Characterization of bounded-exponential-energy gate sets

Characterize which bosonic gate sets, when applied to coherent states, generate states with bounded exponential energy, meaning that some \(s>1\) satisfies \(\langle s^{N}\rangle\leq s^E\) throughout the relevant computation.

Background

The paper’s Solovay–Kitaev theorem is formulated using a sharp photon-number cutoff, whereas energy-constrained norms allow states with high-energy tails. A smooth-cutoff extension would be efficient for states with bounded exponential-energy, and the paper establishes this property for some Gaussian-plus-Kerr evolutions but not for general gate sets.

References

As shown in this work, these include Gaussian + Kerr acting on coherent states, and it is an interesting open question to characterize which gate sets generate states with bounded exponential-energy, starting from coherent states.

— A physical and universal model of bosonic computations with Solovay-Kitaev theorem  (2609.40226 - Rudolph et al., 30 Sep 2026) in Section 1. Results, subsection “Open questions,” item “Generalising the energy-preserving SK theorem”