{"id":"a533837d-340f-4d87-937f-30d37de59aee","arxiv_id":"2505.04304","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Quantum circuits based on Schrodingerisation are constructed for the Black-Scholes equation, with complexity estimates and small-scale statevector demonstrations.","lead":"This paper designs quantum circuits that simulate the Black-Scholes option-pricing equation by converting it into a unitary quantum evolution. It matters because it targets high-dimensional multi-asset pricing, though the d-dimensional formulas need correction and no hardware run is shown.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (4.4) omits the -rU term from the d-dimensional Black-Scholes equation (4.2), so the circuits in Figs. 4.1-4.3 simulate the PDE without discounting; Theorem 5.2 is not supported as stated.","rationale":"The reader's weakest_assumption already identifies the missing discounting term in Eq. (4.4), and an independent derivation from Eq. (4.2) confirms that the omission is real rather than a harmless notational difference. This is the most load-bearing concern because Theorem 5.2 states complexity for Eq. (4.3), which inherits the -rU term from Eq. (4.2); if the implemented Hamiltonian lacks that term, the prepared quantum state solves a different PDE. The 1D sign inconsistency in Eq. (3.7) is related: the correct bottom-right block of C1 is -rI, which is what Eq. (3.10) implicitly uses, not +rI as printed. Since the d-dimensional construction is the main claimed contribution, a correctable but unaddressed omission of discounting makes the d-dimensional claims conditional at best. I do not see a need to change the reader's conditional verdict: the issue is concrete and fixable, but it must be corrected and re-validated before the high-dimensional claim can be accepted as stated.","tokens_in":19506,"tokens_out":12601,"duration_ms":126415,"concrete_test":"Recompute A1 = (A + A†)/2 directly from Eq. (4.2) and compare the coefficient of Dη with Eq. (4.4). If -r I⊗Dη is missing, apply the resulting operator to a spatially constant initial state with r>0, all σ_m=0, and T=1: Eq. (4.2) gives U(T) = e^{-rT} U(0), whereas the Eq. (4.4) Hamiltonian annihilates the spatial derivatives and yields U(T) = U(0). This one-line calculation detects the missing discounting unambiguously.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Start from Eq. (4.2): dU/dτ = Σ_m (r - σ_m^2/2)(D±)_m U + Σ_m (σ_m^2/2)(DΔ)_m U - rU = AU. Hence A1 = (A + A†)/2 contains -rI, and HBS = ηA1 + A2 must contain -η r I⊗Dη. The 1D Eq. (3.10) does contain this term, as -r/Lp I on both the |0> and |1> sectors, and Eq. (3.7)'s bottom-right +rI is a sign typo: it should be -rI. The d-dimensional Eq. (4.4), however, expands HBS only as Σ_m (σ_m^2 ρ_mm/2)(H1)_m⊗Dη + Σ_m (r - σ_m^2/2)(H2)_m⊗I, with no -r I⊗Dη term. Consequently the explicit circuits of Figs. 4.1-4.3 implement the Hamiltonian of the undiscounted PDE, not Eq. (4.2). For example, on a spatially constant vector all difference operators vanish; Eq. (4.2) predicts decay e^{-rT}, while the Hamiltonian in Eq. (4.4) leaves the state unchanged. The closed-form solution used in Example 6.2 contains an explicit e^{-rτ} factor, so that numerical validation cannot reproduce the published circuit unless the discounting is inserted elsewhere. The omission is plausibly fixable by adding the same Ph(-τ r/Lp) phase gates used in Eq. (3.12), but Theorem 5.2 and Figs. 4.1-4.3 must be revised accordingly before the d-dimensional claim can be accepted.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs explicit quantum circuits for the one- and d-dimensional Black-Scholes equations using the Schrödingerisation method. For the 1D case, the authors discretize the log-price PDE, introduce an ancilla dilation to handle boundary terms, derive a Hermitian Hamiltonian H_BS, approximate the time evolution by first-order Lie-Trotter-Suzuki decompositions, and present explicit circuits (Figs. 3.1-3.5) with a complexity estimate (Theorem 5.1). For the d-dimensional case with uncorrelated assets (ρ_mn=0, m≠n), analogous circuits are given (Figs. 4.1-4.3) and a complexity estimate is stated (Theorem 5.2) that is polynomial in d. Numerical experiments on Qiskit are reported for 1D and 2D examples.","tokens_in":19879,"tokens_out":7343,"duration_ms":71861,"significance":"If the central claims were fully supported, the paper would provide one of the few end-to-end explicit quantum circuit constructions for a financial PDE, with complexity polynomial in dimension d and logarithmic in grid size, together with numerical validation. The authors make a genuine effort to give gate-level details and to quantify Trotter, discretization, and measurement errors, which is valuable for a practitioner-oriented quantum simulation paper. The 1D construction follows the established Schrödingerisation pipeline and the 1D numerics are consistent with the exact solution. However, the d-dimensional Hamiltonian in Eq. (4.4) omits the discounting term -rU, so the stated d-dimensional theorem and its circuit implementations are not currently supported.","major_comments":[{"comment":"The d-dimensional Hamiltonian omits the discounting term. Starting from Eq. (4.2), A = Σ_m (r-σ_m²/2)(D±_D)_m + Σ_m (σ_m²ρ_mm/2)(DΔ_D)_m - rI, so A1 = (A+A†)/2 contains -rI. Consequently H_BS = ηA1 + A2 must contain -η r I⊗D_η. Eq. (4.4), however, contains only Σ_m (σ_m²ρ_mm/2)(H1)_m⊗|k⟩⟨k| and Σ_m (r-σ_m²/2)(H2)_m⊗I, with no rI⊗D_η term. A direct consequence is that for a spatially constant vector, Eq. (4.2) predicts decay e^{-rτ}, while the Hamiltonian in Eq. (4.4) leaves the state unchanged. Lemmas 5.4-5.6, Theorem 5.2, and Figs. 4.1-4.3 therefore implement the undiscounted PDE, not Eq. (4.2). This is fixable by adding the same Ph(-τ r/L_p) phase gates used in Eq. (3.12) to the k-controlled part of the d-dimensional circuit, but the circuits and all complexity statements depending on Eq. (4.4) must be revised accordingly.","section":"§4.2, Eq. (4.4)"},{"comment":"The 2D numerical example does not validate the circuits of Figs. 4.1-4.3. Example 6.2 uses correlated assets with ρ12=0.6 and an exact solution containing the factor e^{-rτ}, while the circuits in §4.2 are constructed for the uncorrelated case ρ_mn=0, m≠n, and, as noted above, omit the -rU discounting term. The numerical experiment therefore appears to validate a different (presumably correctly discounted) central-difference Schrödingerisation, not the published circuit construction. If the intent is to validate the proposed circuits, the example must be run with the actual circuits or the claims must be restricted to the uncorrelated case with the discounting term included.","section":"§4 and Example 6.2"},{"comment":"The lower-right block of C1 is written as +rI, but direct calculation from Eq. (3.6) gives C1 = (C+C†)/2 with lower-right block -rI. This sign error is present in a central equation of the derivation. Although the later Hamiltonian in Eq. (3.10) and the phase gates in Eq. (3.12) use the correct -r/L_p sign, the inconsistency must be corrected in Eq. (3.7) and in any statement about the spectrum of C1.","section":"§3.1, Eq. (3.7)"},{"comment":"The recovery condition p⋄ > max(λ_n(C1)T, 0) is imported from reference [10] without verifying the spectral assumptions for the specific C1 constructed here. The proof of Theorem 5.1 assumes p⋄ = O(1), which requires a bound on λ_max(C1)T; no such bound is provided for the eigenvalues of A1 or for the perturbation caused by the boundary matrix B. The authors should either state the precise sufficient condition from [10], prove that it holds for this Hamiltonian, and show that p⋄ can indeed be chosen O(1) uniformly in the discretization parameters, or adjust the complexity statement accordingly.","section":"Remark 3.4 and Theorem 5.1"}],"minor_comments":[{"comment":"The two displayed finite-difference definitions are both labeled (D+u)_j; the second should be the backward difference (D_-u)_j or a similar distinct symbol.","section":"§2, Eq. (2.2)"},{"comment":"The notation ar u(x_1,0),...,ar u(x_{2N_x-2},0) is confusing because the spatial grid has only N_x-1 interior points after the dilation, so the index range should be over the dilated vector components, not over x_j indices. Please clarify the indexing.","section":"§3.2, Eq. (3.9)"},{"comment":"The proof uses |U(0)| and |U(T)| instead of the norms ∥U(0)∥ and ∥U(T)∥ used in the statement; the notation should be made consistent.","section":"Theorem 5.2, proof"},{"comment":"Several arXiv identifiers contain typos, e.g., 'ariXv:2403.19123v3' in [11] and 'ariXv:2402.18398v1' in [18]; please correct them.","section":"References"},{"comment":"In the displayed Trotter error estimate for the first decomposition, the commutator norm is written without a norm symbol around the operator; adding the missing norm would make the bound rigorous and easier to follow.","section":"§5, Lemma 5.1 proof"}],"recommendation":"major_revision","confidential_remarks":"The d-dimensional claim is the main advertised quantum advantage, and the missing -rI⊗D_η term in Eq. (4.4) is a genuine technical error that invalidates Theorem 5.2 and Figs. 4.1-4.3 as stated. The error is local and fixable within the manuscript's scope, so I recommend major revision rather than rejection. I would also ask the editor to ensure that the revised version either proves the recovery-condition bound for the specific C1 or clearly imports it with the necessary hypotheses, since the current manuscript leaves the p⋄ = O(1) assumption unexamined. The reliance on several papers by the same group is not itself problematic, because those papers contain independent derivations, but the applicability of the imported theorem needs to be checked explicitly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nRead arXiv:2505.04304. The useful part is the 1D construction: explicit circuits for Black-Scholes via Schrodingerisation, including the dilation for the inhomogeneous boundary term and the binary-control trick for powers of V1. The numerics in Example 6.1 match the exact solution and the error decreases as np grows. No free parameters are fitted; the core maps come from prior work, so the circularity burden is low.\n\nThe d-dimensional claim has a real hole. Eq. (4.2) contains -rU. When you build A1 and then HBS = A1⊗Dη + A2⊗I, the -rI⊗Dη term has to be there. Eq. (4.4) omits it. On a constant vector all difference operators vanish; Eq. (4.2) predicts decay e^{-rτ}, while the Hamiltonian in (4.4) leaves the state unchanged. Example 6.2 uses an exact solution with an explicit e^{-rτ} factor, so either the numerics inserted the discounting somewhere not shown or they do not match the published circuits. The fix is straightforward, adding the same phase gates used in 1D, but Theorem 5.2 and Figures 4.1-4.3 need revision before the high-dimensional result can be accepted.\n\nTwo smaller things. Eq. (3.7) writes the lower-right block of C1 as +rI; direct calculation gives -rI. The later Hamiltonian (3.10) uses the correct sign, so this is a typo, not a load-bearing error. And Remark 5.2's 'O~(d/ε³)' hides the h^{-4} and the norm ratios; the full theorem is honest about these, but the remark overstates the practical advantage.\n\nNo code was shipped, so the Qiskit experiments are not independently reproducible from the text. That is minor for a theory/circuit paper.\n\nOverall: the 1D part is a solid application of a known framework. The d-dimensional part is not supported as written, but the missing term is identifiable and fixable. I would send it to peer review; a good referee can force the correction. I would cite the 1D construction, not the dD theorem until it is fixed.","headline":"Solid 1D Schrodingerisation circuit for Black-Scholes; the d-dimensional Hamiltonian drops the -rU discounting term, so Theorem 5.2 needs a fix before the high-dimensional claim holds.","tokens_in":20409,"tokens_out":4414,"would_cite":true,"duration_ms":41686,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P68","91G20","65M06"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper constructs explicit quantum circuits for the Black-Scholes equation whose gate count grows polynomially in the number of assets, breaking the curse of dimensionality in the uncorrelated case.","keywords":["Black-Scholes equation","Schrödingerisation","quantum circuits","option pricing","quantum simulation of PDEs","finite difference methods","high-dimensional pricing","Trotter-Suzuki decomposition"],"falsifier":"Simulate the paper's two-dimensional cash-or-nothing example with $r=0$ and with $r>0$, comparing the circuit output to the exact solution $c e^{-r\\tau}B(d_x,d_y,\\rho)$. If the $r>0$ error is systematically larger by a factor tied to $1-e^{-r\\tau}$ than the $r=0$ case, the omitted discounting term is present in the circuits and the $d$-dimensional theorem describes a different PDE.","tokens_in":19293,"feed_emoji":"📈","tokens_out":11315,"duration_ms":98142,"temperature":0.7,"pith_summary":"This paper claims that the Black-Scholes equation of option pricing can be simulated by explicit quantum circuits, and that in $d$ spatial dimensions those circuits avoid the exponential scaling that classical grid methods suffer. The central mechanism is Schrödingerisation: a warped-phase change of variables followed by a Fourier transform converts the non-unitary diffusion equation into a unitary Schrödinger equation whose Hamiltonian is assembled from finite-difference shift operators. For a one-dimensional European call the proof gives a gate count of $\\widetilde{O}(T^{2}\\Vert u(0)\\Vert^{3}/(h^{4}\\varepsilon^{3}\\Vert u(T)\\Vert^{3}))$, and for $d$ uncorrelated underlying assets the count is $\\widetilde{O}(d\\,T^{2}\\Vert U(0)\\Vert^{3}\\sum_{m} c_{m}^{2}/(h^{4}\\varepsilon^{3}\\Vert U(T)\\Vert^{3}))$, polynomial in $d$. The paper also reports one- and two-dimensional numerical experiments that reproduce the analytic option price.","feed_headline":"Explicit quantum circuits price options with cost polynomial in d","feed_subtitle":"The d-asset Black-Scholes equation becomes a qubit circuit with polynomial-in-d cost—no exponential grid.","key_machinery":"The workhorse is the warped phase transformation $v(\\tau,p)=e^{-p}\\bar u(\\tau)$ for $p>0$, extended symmetrically to $p<0$, followed by a Fourier transform in $p$. This converts a non-unitary ODE into $\\partial_{\\tau}\\hat v=i(\\eta C_{1}+C_{2})\\hat v$ with Hermitian $H_{\\mathrm{BS}}=\\eta C_{1}+C_{2}$. The finite-difference operators $D_{\\pm}$ and $D_{\\Delta}$ are written as sums of shift operators $S_{\\pm}=\\sum_{j} I\\otimes\\cdots\\otimes\\sigma_{01}\\otimes\\cdots\\otimes\\sigma_{10}$, and each shift term becomes a one-parameter unitary block $\\exp(i\\gamma\\tau(e^{i\\lambda}s_{j}^{-}+e^{-i\\lambda}s_{j}^{+}))=B_{j}(\\lambda)\\,\\mathrm{CRZ}_{1\\ldots j-1}^{j}(-2\\gamma\\tau)\\,B_{j}(\\lambda)^{\\dagger}$, built from Hadamards, phase gates, CNOT ladders, and a multi-controlled rotation. Lie-Trotter-Suzuki splitting assembles these blocks into the full evolution, and the binary representation of the index $k$ reduces many controlled applications to a logarithmic number of controlled powers. Recovery of $u(T)$ uses the inverse quantum Fourier transform and a projection onto $p>0$, relying on the condition $p^{\\star}>\\lambda_{\\max}(C_{1})T$ imported from the theory of Schrödingerisation with inhomogeneous terms.","core_discovery":"On the paper's own terms, the central discovery is that the finite-difference Black-Scholes Cauchy problem can be embedded in a Hermitian evolution and then assembled from standard one- and two-qubit gates. After the log-price substitution, the PDE becomes a constant-coefficient convection-diffusion equation; central differencing yields a non-Hermitian ODE of the form $\\frac{du}{d\\tau}=Au+b$. Dilating the state with an auxiliary recovery variable and applying Schrödingerisation gives a Hermitian Hamiltonian $H_{\\mathrm{BS}}=C_{1}\\otimes D_{\\eta}+C_{2}\\otimes I$. The circuits approximate $\\exp(i\\tau H_{\\mathrm{BS}})$ by Trotter products of Bell-basis blocks $W_{j}(\\gamma\\tau,\\lambda)$, controlled phase gates, and controlled rotations; binary decomposition of the index $k$ keeps the circuit logarithmic in the number of $p$-grid points. Theorems 5.1 and 5.2 assert that, with these circuits plus a quantum Fourier transform and one projection, the solution state is prepared to precision $\\varepsilon$ with the gate counts quoted above, and that the count for $d$ uncorrelated assets is polynomial in $d$, not exponential.","pith_inferences":["If the $d$-dimensional Hamiltonian is corrected to include the discounting term, the same circuit architecture should still work with an extra phase factor; a direct check would be to compare the $d=2$ circuit output with and without the $-rU$ term.","The factor $\\Vert U(0)\\Vert^{3}/\\Vert U(T)\\Vert^{3}$ means the cost blows up when the option value decays; amplitude amplification could reduce the measurement overhead $O(\\Vert U(0)\\Vert^{2}/\\Vert U(T)\\Vert^{2})$ to a square-root scaling, improving the $\\varepsilon$-dependence.","For correlated assets, the paper notes that $O(d^{2})$ coupling terms enter; because each coupling is a tensor product of one-dimensional shift operators, the circuit construction should extend polynomially in $d$, suggesting the practical speedup survives for full correlation matrices.","Alternative spatial discretizations, such as higher-order finite differences or spectral methods, could lower the $h^{-4}$ factor that dominates the quoted complexity, since the Schrödingerisation machinery is agnostic to the chosen discretization."],"forward_implications":["The one-dimensional European-call circuit has a proved complexity of at most $\\widetilde{O}(T^{2}\\Vert u(0)\\Vert^{3}/(h^{4}\\varepsilon^{3}\\Vert u(T)\\Vert^{3}))$ single-qubit and CNOT gates, so the circuit depth remains polynomial in $1/h$, $T$, and $1/\\varepsilon$.","In the uncorrelated $d$-asset case the gate count grows polynomially in $d$, whereas storing the classical finite-difference state would cost $O(N_x^d)$ amplitudes, giving an exponential memory saving for $d\\ge 3$.","Cash-or-nothing options can be handled without the dilation step, because their boundary values vanish and the source term $b$ is zero.","With smooth initial data in the auxiliary $p$ variable, the numerical experiments exhibit second-order convergence in the $p$-discretization, matching the error analysis in the paper."],"supporting_citations":[{"why":"Supplies the Bell-basis circuit construction that turns finite-difference shift operators into one- and two-qubit gates.","marker":"[8]"},{"why":"Provides the recovery condition $p^{\\star}>\\lambda_{\\max}(C_{1})T$ and the dilation used for inhomogeneous boundary terms.","marker":"[10]"},{"why":"Gives the scalable Bell-basis approach and the error analysis for wave- and heat-type PDE circuits.","marker":"[18]"},{"why":"Supplies the commutator-scaling Trotter error bound used in Lemmas 5.1 and 5.4.","marker":"[6]"},{"why":"Provides the multi-controlled gate decompositions used to count CNOT gates.","marker":"[20]"},{"why":"Underlies the $p$-space discretization error estimate and the relative error $\\Delta p$ used in the numerical discussion.","marker":"[9]"},{"why":"Supplies the smoother initial data $g(p)$ used to reach second-order convergence in the experiments.","marker":"[11]"}],"fun_headline_variants":["Quantum circuits for Black-Scholes beat exponential cost","Polynomial quantum circuits for option pricing","No exponential grid: quantum circuits for d-asset options","Quantum advantage: pricing options with polynomial cost"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The $d$-dimensional result assumes that Eq. (4.4), the Hamiltonian the circuits implement, is the correct discretization of Eq. (4.2); as written it drops the $-rU$ discounting term that the 1D construction keeps.","fun_headline_variants_meta":{"raw":{"variants":["Quantum circuits for Black-Scholes beat exponential cost","Polynomial quantum circuits for option pricing","No exponential grid: quantum circuits for d-asset options","Quantum advantage: pricing options with polynomial cost"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0004,"raw_usage":{"total_tokens":2070,"prompt_tokens":906,"completion_tokens":1164,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":522,"completion_tokens_details":{"reasoning_tokens":1106}},"tokens_in":522,"tokens_out":1164,"duration_ms":8929,"temperature":1.0,"reasoning_tokens":1106,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T23:33:16.014621+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the paper's two-dimensional cash-or-nothing example with $r=0$ and with $r>0$, comparing the circuit output to the exact solution $c e^{-r\\tau}B(d_x,d_y,\\rho)$. If the $r>0$ error is systematically larger by a factor tied to $1-e^{-r\\tau}$ than the $r=0$ case, the omitted discounting term is present in the circuits and the $d$-dimensional theorem describes a different PDE.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Bell-basis circuit construction that turns finite-difference shift operators into one- and two-qubit gates."},{"cited_title":"Childs, Y","cited_arxiv_id":null,"evidence_quote":"Supplies the commutator-scaling Trotter error bound used in Lemmas 5.1 and 5.4."},{"cited_title":"Vale, T.M.D","cited_arxiv_id":null,"evidence_quote":"Provides the multi-controlled gate decompositions used to count CNOT gates."},{"cited_title":"Jin and N","cited_arxiv_id":null,"evidence_quote":"Underlies the $p$-space discretization error estimate and the relative error $\\Delta p$ used in the numerical discussion."}],"review_version":1}