{"id":"8bc5c617-d64a-4320-8795-9074ad2f4a5e","arxiv_id":"2608.04601","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A continuous-variable quantum annealing formulation maps discretized linear differential equations to a quadratic bosonic Hamiltonian, and numerical simulations of two second-order equations match the analytic solutions.","lead":"The authors recast linear differential equations as quadratic optimization problems and encode the variables in bosonic cavity modes, then simulate a continuous-variable quantum annealing protocol. The paper is a proof-of-principle that this route can reproduce analytic solutions for two simple second-order equations, without needing to binary-encode each variable into many qubits.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The operator substitution is exact for these quadratic bosonic Hamiltonians, but the paper omits the proof and key numerical details; CONDITIONAL remains the right verdict.","rationale":"The reader's weakest assumption identifies the operator substitution of Eq. (4) as the load-bearing premise. That is indeed the right place to look: the entire Hamiltonian construction and the subsequent readout depend on it. However, closer analysis shows that, for the specific Hamiltonians in this paper, the substitution is not a heuristic approximation but an exact representation. The Hamiltonian H_P has only normal-ordered a_i†a_j terms and linear drives; it contains no pairing terms a_i a_j or a_i†a_j†. Such a Hamiltonian is quadratic and number-conserving, so its ground state is a coherent state. Completing the square reveals the ground-state displacement is exactly (C^T C)^{-1}C^T b, the least-squares solution of the discretized equation. Hence the central mathematical construction is sound. The legitimate remaining concerns are (i) the paper does not supply this proof, so the validity of the mapping is not evident and a reader cannot gauge when it might fail (e.g., if a different substitution or a cost function with negative-definite parts introduced pairing terms, the ground state could become squeezed and the readout would no longer equal the classical minimizer); and (ii) the numerical evidence is under-specified: no Fock truncation, no time-step/error-control parameters, no pointwise or L2 error between the QA output and the analytic solution, only visual agreement and expectation values of H_P. These are reproducibility and completeness gaps rather than refutations. The reader's CONDITIONAL verdict is therefore appropriate and should not be changed; the conditions are that the authors add the coherent-ground-state proof and a reproducible numerical error analysis. My agreement is 'partial' because I identify the same root area as the reader but conclude the mapping is actually exact for this Hamiltonian class, so the concern lands as a missing-justification issue rather than a correctness failure.","tokens_in":9117,"tokens_out":14972,"duration_ms":217976,"concrete_test":"For both examples, compute the exact ground state of H_P using a Gaussian-state solver (or exact diagonalization in a truncated Fock basis with increasing truncation N_max) and verify that ⟨(a_i+a_i†)/2⟩ equals (C^T C)^{-1}C^T b to machine precision. Then rerun the annealing simulation with the same truncation and report the L2 error between the final ⟨x_i⟩ and the analytic solution for T ∈ {100, 1000, 10000} and M ∈ {5, 7, 9}, along with the chosen N_max. If the L2 error decreases with T and converges to the Gaussian-state prediction, the central claim is confirmed and the missing proof is the only defect; if the L2 error does not decrease, the numerical support for the claim is inadequate.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim relies on Eq. (4): replacing θ_i^2 by a_i†a_i and θ_iθ_j by (a_i†a_j + a_i a_j†)/2 should produce a problem Hamiltonian whose ground-state quadrature expectations solve the discretized equation. The reader worries this substitution is not an exact quantization. For the Hamiltonians actually constructed here, the substitution is in fact exact, for a reason the paper does not explain: H_P in Eq. (23) is a quadratic, number-conserving bosonic Hamiltonian with only linear drive terms and no a_i a_j or a_i† a_j† pairing. Such a Hamiltonian can be completed to H_P = (a + ½Q^{-1}r)† Q (a + ½Q^{-1}r) + const, so its ground state is the multi-mode coherent state with ⟨a_i⟩ = -½(Q^{-1}r)_i = (C^T C)^{-1}C^T b, precisely the least-squares solution of Cx = b. Thus the mapping is not approximate for this Hamiltonian class. The load-bearing gap is therefore one of proof and reproducibility, not of correctness: the paper does not state why the ground state is coherent, does not report the Fock-space truncation used in the QuTiP simulations, and gives no quantitative error between the final ⟨x_i⟩ and the analytic solution; it only shows a plot and the residual ⟨H_P⟩. Without the proof, a reader cannot tell whether the readout is reliable outside the two toy examples, and without the truncation and error metrics the numerical claim cannot be independently checked.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript extends continuous-variable quantum annealing (CV-QA) to linear ordinary differential equations. The authors discretize the time interval, approximate derivatives with central finite differences to obtain a linear system Cx=b, square the residual to produce the quadratic cost L(x)=(Cx-b)^T(Cx-b), and map this cost to a bosonic problem Hamiltonian using the substitution rules of Eq. (4) with quadrature readout. QuTiP simulations for two second-order boundary-value problems show that for M=5,7,9 and sufficiently long annealing times, the readout quadratures appear to agree with the analytic solutions and the final problem-Hamiltonian expectation decreases with T. The paper concludes that this establishes a potential route to solving differential equations without the qubit overhead of binary encodings.","tokens_in":9414,"tokens_out":11191,"duration_ms":126625,"significance":"The central derivation is transparent and the matrices C and b for the two examples are correct; the residual formulation in Eq. (12) is a clean least-squares approach. If the ground-state readout claim is proven, the result would be a useful proof-of-principle extension of CV-QA from linear regression (Ref. [18]) to differential equations, potentially avoiding the precision-dependent qubit overhead of QUBO-based approaches. The paper is honest about the restricted conditions examined. The current significance is limited by two gaps: no proof that the ground state of the constructed Hamiltonian has quadrature expectations minimizing L(x), and numerical evidence that is only qualitative (plots plus a residual whose definition is ambiguous). Both are fixable in revision.","major_comments":[{"comment":"The paper does not prove that the ground state of the problem Hamiltonian has quadrature expectations minimizing L(x); it simply inherits the substitution rules from Ref. [18]. For the Hamiltonians actually constructed here this is true and should be shown: H_P is a quadratic, number-conserving bosonic Hamiltonian with only linear drive terms and no pairing terms, so it can be completed to H_P = (a + 1/2 Q^{-1} r)^\\dagger Q (a + 1/2 Q^{-1} r) - (1/4) r^T Q^{-1} r + c, whose ground state is a multi-mode coherent state with \\langle x_i \\rangle = - (1/2)(Q^{-1} r)_i = (C^T C)^{-1} C^T b. The authors should add this argument, discuss the case where Q is singular (pseudo-inverse, possible ground-state degeneracy), and clarify that the general substitution Eq. (4) is not an exact quantization for arbitrary cost functions. Without this proof, the central claim rests entirely on two numerical examples.","section":"Sec. II A, Eq. (4); Sec. III, Eq. (23)"},{"comment":"The quantity plotted as the expectation value of \\hat H_P is inconsistent with the definition in Eq. (23), which includes the additive constant c = b^T b from Eq. (14). For Eq. (16) with M=7, c is approximately 2229, so \\langle \\hat H_P \\rangle cannot approach zero if Eq. (23) is used literally. If the simulation omitted the constant term, as is standard for optimization, the paper should state this explicitly and define the residual operator actually plotted; otherwise the convergence criterion is ill-posed. This matters because Figs. 2 and 5 are the main quantitative evidence for convergence.","section":"Sec. IV B, Eq. (23); Figs. 2, 3, 5, 6"},{"comment":"The numerical results are not sufficiently specified or quantified. The paper states that simulations were implemented in QuTiP but does not report the Fock-space truncation dimension, the solver or tolerance settings, or a convergence check in the truncation dimension. It also gives no quantitative error between the final \\langle x_i \\rangle and the analytic solution, e.g., max or RMS deviation over the grid for each M and T; Figs. 1 and 4 show only visual agreement. Please add such error metrics, report the truncation dimension, and state that the closed-system evolution is deterministic, in which case error bars are not required. Without these additions, the conclusion that the solutions converge toward the analytical solutions cannot be independently checked.","section":"Sec. IV A and Figs. 1-6"}],"minor_comments":[{"comment":"The phrase 'Section 3' should be 'Section III' for consistent nomenclature.","section":"Sec. IV B"},{"comment":"The acronym CV-QA is used in the abstract and introduction but is never defined; define it at first use.","section":"Abstract and Sec. I"},{"comment":"In the version provided, the captions contain garbled text rendering 'analytic solution'; please check the production files before submission.","section":"Figs. 1 and 4 captions"},{"comment":"A short discussion of why M=7 is sufficient and how the discretization error behaves with M would help separate the finite-difference error from the annealing error.","section":"Sec. IV A"},{"comment":"Since Q = C^T C is symmetric by construction, the symmetrized coefficient (Q_ij + Q_ji)/2 in Eq. (5) could be simplified to Q_ij, and the paper can note this.","section":"Sec. II A, Eq. (5)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the scope of a quantum-information journal and the core idea is viable as a proof of principle. The concerns are fixable with a short proof of the ground-state readout and a more complete numerical report; I do not see grounds for rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read it. The new thing is narrow but real: the first continuous-variable quantum annealing formulation for linear ODEs, built from finite differences, a squared residual, and the bosonic substitution rules of Ref. [18]. The reduction from Eq. (12) to Eq. (23) is clean; the C and b matrices for both examples check out, and the authors compare against analytic solutions with no fitted parameters, so the circularity concern does not land. Within the stated conditions (M=7, T=10000), the solutions do converge to the analytic curves.\n\nThe softness is in the load-bearing step. Eq. (4) is taken from Ref. [18] without proof, and the reader cannot tell why the ground state readout should give the least-squares vector. Your stress-test note is right: for this Hamiltonian class the substitution is exact, not approximate. H_P is quadratic, number-conserving, with linear drives only, so it completes to a displaced quadratic form; the ground state is a multi-mode coherent state and its quadrature expectations are exactly (C^T C)^{-1} C^T b. That fact is never stated. Without it, a reader cannot judge whether the method is reliable beyond the two toy examples. So the gap is an omitted proof, not a wrong derivation.\n\nThe numerics are also under-reported. No Fock truncation dimension, no code, no quantitative error between the final ⟨x_i⟩ and the analytic values, only plots and the residual ⟨H_P⟩. The residual plots do what they claim, but they don't let a reader reproduce or compare. One more overreach: the abstract says \"without the discretization overhead.\" What they avoid is binary encoding of the variables; time is discretized as finely as in any finite-difference method. The conclusion should say variable-encoding overhead.\n\nNone of this is fatal. The paper is a legitimate, incremental extension of Ref. [18] to a new problem class, and it is honest about being a proof of principle. A serious referee should see it, but the referee should ask for the completion argument and the missing numerical details before publication.","headline":"A clean but thin proof of principle: the ODE-to-CV-QA mapping is correct and the substitution is exact for this Hamiltonian class, but the proof and numerics are under-reported.","tokens_in":9961,"tokens_out":2737,"would_cite":false,"duration_ms":32990,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A linear differential equation can be discretized with finite differences and turned into a quadratic cost over continuous variables; the paper claims that continuous-variable quantum annealing on bosonic modes finds the ground state…","keywords":["continuous-variable quantum annealing","linear differential equations","bosonic modes","finite-difference method","quadratic optimization","cavity modes","quantum annealing","adiabatic evolution"],"falsifier":"Compare the ground-state quadrature readouts against the classical least-squares solution for a discretized equation where the classical residual has a unique minimum and the Hamiltonian is known to have a squeezed ground state; a detectable deviation beyond the finite-difference error would show the mapping does not preserve the optimization problem.","tokens_in":8925,"feed_emoji":"⚛️","tokens_out":6309,"duration_ms":69671,"temperature":0.7,"pith_summary":"The paper tries to establish that linear differential equations can be solved by continuous-variable quantum annealing, in which each unknown grid value is a bosonic mode rather than a register of qubits. The authors discretize the equation with finite differences, square the residual to get a quadratic cost, and map that cost to a Hamiltonian whose ground state's quadrature expectations give the solution. Numerical simulation of two second-order linear ordinary differential equations with fixed endpoints reproduces the analytical solutions under the tested conditions. The significance would be a route to differential-equation solvers that avoid the precision-driven qubit overhead of binary encoding.","feed_headline":"Differential equations solved by bosonic quantum annealer","feed_subtitle":"Quadrature expectations of the annealer's ground state match analytic solutions for two second-order ODEs.","key_machinery":"The central object is the operator substitution rule of Eq. (4), which converts the quadratic cost $L(\\theta)=\\theta^T Q\\theta + r^T\\theta + c$ into a bosonic Hamiltonian built from number operators $a_i^\\dagger a_i$, beamsplitter-style couplings $(a_i^\\dagger a_j + a_i a_j^\\dagger)/2$, and linear drives $(a_i+a_i^\\dagger)/2$. The finite-difference matrix $C$ from the differential equation enters through $Q=C^T C$ and $r=-2C^T b$, so the classical residual is exactly the quadratic cost. The driver Hamiltonian $H_D=d\\sum_i a_i^\\dagger a_i$ has the vacuum as its ground state, and adiabatic evolution from vacuum to $H_P$ is intended to leave the system in $H_P$'s ground state, whose quadrature expectations are read out as $x(\\tau_i)$.","core_discovery":"The central claim is that the squared residual $L(x)=(Cx-b)^T(Cx-b)$ of a finite-difference discretization of a linear differential equation is a quadratic form, and applying the operator substitution $\\theta_i^2 \\to a_i^\\dagger a_i$, $\\theta_i\\theta_j \\to (a_i^\\dagger a_j + a_i a_j^\\dagger)/2$, $\\theta_i \\to (a_i+a_i^\\dagger)/2$ yields a bosonic problem Hamiltonian whose ground state, read out through $\\langle (a_i+a_i^\\dagger)/2\\rangle$, solves the discretized equation. The paper demonstrates this for $d^2x/d\\tau^2=2$ and $d^2x/d\\tau^2+5\\,dx/d\\tau=-1$ with $x(0)=0$, $x(1)=1$, using $M=7$ grid intervals, annealing time $T=10000$, and a numerical simulation library; in both cases the annealed values are close to the analytical solutions.","pith_inferences":["A direct extension the paper does not spell out is that the same squared-residual construction applies to systems of coupled linear differential equations, since the finite-difference matrix $C$ simply grows to cover all variables and the cost remains quadratic.","The numerical trend that $\\langle H_P\\rangle$ stops approaching zero as $M$ grows suggests that the bosonic encoding trades qubit count for sensitivity to adiabaticity; the practical overhead may reappear as longer annealing times rather than disappearing.","Because the substitution rule omits the $a^2$ and $a^{\\dagger 2}$ terms that an exact quantization of $\\theta_i^2$ would contain, a squeezed or non-coherent ground state could in principle spoil the readout; checking the second moments of the final state would test reliability beyond the two examples.","For nonlinear differential equations the residual is no longer quadratic, so the method as stated does not extend; whether a polynomial residual could be mapped by higher-order operator products is an open question the paper leaves implicit."],"forward_implications":["A linear ordinary differential equation of any finite order can be turned into a quadratic cost by finite differences, so the construction is not limited to the two example equations.","Because each variable is encoded in one bosonic mode, the number of physical modes equals the number of unknown grid points, rather than the number of grid points times the bits per variable.","Increasing the annealing time $T$ drives the final expectation value of $H_P$ toward zero, meaning longer annealing improves satisfaction of the discretized equations.","Under conditions where annealing reaches the ground state, the results are relatively insensitive to the driver strength $d$.","The two tested second-order equations reproduce their analytical solutions at $M=7$ and $T=10000$, indicating that a moderate grid can suffice for these cases."],"supporting_citations":[{"why":"Supplies the operator substitution rules and the cavity-based continuous-variable annealing setup that the paper applies to differential equations.","marker":"[18]"},{"why":"Provide the quadratic-optimization reformulation of linear problems that is the paper's starting point when converting a discretized equation into a cost function.","marker":"[19, 20]"},{"why":"Provides a prior annealing-based approach to differential equations that this work extends to continuous variables.","marker":"[25]"},{"why":"Supply the simulation library used for all numerical experiments in the paper.","marker":"[29, 30]"}],"fun_headline_variants":["Bosonic annealer solves ODEs without discretization","Continuous-variable annealer solves ODEs","Annealing bosons to solve linear ODEs","Quantum annealer for ODEs uses bosonic modes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The method assumes that the quantum ground state of the bosonic Hamiltonian, after the variable-to-operator substitution, has average readings that match the minimum of the original classical equation residual; this is inherited from earlier work and is not proved for differential equations here.","fun_headline_variants_meta":{"raw":{"variants":["Bosonic annealer solves ODEs without discretization","Continuous-variable annealer solves ODEs","Annealing bosons to solve linear ODEs","Quantum annealer for ODEs uses bosonic modes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000631,"raw_usage":{"total_tokens":2891,"prompt_tokens":898,"completion_tokens":1993,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":514,"completion_tokens_details":{"reasoning_tokens":1929}},"tokens_in":514,"tokens_out":1993,"duration_ms":17745,"temperature":1.0,"reasoning_tokens":1929,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T21:01:41.029954+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compare the ground-state quadrature readouts against the classical least-squares solution for a discretized equation where the classical residual has a unique minimum and the Hamiltonian is known to have a squeezed ground state; a detectable deviation beyond the finite-difference error would show the mapping does not preserve the optimization problem.","supporting_citations":[{"cited_title":"Karimi and P","cited_arxiv_id":null,"evidence_quote":"Supplies the operator substitution rules and the cavity-based continuous-variable annealing setup that the paper applies to differential equations."}],"review_version":1}