The Berry phase is a fundamental quantity in the classification of topological phases of matter. In this paper, we present a new quantum algorithm and several complexity-theoretical results for the Berry phase estimation (BPE) problems. Our new quantum algorithm achieves BPE in a more general setting than previously known quantum algorithms, with a theoretical guarantee. For the complexity-theoretic results, we consider three cases. First, we prove $\mathsf{BQP}$-completeness when we are given a guiding state that has a large overlap with the ground state. This result establishes an exponential quantum speedup for estimating the Berry phase. Second, we prove $\mathsf{dUQMA}$-completeness when we have $\textit{a priori}$ bound for ground state energy. Here, $\mathsf{dUQMA}$ is a variant of the unique witness version of $\mathsf{QMA}$ (i.e., $\mathsf{UQMA}$), which we introduce in this paper, and this class precisely captures the complexity of BPE without the known guiding state. Remarkably, this problem turned out to be the first natural problem contained in both $\mathsf{UQMA}$ and $\mathsf{co}$-$\mathsf{UQMA}$. Third, we show $\mathsf{P}^{\mathsf{dUQMA[log]}}$-hardness and containment in $\mathsf{P}^{\mathsf{PGQMA[log]}}$ when we have no additional assumption. These results advance the role of quantum computing in the study of topological phases of matter and provide a pathway for clarifying the connection between topological phases of matter and computational complexity.

翻译：贝里相位是拓扑物相分类中的一个基本物理量。本文针对贝里相位估计问题，提出了一种新的量子算法并给出了若干复杂性理论结果。我们提出的新量子算法在比现有量子算法更一般的设定下实现了贝里相位估计，并具有理论保证。在复杂性理论方面，我们研究了三种情况：首先，当给定与基态具有较大重叠的引导态时，我们证明了该问题是𝖡𝖰𝖯-完全的，这一结果确立了贝里相位估计的指数级量子加速。其次，当存在基态能量的先验界时，我们证明了该问题是𝖽𝖴𝖰𝖬𝖠-完全的。其中𝖽𝖴𝖰𝖬𝖠是本文引入的𝖴𝖰𝖬𝖠（即唯一见证版本的𝖰𝖬𝖠）的变体，该复杂性类精确刻画了在缺乏已知引导态时贝里相位估计的计算复杂性。值得注意的是，该问题成为首个被证明同时属于𝖴𝖰𝖬𝖠与𝖼𝗈-𝖴𝖰𝖬𝖠的自然问题。第三，在无额外假设的情况下，我们证明了该问题的𝖯^𝖽𝖴𝖰𝖬𝖠[𝗅𝗈𝗀]-困难性，并证明其包含于𝖯^𝖯𝖦𝖰𝖬𝖠[𝗅𝗈𝗀]中。这些研究成果推进了量子计算在拓扑物相研究中的作用，并为阐明拓扑物相与计算复杂性之间的联系提供了路径。