Grover’s algorithm and the Schrödinger equation

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, connected to Grover’s algorithm, 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 O(√N)) compared to classical implementations with O(N²), 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.

Arxiv link: arXiv:2506.13534

Bio:
Grzegorz is a physicist specializing in quantum information, in particular, entanglement and quantum channels. He started the journey with this branch of physics doing Bachelor and Master under supervision of Andrzej Dragan at University of Warsaw, studying properties of quantum protocols in the vicinity of black holes. Then, he moved on to a more applied branch of quantum science while doing PhD at Centre for Theoretical PAS, guided by Karol Życzkowski. During his PhD, he solved several problems related to multipartite quantum entanglement and quantum designs. One of those solutions, dubbed as entangled officers of Euler, was awarded a golden prize by National Quantum Information Centre (KCIK). Subsequently, also the entire thesis was recognized as the best PhD thesis in quantum information, defended in Poland in years 2021/22, awarded by the same institution. After the defense, he moved on to ICFO in Barcelona, joining the Quantum Optics Theory group of Maciej Lewenstein. He spent there two years, responsible for the branch of the group devoted to advancing the research on experimentally-motived quantum technologies. On coming back to Poland he joined newly created Quantum Computing group at NASK PIB. In October 2024, he became part of BEIT, and is exploring topics at the cross-section of quantum algorithms in chemistry and quantum information.