Quantum Filter Diagonalization: Quantum Eigendecomposition without Full Quantum Phase Estimation

TL;DR

Quantum Filter Diagonalization (QFD) introduces a hybrid quantum-classical approach to approximate eigendecomposition of sparse Pauli Hamiltonians by building a variational subspace from time-propagated guess states and solving a Rayleigh-Ritz problem classically. Matrix elements are efficiently estimated on quantum hardware using a one-ancilla extended swap test, enabling parallel evaluation and scalable handling of ground, excited, and transition properties. The method is demonstrated on an 8-qubit AIEM system, showing accurate reproduction of absorption spectra with modest time-grid sizes and tolerable Trotter errors, highlighting potential advantages for near-term quantum devices. QFD sits between VQE and PEA in the algorithmic landscape, offering a feasible route to eigenvalue estimates without the deep circuits required by full phase estimation while enabling meaningful observables through the subspace formalism.

Abstract

We develop a quantum filter diagonalization method (QFD) that lies somewhere between the variational quantum eigensolver (VQE) and the phase estimation algorithm (PEA) in terms of required quantum circuit resources and conceptual simplicity. QFD uses a set of of time-propagated guess states as a variational basis for approximate diagonalization of a sparse Pauli Hamiltonian. The variational coefficients of the basis functions are determined by the Rayleigh-Ritz procedure by classically solving a generalized eigenvalue problem in the space of time-propagated guess states. The matrix elements of the subspace Hamiltonian and subspace metric matrix are each determined in quantum circuits by a one-ancilla extended swap test, i.e., statistical convergence of a one-ancilla PEA circuit. These matrix elements can be determined by many parallel quantum circuit evaluations, and the final Ritz estimates for the eigenvectors can conceptually be prepared as a linear combination over separate quantum state preparation circuits. The QFD method naturally provides for the computation of ground-state, excited-state, and transition expectation values. We numerically demonstrate the potential of the method by classical simulations of the QFD algorithm for an N=8 octamer of BChl-a chromophores represented by an 8-qubit ab initio exciton model (AIEM) Hamiltonian. Using only a handful of time-displacement points and a coarse, variational Trotter expansion of the time propagation operators, the QFD method recovers an accurate prediction of the absorption spectrum.

Figures (3)

• Figure 1: Eigenspectrum and Gershgorin circle theorem analysis for $N=8$ BChl-a AIEM Hamiltonian. The eigenvalues of $\hat{H}$ are given by small x symbols on the x axis. The Gershgorin disks are presented as blue circles. The Gershgorin disks for the first and last row of the Hamiltonian, corresponding to the all-ground-state $|00\ldots \rangle$ configuration and all-excited-state $|11\ldots \rangle$ configuration, are presented as red disks.
• Figure 2: Test of simulated QFD with exact representation of the time propagation operators vs. full configuration interaction (FCI) and configuration interaction singles (CIS). The notation QFD-$k_{\mathrm{max}}$ means that the QFD time grid is truncated at $k_{\mathrm{max}}$, e.g., QFD-1 has $k_{\mathrm{max}} = 1$ and thus $k \in [-1, 0, +1]$. Top - Simulated absorption spectrum of $N=8$ linear stack BChl-a test case (geometry depicted in inset), computed from the excitation energies and oscillator strengths of the lowest 8 electronic transitions, depicted as vertical sticks. The envelope of the absorption spectrum is sketched by broadening the contribution from each transition with a Lorentzian with width of $\delta=0.15$ eV. Middle - errors in excitation energies. Bottom - errors in oscillator strengths. Middle and bottom - thin lines are a guide for the eye.
• Figure 3: Test of simulated QFD with Trotterized representation of the time propagation operators vs. full configuration interaction (FCI) and configuration interaction singles (CIS). One Trotter step per $k$ point is used. The notation QFD-$k_{\mathrm{max}}$ means that the QFD time grid is truncated at $k_{\mathrm{max}}$, e.g., QFD-1 has $k_{\mathrm{max}} = 1$ and thus $k \in [-1, 0, +1]$. Top - Simulated absorption spectrum of $N=8$ linear stack BChl-a test case (geometry depicted in inset), computed from the excitation energies and oscillator strengths of the lowest 8 electronic transitions, depicted as vertical sticks. The envelope of the absorption spectrum is sketched by broadening the contribution from each transition with a Lorentzian with width of $\delta=0.15$ eV. Middle - errors in excitation energies. Bottom - errors in oscillator strengths. Middle and bottom - thin lines are a guide for the eye.