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Energy, Bosons and Computational Complexity

T0 review · 2 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read Average photon number is a computational resource: with exponential energy, bosonic circuits solve NP; with unbounded energy, they reach PTOWER.

desk verdict The lower-bound half of this paper is genuinely new and valuable; the PP upper-bound proof has a fixable but real squeezing-budget gap. read the letter →

arxiv 2510.08545 v2 pith:6NJGUK3R submitted 2025-10-09 quant-ph

Dorian Rudolph , Arsalan Motamedi , Dhruva Sambrani , Hamid Reza Naeij , Ulysse Chabaud , Sevag Gharibian , Saeed Mehraban This is my paper · ORCID
classification quant-ph MSC 68Q1281P6868Q17 PACS 03.67.Lx
keywords bosoniccomputationcontinuous-variablequantumcomputingaveragephotonnumberenergycomplexityCVBQPPTOWERadiabaticSolovay–Kitaevtheorem
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Treating average photon number as a resource, this paper argues that energy — not just time or mode count — determines what bosonic continuous-variable quantum computers can do. It proves that with just exponential energy and a constant number of modes, certain finite gate sets make the class CVBQP contain NP, and that dropping the energy restriction pushes the power all the way to PTOWER, the class of problems solvable in iterated-exponential time. On the upper side, it shows that polynomial-energy CV computations lie in BQP/poly (or BQP under a block-encoding assumption), and that the commonly studied Gaussian-plus-cubic-phase gate set with exponential energy lies in PP, improving the previous PSPACE upper bound. It also proves some bosonic gate sets reach infinite energy in constant time, and that deciding such properties is undecidable. Together the bounds rule out an efficient continuous-variable analogue of the Solovay–Kitaev theorem for the Gaussian and cubic phase gate set.

What carries the argument

The load-bearing construction is the time-independent Hamiltonian H = P̂₀ + A(X̂₀)N̂₄ₖ₊₃ of Eq. (88), which encodes a time-dependent adiabatic evolution A(t) using a continuous position clock and a high-energy register that rescales time to constant physical duration. Its logical restriction A(t) preserves photon number and has a unique ground state with spectral gap Ω(n_max^{-2}) by mapping the problem to the Laplacian of a 'whiskered grid graph' (Lemma 4.7). Other essential pieces: the photon-number-controlled squeezer that doubles energy per application and produces the astronomical input energies needed for PTOWER; the detuned degenerate parametric amplifier gadget (Hamiltonian G(c), Eq.

What would settle it

Compute the low-lying spectrum of the full unbounded Hamiltonian of Eq. (88) on a sequence of increasing photon-number truncations: if the gap between the two lowest eigenvalues shrinks to zero (or many eigenvalues intrude below the claimed bound) as the cutoff grows, Lemma 4.7's gap analysis fails in the infinite-dimensional limit and the PTOWER containment collapses. Alternatively, exhibit a CV circuit with energy o(ε^{-1}) that solves the ε-BeamSplitPrec problem, contradicting the energy hierarchy theorem.

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Extended reading notes

Core claim

The central discovery is that the energy of a bosonic computation, measured as the expected photon number ⟨ψ|N|ψ⟩, is a genuinely computational resource with a strict hierarchy: more energy buys strictly more power. Concretely, the paper constructs finite gate sets for which polynomial-time CVBQP with exponential energy and O(1) modes solves NP (Theorem 4.1), with O(k) modes solves NTIME(exp^(k)), and with no energy bound contains PTOWER (Theorems 4.2 and 4.3). The proof goes through a CV adiabatic algorithm for Diophantine equations whose evolution time is made constant by a photon-number-preserving Hamiltonian with a provably polynomial spectral gap over a 'whiskered grid graph'; the high-

Load-bearing premise

The NP and PTOWER lower bounds hinge on the assumption that the spectral gap and unique ground state proven for the adiabatic Hamiltonian on a finite-dimensional logical subspace survive when the same Hamiltonian is embedded in the full infinite-dimensional operator of Eq. (88), whose essential self-adjointness is only sketched in the paper.

Editorial extensions

If this is right

  • Exponential energy with O(1) modes suffices for CVBQP to contain NP, giving formal evidence that constant-mode, exponential-energy setups (such as recent CV factoring proposals) are computationally very strong.
  • Without energy restrictions, CVBQP with a suitable finite gate set contains PTOWER; any efficient simulation of these circuits by Gaussian-and-cubic circuits would violate the time hierarchy theorem.
  • Polynomial-energy CV computations are simulable by BQP/poly, and by BQP for gate sets with efficient block encodings (including Kerr, Gaussians, Jaynes–Cumming, and cubic phase), so low energy does not grant quantum advantage beyond BQP.
  • The Gaussian-plus-cubic gate set with exponential energy lies in PP, improving the known PSPACE bound and placing a common non-Gaussian gate set 'not too far' from BQP.
  • Finite-energy but otherwise unconstrained 'physical' CVBQP computations are decidable (in R); undecidability sets in only with unbounded or infinite-energy states.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the spectral-gap/unbounded-operator analysis is later completed, the adiabatic Diophantine construction would become the first rigorous CV adiabatic algorithm for NP/PTOWER with bounded runtime; until then, the lower bounds rest on an infinite-dimensional gap assumption that needs independent checking.
  • The PP upper bound for Gaussian+cubic suggests any quantum advantage from this gate set with exponential energy would not reach beyond the counting hierarchy; a natural next step is to test whether the bound can be tightened to BQP or whether a matching NP-hardness lower bound can be shown for the same gate set.
  • The energy hierarchy theorem (ε-BeamSplitPrec) offers a concrete experimental probe: with energy budget E=o(ε^{-1}) a single query should fail to detect the beamsplitter angle, while E=O(ε^{-2}) should succeed — a distinction testable in photonic platforms.
  • The infinite-energy-in-finite-time results imply that general polynomial-Hamiltonian gate sets are not physically realisable to arbitrary precision, so experimental roadmaps should restrict to energy-constrained or block-encoded gate sets to avoid unphysical regimes.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper studies energy (average photon number) as a computational resource in continuous-variable bosonic circuits. It proves three groups of results: (1) quantitative energy-growth bounds for specific gate sets, including doubly exponential growth for Gaussian+cubic-phase gates and infinite-energy states in constant time; (2) complexity lower bounds, showing that certain finite gate sets with exponential energy and O(1) modes contain NP, and with more modes/energy contain ELEMENTARY and even PTOWER; (3) simulation upper bounds, including CVBQP with polynomial energy in BQP/poly, decidability of physical CV computations with finite energy, and containment in PP of Gaussian+cubic-phase circuits with exponential energy. The paper further combines the upper and lower bounds to argue against efficient continuous-variable Solovay–Kitaev theorems for the Gaussian+cubic gate set.

Significance. If the main theorems are fully established, this is a substantial contribution: it makes precise how energy can act as a computational resource in infinite-dimensional bosonic systems, connects to the BCCK factoring setup, gives new upper bounds (PP) for a physically relevant gate set, and provides a novel no-go argument for CV Solovay–Kitaev. The energy-growth calculations in Section 3 are self-contained and checkable, and the BQP/poly simulation of Section 5.1 is a useful general tool. The paper is also honest in flagging some of its own limitations, notably the deferred unbounded-operator analysis in Appendix A and the conditional nature of the exponential-energy promise in the PP result.

major comments (2)
  1. [§4.1.5–4.1.9, Lemmas 4.7–4.15, Appendix A] The NP/PTOWER lower bounds rest on spectral-gap and unbounded-operator claims that are not fully proved. Lemma 4.7 establishes the gap Ω(n_max^{-2}) only for A(t) restricted to a single logical subspace Hlog_n; the full Hamiltonian in Eq. (88) is unbounded and acts on infinitely many photon-number invariant sectors, and the paper does not give a uniform gap bound or a rigorous adiabatic theorem for this unbounded operator. Lemma 4.10's essential self-adjointness argument is a sketch ('It suffices to check...'), and Appendix A explicitly says a more complete unbounded-operator treatment is planned for a future revision. Since Lemma 4.15 and the E_T fast-forwarding are load-bearing for Theorems 4.1–4.3, these containments are conditional on completing that analysis.
  2. [§5.5.1, proof of Theorem 5.27, Eqs. (314), (322)–(327)] The inductive invariant that all coefficients and Gaussian parameters remain ≤exp(n) under adaptive homodyne postselection is underproved. The update rule for the mean vector in Eq. (327) is asserted without derivation, and the coefficient update in Eq. (322) is only bounded by combining Lemma 5.32 with the normalization product in Eq. (323). A single teleportation step can multiply a coefficient by exp(n)/√Z ≈ exp(n/2) if ξ=exp(n), so after T=poly(n) cubic gates the coefficients can grow to exp(poly(n)), not necessarily ≤exp(n). The text may intend a global rescaling, but this is not stated or proved. Since the PP simulation requires each branch to be generated and evaluated in polynomial time, this invariant needs a complete proof or a revised parametrization.
minor comments (5)
  1. [Appendix A] The sentence 'We plan to include a more complete overview over the required theory of unbounded operators in a future revision' is inappropriate for a submitted manuscript if it covers load-bearing technical material. Either complete the treatment or explicitly mark the affected theorems as conditional.
  2. [Theorem 5.27] The phrase 'using one query to a PP oracle' is redundant since P^{PP[1]}=PP; the statement should simply say the problem is in PP.
  3. [Eq. (315)] The notation (^n⊗I) appears to denote the Fock-state projector |n⟩⟨n|⊗I, not the number operator; this should be clarified.
  4. [Theorem 5.20 and Lemma 5.9] The symbol ε is used both for the teleportation error in Lemma 5.9 and for the width parameter in the Gaussian decomposition of Theorem 5.20. These are different parameters and should be renamed to avoid confusion.
  5. [Lemma 5.21] Typo: 'Reimann' should be 'Riemann'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: lower and upper bounds are new constructions, not re-definitions of their inputs.

full rationale

The paper's derivation chain is not circular. The CV model is inherited from prior work [CJMM25], but none of the advertised containments (NP⊆CVBQP[G], PTOWER⊆CVBQP[G], CVBQP_poly⊆BQP/poly, CVBQP[X^3]_exp⊆PP) is obtained by fitting parameters to the target result or by assuming the conclusion. The NP/PTOWER lower bounds are built from a novel time-independent adiabatic Hamiltonian with an explicit spectral-gap analysis (Lemma 4.7) and error bounds (Lemma 4.14), and the PP upper bound uses an explicit teleportation gadget plus a Gaussian-rank decomposition of cubic phase states (Theorem 5.20). Self-citations to [CJMM25], [URC25], and [ABC25] supply background lemmas and technical ingredients, but they are not uniqueness theorems invoked to forbid alternatives, and the central results do not reduce to those citations. The skeptic-flagged mismatch between the squeezing requirement in Lemma 5.9 (ξ>c(E/ε)exp(½log²(2/δ)log(1/δ))) and the ξ=exp(n) regime used in Corollary 5.30 is a quantitative proof-consistency issue, not a circularity: it concerns whether the stated error bound is achieved with the stated parameters, not whether the output was assumed as input. Likewise, Appendix A's statement that a fuller unbounded-operator treatment is planned for a future revision flags a rigor gap in Lemma 4.10, but an omitted proof is not an equivalence-by-construction. Therefore no circular step is exhibited, and the appropriate circularity score is 0.

Assumptions & free parameters 3 free parameters · 8 assumptions · 0 invented entities

The central claims rest on standard mathematical facts (MRDP, adiabatic theorem, spectral graph theory, time hierarchy) and on two explicit modeling assumptions: block-encodability of gates and the exponential-energy proxy-state promise in Theorem 5.27. No new physical entities are postulated. The hand-chosen scales (squeezing, energy registers, teleportation squeezing) are construction parameters, not fitted degrees of freedom.

free parameters (3)
  • Squeezing parameter r in lower-bound construction = poly(n)
    Chosen large enough so that squeezed states approximate position eigenstates with exponentially small error (Lemmas 4.14, 4.17). Not data-fitted, but a hand-picked scale required for the adiabatic proof.
  • Adiabatic time-register energy E_T = sufficiently large, E^{O(k)}
    Used in Lemma 4.15 to make the adiabatic evolution error E^{-Ω(1)} in constant physical time. Its required value depends on the spectral gap and derivative bounds of the constructed Hamiltonian.
  • Teleportation squeezing ξ in Theorem 5.27 = ≤ exp(n)
    Chosen at the exponential-energy scale to satisfy Lemma 5.9's cubic gate teleportation accuracy. It is an input to the PP simulation, not an empirical fit.
assumptions (8)
  • standard math Matiyasevich–Robinson–Davis–Putnam theorem: Diophantine equations are undecidable
    Used in Theorems 4.1–4.3 and 4.33–4.35 to build Hamiltonians from Diophantine equations.
  • standard math Adleman–Manders NP-completeness of binary quadratic Diophantine equations and solution bounds
    Provides the NP-complete problem L_MA and the bounded-search-space formulation used for the NP lower bound (Section 4.1.1).
  • standard math Adiabatic theorem with error bounds (Reichardt form)
    The lower-bound algorithms rely on Theorem 4.9 to convert spectral gap and derivative bounds into evolution-time guarantees.
  • standard math Spectral gap lower bound for whiskered grid graphs (BPS07 formula)
    Lemma 4.7 uses this graph-theoretic result to claim Ω(n_max^{-2}) gap for the adiabatic Hamiltonian.
  • standard math Time hierarchy theorem and PTOWER ⊄ EXPSPACE
    Used in the claimed no-go for CV Solovay–Kitaev in Section 1.1.
  • standard math Essential self-adjointness criteria (Stone, Nelson, Kato–Rellich)
    Required to make the unitary evolutions of polynomial Hamiltonians well-defined; used in Lemmas 4.10, 4.12 and Propositions 3.12–3.14.
  • domain assumption Assumption 5.1: polynomial-time block encodings for the gate set at bounded energy
    Needed to strengthen CVBQP_poly ⊆ BQP/poly to CVBQP_poly ⊆ BQP; listed as holding for Kerr, Gaussian, Jaynes–Cummings, and cubic gates.
  • domain assumption Proxy-state promise in Theorem 5.27: a state of energy exp(n) exists that is exponentially close to the true state
    This promise underpins the PP simulation and is stated as an assumption rather than derived; it holds when the energy is itself bounded by exp(n), but is not automatic for all 'effective exponential energy' circuits.

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Pith. "Pith review of Energy, Bosons and Computational Complexity." pith.science (2026). https://pith.science/paper/6NJGUK3R

@misc{pith2026251008545,
  author       = {Pith},
  title        = {Pith review of: Energy, Bosons and Computational Complexity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6NJGUK3R}},
  note         = {Machine review of arXiv:2510.08545}
}
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

Figures reproduced from arXiv: 2510.08545 by the authors.

Figure 1
Figure 1. The adiabatic Hamiltonian A(t) in a logical subspace. Note that the weights refer to weights in the weighted history state picture [BC18] (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). we prepare the vacuum state |0, 1⟩L . We also modify W0 in the next step, so that at t = 0 the ground state will be close… view at source ↗
Figure 2
Figure 2. Graph structure of A(t)↾Hlog 6,4 , where the red edges are weighted as in [PITH_FULL_IMAGE:figures/full_fig_p026_2.png] view at source ↗
Figure 3
Figure 3. The circuit solving the ε-BeamSplitPrec problem with O(ε −2 ) energy. Note that the first gate is a two-mode squeezer with a parameter ξ that produces large enough photon number on average (c.f. (167)). The second gate is a beam-splitter, with angle either ε (No case) or angle 0 (Yes case). The measurement is performed in the number basis. Note that we accept if n = 0 as the measurement outcome, and otherwise, we re… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: The cubic teleportation gadget introduced in [ [PITH_FULL_IMAGE:figures/full_fig_p050_4.png]

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Works this paper leans on

2 extracted references · 1 linked inside Pith

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    Gaus- sian Boson Sampling

    New York, NY, USA: Association for Computing Machinery, 2019, pp. 193–204. doi:10.1145/3313276.3316366. [Hal13] B. C. Hall.Quantum Theory for Mathematicians. Springer New York, 2013.doi:10 . 1007/978-1-4614-7116-5. [HJ12] R. A. Horn and C. R. Johnson.Matrix Analysis. 2nd ed. Cambridge University Press, 2012. 71 [HKS+17] C. S. Hamilton, R. Kruse, L. Sanson...

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