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Quantum Solvers for Nonlinear Matrix Equations in Quantum Chemistry

P. Rodenas-Ruiz, A. Zhao, J. Lee

arXiv preprint (quant-ph, physics.chem-ph), 2026

arXiv:2605.16189 · PDF · BibTeX

Abstract

We present a quantum algorithm for solving algebraic Riccati equations, with applications to quantum-chemical random-phase approximation (RPA) and higher-order RPA theories. Our method block-encodes stabilizing Riccati solutions via Riesz projectors onto invariant subspaces of an associated non-normal matrix, implemented using contour-integral resolvents and quantum singular value transformations. Applied to m-particle, m-hole RPA, our algorithm yields a block-encoding of the amplitude solution and estimates the electronic correlation-energy density with it. Under localized-orbital sparsity assumptions, the end-to-end cost scales linearly with system size and polynomially with excitation rank m, suggesting an exponential advantage in m over plausible classical local-correlation heuristics. More broadly, this work provides a framework for quantum algorithms for nonlinear matrix equations in quantum chemistry and opens a possible route toward developing quantum algorithms for coupled-cluster theory.

Cite (BibTeX)

@misc{rodenasruiz2026quantum,
  title = {{Quantum Solvers for Nonlinear Matrix Equations in Quantum Chemistry}},
  author = {Rodenas-Ruiz, P. and Zhao, A. and Lee, J.},
  year = {2026},
  month = may,
  howpublished = {arXiv preprint (quant-ph, physics.chem-ph)},
  eprint = {2605.16189},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.16189},
  abstract = {We present a quantum algorithm for solving algebraic Riccati equations, with applications to quantum-chemical random-phase approximation (RPA) and higher-order RPA theories. Our method block-encodes stabilizing Riccati solutions via Riesz projectors onto invariant subspaces of an associated non-normal matrix, implemented using contour-integral resolvents and quantum singular value transformations. Applied to m-particle, m-hole RPA, our algorithm yields a block-encoding of the amplitude solution and estimates the electronic correlation-energy density with it. Under localized-orbital sparsity assumptions, the end-to-end cost scales linearly with system size and polynomially with excitation rank m, suggesting an exponential advantage in m over plausible classical local-correlation heuristics. More broadly, this work provides a framework for quantum algorithms for nonlinear matrix equations in quantum chemistry and opens a possible route toward developing quantum algorithms for coupled-cluster theory.}
}

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