Energy, Bosons and Computational Complexity

TL;DR

This work treats energy as a fundamental resource in continuous-variable bosonic quantum computation, revealing striking links between photon-number energy and computational power. It develops a tripartite framework: (i) quantifying energy growth across CV gate sets, including constructions that yield infinite energy in finite time and even doubly exponential increases; (ii) establishing sharp lower bounds where energy enables NP- and tower-time computations, along with undecidability results for high-energy CV evolutions; and (iii) proving upper bounds showing decidability and DV-like simulability under realistic energy constraints, with PP-containing simulations for Gaussian+cubic gates and frameworks to bound energy via truncation, block-encoding, and Gaussian-rank decompositions. The results yield no-go theorems for a universal CV Solovay–Kitaev style theorem in general, while also identifying regimes (polynomial energy, certain gate sets) where CV computations remain efficiently simulable by quantum/classical means. Collectively, the paper maps the boundary between physically plausible CV implementations and computationally intractable regimes, highlighting how energy budgets and gate sets shape the landscape of continuous-variable quantum complexity and offering concrete methods to simulate or constrain CV computations in practice.

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.

Figures (3)

• Figure 1: The adiabatic Hamiltonian $A(t)$ in a logical subspace. Note that the weights refer to weights in the weighted history state picture Bausch_2018 ($A(t)$ is not a Laplacian). The weights below the red vertices refer the relative weight of those states compared to the base chain in the ground state of $A(t)$.
• Figure 2: Graph structure of $A(t){\upharpoonright}_{\mathcal{H}^{\mathrm{log}}_{6,4}}$, where the red edges are weighted as in \ref{['fig:walk']}.
• Figure 4: The cubic teleportation gadget introduced in GKP2001ghose2007non. The state $\ket{V^{(3)}(\theta);\xi} = V^{(3)}(\theta)\ket{S_\xi}$ is the cubic phase gate applied to a finitely-squeezed state. We highlight that the measurement is in the $X$ quadrature basis (giving output $q\in\mathbb R$), and $G$ and $\mathrm{SUM}$ are Gaussian gates. In \ref{['lem:cubic-teleportation']} we prove that this gadget works with high success probability $1-\delta$ and accuracy $\varepsilon$ with finite squeezing parameter $\xi\leq \mathsf{poly}(E,\varepsilon^{-1}) \, \mathsf{qpoly}(\delta^{-1})$, with $E$ being the energy of the input state. We can also flag if the implementation if faulty (occuring with probability $\delta$).

Theorems & Definitions (188)

• Theorem 1.1: Energy growth rates (informal; see each bullet point for formal reference)
• Theorem 1.2: Complexity class lower bounds (informal; see each bullet point for formal reference)
• Theorem 1.3: Energy hierarchy
• Theorem 1.4: Undecidable properties (informal; see each bullet point for formal reference)
• Theorem 1.5: $\BQP$ simulations of $\CVBQP$ (informal; \ref{['thm:polyenergypolysim']})
• Theorem 1.6: Informal; see \ref{['thm:decidable']}
• Theorem 1.7: Informal; see \ref{['thm:XCubedInPP']}
• Lemma 2.1
• Definition 2.2: chabaudBosonicQuantumComputational2025
• Remark 2.3