Low-energy and average-case guarantees for extrapolated product formulae

Establish whether extrapolated product formulae inherit the improved performance of ordinary product formulae on input states supported in low-energy subspaces and the stronger average-case guarantees known for product-formula simulation relative to worst-case bounds.

Background

The paper shows that extrapolated product formulae can obtain favorable gate-complexity bounds and can exploit several structural properties, including locality and Hamiltonian symmetries. The discussion compares these results with known properties of ordinary product formulae.

For ordinary product formulae, prior work establishes improved behavior for input states in low-energy subspaces and stronger average-case guarantees than worst-case analyses provide. The paper explicitly leaves unresolved whether these two advantages persist after product-formula extrapolation.

References

It is known for $k$-local systems product formulae perform better on input states that lie in a low-energy subspace , and that product formulae have stronger average-case guarantees than worst-case bounds provide . Whether the same applies for extrapolated product formulae is an open question which we leave for future work.

— Resource-efficient quantum eigenvalue transform with commutator scaling  (2608.13862 - Mazumder et al., 14 Aug 2026) in Section 1, Discussion subsection