Trotter Error and Orbital Transformations in Quantum Phase Estimation

TL;DR

This paper investigates how orbital transformations affect Trotter errors in quantum phase estimation, finding that reducing Trotter error via orbital basis changes is challenging, but localised orbitals do not necessarily cause large errors in molecular calculations.

Contribution

The study explores three strategies for reducing Trotter errors through orbital transformations, providing insights into their effectiveness and limitations.

Findings

Localised orbitals do not produce large Trotter errors in molecular calculations.

Reducing Trotter error via orbital transformations is generally challenging.

Analytical expressions suggest possible ways to decrease Trotter error, but practical recipes are elusive.

Abstract

Quantum computation with Trotter product formulae is straightforward and requires little overhead in terms of logical qubits. The choice of the orbital basis significantly affects circuit depth, with localised orbitals yielding lowest circuit depths. However, literature results point to large Trotter errors incurred by localised orbitals. Here, we therefore investigate the effect of orbital transformations on Trotter error. We consider three strategies to reduce Trotter error by orbital transformation: (i) The a priori selection of an orbital basis that produces low Trotter error. (ii) The derivation of an orbital basis that produces a ground state energy free of Trotter error (as we observed that the Trotter error is a continuous function in the Givens-rotation parameter, from which continuity of this error upon orbital transformation can be deduced). (iii) Application of propagators that change the computational basis between Trotter steps. Our numerical results show that reliably reducing Trotter error by orbital transformations is challenging. General recipes to produce low Trotter errors cannot be easily derived, despite analytical expressions which suggest ways to decrease Trotter error. Importantly, we found that localised orbital bases do not produce large Trotter errors in molecular calculations, which is an important result for efficient QPE set-ups.