Quantum Solvers for Nonlinear Matrix Equations in Quantum Chemistry

TL;DR

This paper introduces a quantum algorithm for solving nonlinear matrix equations in quantum chemistry, specifically algebraic Riccati equations, with potential exponential advantages over classical methods in certain regimes.

Contribution

The authors develop a quantum algorithm that encodes Riccati solutions and estimates electronic energies, offering a scalable approach under sparsity assumptions and paving the way for quantum coupled-cluster algorithms.

Findings

End-to-end cost scales linearly with system size under sparsity assumptions.

Algorithm provides a block-encoding of the amplitude solution for RPA.

Potential exponential advantage in excitation rank m over classical heuristics.

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.