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

TL;DR

This paper evaluates the resource efficiency of a quantum eigenvalue estimation algorithm when applied to non-Hermitian transcorrelated electronic Hamiltonians, demonstrating potential accuracy and resource advantages in quantum chemistry calculations.

Contribution

It provides the first detailed complexity analysis of QEVE on transcorrelated Hamiltonians, comparing its T gate cost to standard methods across different basis sets.

Findings

QEVE's T gate count is between standard qubitization in cc-pVTZ and cc-pVQZ basis sets.

Transcorrelated energies are more accurate than cc-pVQZ for Li and Be.

For larger atoms, transcorrelated energy errors increase, surpassing cc-pVDZ accuracy.

Abstract

In electronic structure calculations, the transcorrelated method consists in transforming the Hamiltonian so as to remove the Coulomb cusp in its eigenfunctions. As a result, the wavefunction can be described more accurately without increasing the size of the basis set. However, the transcorrelated Hamiltonian is non-Hermitian and non-normal, which makes many common quantum algorithms inapplicable. Recently, a quantum eigenvalue estimation algorithm (QEVE) was proposed for non-Hermitian Hamiltonians with real spectra [FOCS 65, 1051 (2024)]. Although the asymptotic scaling of this algorithm with the desired accuracy is shown to be optimal, the constant factor in its complexity scaling has not been analyzed. Here we investigate the cost of QEVE 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, with the xTC approximation, the T gate count of QEVE in the minimal STO-6G basis is between those of standard qubitization in the cc-pVTZ and cc-pVQZ bases. The accuracy of the transcorrelated energy differs between systems: for Li and Be, it is more accurate than the cc-pVQZ energy, while for larger atoms, the error gradually increases, exceeding the cc-pVDZ level for O, F, and Ne.