@unpublished{61776,
  abstract     = {{We investigate the role of energy, i.e. average photon number, as a resource
in the computational complexity of bosonic systems. We show three sets of
results: (1. Energy growth rates) There exist bosonic gate sets which increase
energy incredibly rapidly, obtaining e.g. infinite energy in finite/constant
time. We prove these high energies can make computing properties of bosonic
computations, such as deciding whether a given computation will attain infinite
energy, extremely difficult, formally undecidable. (2. Lower bounds on
computational power) More energy ``='' more computational power. For example,
certain gate sets allow poly-time bosonic computations to simulate PTOWER, the
set of deterministic computations whose runtime scales as a tower of
exponentials with polynomial height. Even just exponential energy and $O(1)$
modes suffice to simulate NP, which, importantly, is a setup similar to that of
the recent bosonic factoring algorithm of [Brenner, Caha, Coiteux-Roy and
Koenig (2024)]. For simpler gate sets, we show an energy hierarchy theorem. (3.
Upper bounds on computational power) Bosonic computations with polynomial
energy can be simulated in BQP, ``physical'' bosonic computations with
arbitrary finite energy are decidable, and the gate set consisting of Gaussian
gates and the cubic phase gate can be simulated in PP, with exponential bound
on energy, improving upon the previous PSPACE upper bound. Finally, combining
upper and lower bounds yields no-go theorems for a continuous-variable
Solovay--Kitaev theorem for gate sets such as the Gaussian and cubic phase
gates.}},
  author       = {{Chabaud, Ulysse and Gharibian, Sevag and Mehraban, Saeed and Motamedi, Arsalan and Naeij, Hamid Reza and Rudolph, Dorian and Sambrani, Dhruva}},
  booktitle    = {{arXiv:2510.08545}},
  title        = {{{Energy, Bosons and Computational Complexity}}},
  year         = {{2025}},
}