Quantum algorithm for solving generalized eigenvalue problems with application to the Schrödinger equation

Abstract

Accurate computation of multiple eigenvalues of quantum Hamiltonians is essential in quantum chemistry, materials science, and molecular spectroscopy. Estimating excited-state energies is challenging for classical algorithms due to exponential scaling with system size, posing an even harder problem than ground-state calculations. We present a quantum algorithm for estimating eigenvalues and singular values of parameterized matrix families, including solving generalized eigenvalue problems that frequently arise in quantum simulations. Our method uses quantum amplitude amplification and phase estimation to identify matrix eigenvalues by locating minima in the singular value spectrum. We demonstrate our algorithm by proposing a quantum-computing formulation of the pseudospectral collocation method for the Schrödinger equation. We estimate fault-tolerant quantum resource requirements for the quantum collocation method, showing favorable scaling in the size of the problem $N$ (up to $\tilde{\mathcal{O}}(\sqrt{N})$) compared to classical implementations with $\tilde{\mathcal{O}}(N)$, for certain well-behaved potentials. Additionally, unlike the standard collocation method, which results in a generalized eigenvalue problem requiring matrix inversion, our algorithm circumvents the associated numerical instability by scanning a parameterized matrix family and detecting eigenvalues through singular value minimization. This approach is particularly effective when multiple eigenvalues are needed or when the generalized eigenvalue problem involves a high condition number. In the fault-tolerant era, our method may thus be useful for simulating high-dimensional molecular systems with dense spectra involving highly excited states, such as those encountered in molecular photodynamics or quasi-continuum regimes in many-body and solid-state systems.

21 pages, revised version

References in corpus (21)

• Quantum algorithm for solving linear systems of equations
• Quantum Amplitude Amplification and Estimation
• The Variational Quantum Eigensolver: a review of methods and best practices
• Optimal Hamiltonian Simulation by Quantum Signal Processing
• Encoding Electronic Spectra in Quantum Circuits with Linear T Complexity
• Even more efficient quantum computations of chemistry through tensor hypercontraction
• Quantum computing enhanced computational catalysis
• Is there evidence for exponential quantum advantage in quantum chemistry?
• Fixed-point quantum search with an optimal number of queries
• A different kind of quantum search
• Quantum arithmetic with the Quantum Fourier Transform
• Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems
• Efficient Fully-Coherent Quantum Signal Processing Algorithms for Real-Time Dynamics Simulation
• A Refined Laser Method and Faster Matrix Multiplication
• Block-encoding structured matrices for data input in quantum computing
• Quantum Multiple Eigenvalue Gaussian filtered Search: an efficient and versatile quantum phase estimation method
• Quantum Computation of Hydrogen Bond Dynamics and Vibrational Spectra
• A Query-based Quantum Eigensolver
• Task-based, GPU-accelerated and Robust Library for Solving Dense Nonsymmetric Eigenvalue Problems
• Quantum Simulation of the First-Quantized Pauli-Fierz Hamiltonian
• Refining resource estimation for the quantum computation of vibrational molecular spectra through Trotter error analysis