Evaluating higher-order product formulae for molecular ground-state energy estimation

Abstract

We evaluate deterministic higher-order product formulae for molecular ground-state energy estimation. Motivated by recent fault-tolerant architectures in which non-Clifford operations may be generated more locally and cheaply than in conventional assumptions, we re-examine such formulae as practical candidates for quantum chemistry. Using one-dimensional hydrogen chains from $\mathrm{H}_2$ to $\mathrm{H}_{15}$ as benchmarks, we estimate both the total gate count and the depth of $R_Z$-rotation layers required to reach a target energy error. To make this comparison feasible at larger system sizes, we use a perturbative method to estimate the eigenvalue error induced by each product formula and thereby evaluate the cost of the corresponding phase-estimation procedure. Among the previously considered formulae, the eighth-order construction introduced by Morales et al. [M. E. S. Morales et al., "Greatly improved higher-order product formulae for quantum simulation," arXiv:2210.15817v2 (2024)] minimizes both cost metrics in the benchmark at a chemically relevant target error. We also find that increasing the formal order does not automatically reduce the total cost: near chemical accuracy, the tenth-order formula introduced in the same work can be less efficient than the eighth-order one. Motivated by this observation, we construct a new fourth-order formula; it achieves the lowest total gate count among the formulae considered for all H-chain instances near chemical accuracy and over much of the 0.1-10 mHa target-error window for most instances, while also reducing the $R_Z$-layer depth. These results clarify how deterministic higher-order product formulae should be selected for molecular ground-state energy estimation.