Energy, Bosons and Computational Complexity

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.