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  • Open Access

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

Grzegorz Rajchel-Mieldzioć, Szymon Pliś, and Emil Zak*

  • BEIT sp. z o.o., ul. Wadowicka 8a, 30-415 Kraków, Poland

  • *Contact author: emil@beit.tech
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Phys. Rev. Research 8, 023192 – Published 20 May, 2026

DOI: https://doi.org/10.1103/td1z-jffj

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 parametrized 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 Õ(N)] compared to classical implementations with Õ(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 parametrized 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 quasicontinuum regimes in many-body and solid-state systems.

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Phys. Rev. Research 8, 023192– Published 20 May, 2026
  • Received 18 July 2025
  • Accepted 11 April 2026
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