Fine-grained spectral properties of quantum Hamiltonians, including both eigenvalues and their multiplicities, provide useful information for characterizing many-body quantum systems as well as for understanding phenomena such as topological order. Extracting such information with small additive error is $\#\textsf{BQP}$-complete in the worst case. In this work, we introduce QFAMES (Quantum Filtering and Analysis of Multiplicities in Eigenvalue Spectra), a quantum algorithm that efficiently identifies clusters of closely spaced dominant eigenvalues and determines their multiplicities under physically motivated assumptions, which allows us to bypass worst-case complexity barriers. QFAMES also enables the estimation of observable expectation values within targeted energy clusters, providing a powerful tool for studying quantum phase transitions and other physical properties. We validate the effectiveness of QFAMES through numerical demonstrations, including its applications to characterizing quantum phases in the transverse-field Ising model and estimating the ground-state degeneracy of a topologically ordered phase in the two-dimensional toric code model. We also generalize QFAMES to the setting of mixed initial states. Our approach offers rigorous theoretical guarantees and significant advantages over existing subspace-based quantum spectral analysis methods, particularly in terms of the sample complexity and the ability to resolve degeneracies.

翻译：量子哈密顿量的精细谱特性，包括特征值及其多重性，为表征多体量子系统以及理解拓扑序等现象提供了有用信息。在最坏情况下，以微小加性误差提取此类信息是 $\#\textsf{BQP}$-完全的。在本工作中，我们提出了QFAMES（量子滤波与特征值谱多重性分析），这是一种量子算法，能够在物理动机的假设下高效识别紧密相邻的主导特征值簇并确定其多重性，从而绕过了最坏情况下的复杂性壁垒。QFAMES还能够估计目标能量簇内的可观测量期望值，为研究量子相变及其他物理性质提供了有力工具。我们通过数值演示验证了QFAMES的有效性，包括其在表征横场伊辛模型中的量子相以及估计二维环面编码模型中拓扑有序相的基态简并度方面的应用。我们还将QFAMES推广到混合初始态的情形。我们的方法提供了严格的理论保证，并且在样本复杂度及分辨简并度的能力方面，相较于现有的基于子空间的量子谱分析方法具有显著优势。