$. To prove this result, we introduce an encoding that maps the output of any quantum computation to a Berry phase that is exactly $0$ or $π$. Second, we extend this hardness to physically motivated Hamiltonians on a 2D square lattice, including Heisenberg and XY interactions. Third, we provide a classical polynomial-time algorithm for geometrically local Hamiltonians on fixed-dimensional lattices with constant spectral gaps. Therefore we identify that the spectral gap is a key resource for the classical tractability, rather than precision. These results establish a super-polynomial quantum advantage for computing a quantized topological invariant that is unchanged under symmetry- and gap-preserving deformations, and provide new steps towards practical quantum advantage in quantum physics.
Comments24 pages, 1 figure
