{"id":"39b331a7-814a-4bf4-9d4b-02e8c9a50918","arxiv_id":"2510.08545","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Using more photon energy in bosonic quantum circuits can confer staggering computational power—up to NP with a few modes and PTOWER with no energy cap—while polynomial-energy circuits remain inside BQP.","lead":"This paper treats average photon number—energy—as a formal computational resource in bosonic quantum computing. It proves that high-energy bosonic circuits can become enormously powerful, even undecidable, while bounded-energy circuits stay within standard quantum complexity classes.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"PP simulation of cubic+CVBQP overclaims teleportation precision: Lemma 5.9/Lemma 5.10 require squeezing ξ=exp(Ω(n²)) for error exp(-n), not the ξ=exp(n) used in Corollary 5.30 and Theorem 5.27.","rationale":"The reader's weakest assumption concerned the adiabatic lower-bound construction: the spectral-gap analysis of Lemma 4.7 is performed on a logical subspace while the full Hamiltonian in Eq. (88) is unbounded and has multiple zero-energy sectors, and Appendix A defers unbounded-operator details. That is a legitimate concern and deserving of independent scrutiny. However, I find a more concrete and directly falsifiable weakness in the upper-bound leg: the PP simulation of cubic+Gaussian circuits with exponential energy depends on a teleportation lemma whose proved precision bound forces a squeezing parameter far larger than the exponential-energy promise allows. This is not a matter of outside consensus; it is an internal inconsistency between Lemma 5.9's parameter dependence and the values substituted in Corollary 5.30/Theorem 5.27. The lower-bound results (NP, PTOWER) are less affected by this particular flaw and should be evaluated on their own merits. Since the reader already returned CONDITIONAL, my finding sharpens the conditions rather than changing the verdict; the paper would need either a sharper teleportation error bound or a relaxed precision promise for Theorem 5.27 to stand as stated.","tokens_in":75259,"tokens_out":26716,"duration_ms":216156,"concrete_test":"Set E=e^n, ε=e^{-n}, and δ=1/3 or δ=e^{-n} in Lemma 5.10's condition: with a=1/(2ξ²), b=2√2 log(2/δ), σ²=E+1, solve a ≤ (e^{b²}+1)^{-2}σ^{-4}ε² for ξ. Check whether the required ξ satisfies ξ≤exp(n) as assumed in Corollary 5.30. Then recompute R_δ in Theorem 5.20 using that ξ; if R=exp(ω(n)), the PP summation in Theorem 5.27's proof cannot run in polynomial time, so Theorem 5.27 would need a different error-control argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing weak point is the gate-teleportation error budget behind Theorem 5.27, not the adiabatic gap. Lemma 5.9 states that, for input energy E and Euclidean error ε, the cubic teleportation gadget requires squeezing ξ > c(E/ε) exp(½ log²(2/δ) log(1/δ)). The proof via Lemma 5.10 imposes a = 1/(2ξ²) ≤ (e^{b²}+1)^{-2} σ^{-4} ε², with b=O(log(1/δ)) and σ²=E+1. In the regime used by Corollary 5.30 — E=exp(n), ε=1/exp(n), δ=1/exp(n) — this forces log ξ = Ω(n²) (or Ω(n³) if the stated exponent is taken literally), whereas Corollary 5.30 and Eq. (290) assume ξ=exp(n). Even keeping δ constant, the E/ε factor alone forces ξ ≥ exp(2n), exceeding the exp(n) budget. Consequently the Gaussian-rank expansion in Theorem 5.20 has R=exp(Ω(n²)) terms, which cannot be summed by a polynomial-time PP machine; the PP containment and the no-go Solovay–Kitaev corollary built on it are not established as written. This is an internal mismatch between Lemma 5.9's proved bound and the parameter regime used downstream.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":75591,"tokens_out":17770,"duration_ms":144538,"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":[{"comment":"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.","section":"§4.1.5–4.1.9, Lemmas 4.7–4.15, Appendix A"},{"comment":"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.","section":"§5.5.1, proof of Theorem 5.27, Eqs. (314), (322)–(327)"}],"minor_comments":[{"comment":"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.","section":"Appendix A"},{"comment":"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.","section":"Theorem 5.27"},{"comment":"The notation (^n⊗I) appears to denote the Fock-state projector |n⟩⟨n|⊗I, not the number operator; this should be clarified.","section":"Eq. (315)"},{"comment":"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.","section":"Theorem 5.20 and Lemma 5.9"},{"comment":"Typo: 'Reimann' should be 'Riemann'.","section":"Lemma 5.21"}],"recommendation":"major_revision","confidential_remarks":"This is an ambitious paper with several important ideas and some genuinely self-contained contributions. The main technical weak points are the unbounded-operator/gap analysis behind the lower bounds and the teleportation parameter mismatch behind the PP upper bound. The latter appears locally fixable by choosing inverse-polynomial precision parameters and reworking the Gaussian-rank/energy budget, but as written the proof does not go through. I do not see grounds for rejection, but the current version is not publishable until the load-bearing gaps are resolved."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, you should know: this is a serious, dense paper with real new results, but the PP simulation of cubic+CVBQP has a load-bearing numerical mismatch that needs fixing. The lower-bound and energy-growth sections are the real contribution. The adiabatic construction for NP and PTOWER lower bounds, the energy hierarchy theorem, and the infinite-energy/undecidability results are all new and largely self-contained. The energy-growth calculations in Section 3 are checkable and convincing, and the BQP/poly simulation for polynomial energy looks solid. The paper is also honest about its own limits: Appendix A explicitly says a fuller unbounded-operator treatment is planned for a future revision, and that the spectral-gap analysis is plausible rather than machine-checked.\n\nThe soft spot is exactly where the stress-test points. Lemma 5.9 requires a squeezing parameter xi > c(E/eps) exp(1/2 log^2(2/delta) log(1/delta)). Plugging in E=exp(n), eps=delta=exp(-n) gives log xi = Theta(n^3), or at least Theta(n^2) even if delta is kept constant. Corollary 5.30, however, assumes xi=exp(n). That is an internal inconsistency, and as written the proof of Theorem 5.27 does not go through. The stress-test overreaches when it says the resulting Gaussian rank R=exp(Omega(n^2)) cannot be handled by a PP machine: PP has exp(poly(n)) branches, so exp(n^3) terms are still fine. The likely fix is to increase xi to exp(poly(n)) and re-run the rank bounds; that should preserve the PP containment. But the authors need to supply that.\n\nSo the paper deserves peer review. The lower bounds alone justify it. The PP section needs revision, and the no-go Solovay–Kitaev corollary rests on the PP result, so it is not established as written. For a reader working on CV complexity, the energy-growth and hierarchy results are worth citing now; the PP result should be cited with care. I would bring the lower-bound parts to reading group, and note the PP bug to anyone who asks.","headline":"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.","tokens_in":76177,"tokens_out":5246,"would_cite":true,"duration_ms":63288,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68Q12","81P68","68Q17"],"pacs":["03.67.Lx"],"model":"deepseek-v4-flash","headline":"Average photon number is a computational resource: with exponential energy, bosonic circuits solve NP; with unbounded energy, they reach PTOWER.","keywords":["bosonic computation","continuous-variable quantum computing","average photon number","energy complexity","CVBQP","PTOWER","adiabatic quantum computation","Solovay–Kitaev theorem"],"falsifier":"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.","tokens_in":75116,"feed_emoji":"⚡","tokens_out":5179,"duration_ms":44359,"temperature":0.7,"pith_summary":"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.","feed_headline":"Exponential energy lets bosonic circuits capture NP","feed_subtitle":"A new analysis maps average photon number to computational power — polynomial energy stays in BQP, unbounded energy reaches PTOWER.","key_machinery":"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.","core_discovery":"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-","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Infinite energy makes bosonic problems undecidable","Photon count: the hidden resource for bosonic complexity","More photons, more power: energy hierarchy in quantum computing","Polynomial energy stays in BQP, but unbounded reaches PTOWER","No Solovay-Kitaev for Gaussian and cubic phase gates"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Infinite energy makes bosonic problems undecidable","Photon count: the hidden resource for bosonic complexity","More photons, more power: energy hierarchy in quantum computing","Polynomial energy stays in BQP, but unbounded reaches PTOWER","No Solovay-Kitaev for Gaussian and cubic phase gates"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000344,"raw_usage":{"total_tokens":1772,"prompt_tokens":839,"completion_tokens":933,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":583,"completion_tokens_details":{"reasoning_tokens":846}},"tokens_in":583,"tokens_out":933,"duration_ms":7610,"temperature":1.0,"reasoning_tokens":846,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T10:43:06.406044+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}