Accuracy and resource advantages of quantum eigenvalue estimation with non-Hermitian transcorrelated electronic Hamiltonians

TL;DR

This work evaluates the resource costs of computing ground-state energies for transcorrelated electronic Hamiltonians using a non-Hermitian QEVE algorithm and compares them to standard qubitization on non-transcorrelated Hamiltonians. It shows that TC in the STO-6G basis yields more accurate energies than a cc-pVQZ Hamiltonian, while offering comparable T gates and a substantially smaller qubit count. The study analyzes one-norms, term counts, and Jordan condition numbers to assess the practicality of QEVE for non-Hermitian spectra, highlighting $\kappa_S$ as a key challenge. The results suggest QEVE on TC Hamiltonians is a promising route to reduce quantum resources for accurate electronic-structure calculations, with code and data available on GitHub.

Abstract

In electronic structure calculations, the transcorrelated method enables a reduction of the basis set size by incorporating the electron-electron correlations directly into the Hamiltonian. However, the transcorrelated Hamiltonian is non-Hermitian, which makes many common quantum algorithms inapplicable. Recently, a quantum eigenvalue estimation algorithm was proposed for non-Hermitian Hamiltonians with real spectra [FOCS 65, 1051 (2024)]. Here we investigate the cost of this algorithm applied to transcorrelated electronic Hamiltonians of second-row atoms and compare it to the cost of applying standard qubitization to non-transcorrelated Hamiltonians. We find that the ground state energy of the transcorrelated Hamiltonian in the STO-6G basis is more accurate than that of a standard Hamiltonian in the cc-pVQZ basis. The T gate counts of the two methods are comparable, while the qubit count of the transcorrelated method is 2.5 times smaller.