Quantum Filtering and Analysis of Multiplicities in Eigenvalue Spectra

TL;DR

QFAMES introduces a quantum spectral filtering framework that can identify clusters of dominant energy eigenvalues and exactly recover their multiplicities under a uniform overlap condition, while also enabling observable estimation within identified energy clusters. By combining Gaussian energy filtering with cross-correlations from multiple initial states via a generalized Hadamard test, it produces a filtered density matrix $\mathcal{G}(\theta)$ whose peaks reveal eigenvalue centers and whose rank yields degeneracies. The authors provide rigorous guarantees on sample and time complexity, extending to clusters of near-degenerate eigenvalues and to the estimation of projected observables within degenerate subspaces. Numerical demonstrations on TFIM and the toric code illustrate the method’s ability to resolve degeneracies and extract physically meaningful observables, highlighting its potential for studying quantum phase transitions and topological order on near-term devices.

Abstract

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. 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.

Figures (9)

• Figure 1: Illustration of the QFAMES algorithm. Given the Hamiltonian and two sets of initial states prepared by circuits $\{U_l\}_{l\in [L]}$ and $\{V_r\}_{r\in [R]}$, we measure quantities of the form $\mathcal{Z}_{l,r}(t_n) = \braket{0 | U^{\dagger}_l e^{-\mathbf{i} H t_n} V_{r} | 0}$ using the generalized Hadamard test circuit. The collected quantum data are then processed classically to (1) estimate the locations of energy eigenstate clusters through searching and blocking, and (2) compute the multiplicities of the probed clusters.
• Figure 2: Diagram of the QFAMES algorithm for post-processing the 3-tensor generated from quantum data of the generalized Hadamard test circuit. Left/right tensors $U_l$ and $V_r$ prepare initial-state families; the middle leg represents time evolution $e^{-\mathrm{i}Ht}$ and data acquisition of cross-correlators $\mathcal{Z}_{l,r}(t_n)$. A filter $a_T(t)$ and discrete Fourier transform yield $\mathcal{G}(\theta)$, which is post-processed by search-and-blocking to locate dominant eigenvalues $\{\lambda_i^{\star}\}$ and by rank tests to determine multiplicities $\{m_i\}$.
• Figure 3: Illustration of the QFAMES algorithm on the illustrative example. The dashed vertical lines are the locations of the eigenvalues. (a): The magnitude of $G_{i,j}(\theta)=\frac{1}{N}\sum_{n=1}^NZ_{i,j,n}e^{\mathbf{i} \theta t_n}$ for each $0\leq i,j\leq 2$. The gray dashed lines indicate the positions of the eigenvalues. (b): The Frobenius norm $\|G(\theta)\|_F$, which clearly identifies the two distinct dominant eigenvalues. (c) and (d): The singular values of $G(\theta)$ at each $\lambda_i^{\star}$, accurately indicating the corresponding multiplicities when the threshold $\tau=0.1\sqrt{LR}=0.3$.
• Figure 4: QFAMES (\ref{['alg:QFAMES']}) vs QMEGS in the illustrative example as a function of (left): max evolution time $T_{\max} = \sigma T$, and (right): total evolution time $T_{\rm total}$. The dashed line stands for the fitted error proportional to $1 / T$, as predicted by \ref{['eq:location_err']}.
• Figure 5: Illustration of the QFAMES algorithm on the TFIM model. The upper row shows the Frobenius norm of $G(\theta)$ around the ground state energy for $g=0.5$, $1.0$ and $1.5$, from left to right. The dashed lines stand for the 5 lowest lying energy eigenvalues. The lower row shows the singular values of the $G(\theta_1^{\star})$, where $\theta_1^{\star}$ is the leftmost peak found in corresponding Frobenius norm (marked with red stars). The SVD cutoff threshold is set to be 0.2.

Theorems & Definitions (15)

• Theorem 3.2: Main result, Informal version of \ref{['thm:main']}
• Theorem 4.1
• Proposition 5.1
• Proposition B.1: Information-theoretic indistinguishability
• proof
• Definition C.1: Dominant eigenvalue clusters
• Theorem C.2: General version of \ref{['thm:main']}
• Remark C.3
• Remark C.4
• proof