Accuracy and Resource Advantages of Quantum Eigenvalue Estimation with Non-Hermitian Transcorrelated Electronic Hamiltonians

In electronic structure calculations, the transcorrelated method consists of transforming the Hamiltonian so as to remove the Coulomb cusp in its eigenfunctions. As a result, the wave function can be described more accurately without increasing the size of the basis set. However, the transcorrelated Hamiltonian is non-Hermitian and nonnormal, which makes many common quantum algorithms inapplicable. Recently, a quantum eigenvalue estimation (QEVE) algorithm 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 yet been analyzed. Here, we investigate the cost of QEVE applied to transcorrelated electronic Hamiltonians of Li, Be, B, C, and N atoms and compare it to the cost of applying standard qubitization to nontranscorrelated Hamiltonians. We find that with the x TC 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 is between those of the cc-pVDZ and cc-pVTZ energies.