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Quantum Circuits for the Black-Scholes equations via Schr\"{o}dingerisation

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2505.04304 v1 pith:TRTQXTHL submitted 2025-05-07 quant-ph cs.NAmath.NA

Shi Jin , Zihao Tang , Xu Yin , Lei Zhang This is my paper · ORCID
classification quant-phcs.NAmath.NA MSC 81P6891G2065M06
keywords Black-ScholesequationSchrödingerisationquantumcircuitsoptionpricingsimulationofPDEsfinitedifferencemethodshigh-dimensionalTrotter-Suzukidecomposition
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

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

4 major / 5 minor

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.

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 (4)
  1. [§4.2, Eq. (4.4)] 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.
  2. [§4 and Example 6.2] 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.
  3. [§3.1, Eq. (3.7)] 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.
  4. [Remark 3.4 and Theorem 5.1] 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.
minor comments (5)
  1. [§2, Eq. (2.2)] 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.
  2. [§3.2, Eq. (3.9)] 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.
  3. [Theorem 5.2, proof] 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.
  4. [References] Several arXiv identifiers contain typos, e.g., 'ariXv:2403.19123v3' in [11] and 'ariXv:2402.18398v1' in [18]; please correct them.
  5. [§5, Lemma 5.1 proof] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Black-Scholes circuits are an application of prior Schrödingerisation and Trotter-splitting lemmas to a new equation, and no fitted parameter is relabeled as a prediction.

full rationale

The derivation chain is not circular. The Schrödingerisation transform, the recovery condition p* > max(lambda_n(C1)T, 0), and the Bell-basis Wj circuit construction are taken from prior papers by the same group ([8], [10]), but those are general parameter-free results whose stated assumptions do not include the Black-Scholes conclusion being derived here. The present paper's contribution is the application to the Black-Scholes equation and the explicit U1/U2 operator decompositions; it does not fit any parameter to the target solution. The numerical experiments compare circuit output against independent closed-form solutions for European call options and two-asset cash-or-nothing options, rather than against quantities extracted from the circuit itself. The complexity bounds in Theorems 5.1 and 5.2 follow from Trotter error estimates and state-preparation/measurement norms, not from assuming the stated gate counts. Two non-circularity issues should nevertheless be flagged: (i) Remark 3.4 and Theorem 5.1 import the recovery condition from [10] without repeating the proof, which is an omitted-support gap rather than a circular reduction; (ii) Eq. (4.4) appears to omit the -rU discounting term present in Eq. (4.2), whereas the 1D Hamiltonian (3.10) retains it, which is a correctness/consistency concern for Theorem 5.2 rather than an equivalence-by-construction. Neither issue makes the predictions equal to their inputs.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No data-fitted constants appear; the only numerical choices are algorithm discretization parameters such as h, Lp, Np, and the time step, which are not fitted to target outputs. The central claim rests on the Schrodingerisation recovery theorem from [10], finite-difference discretization, Trotter error bounds, and the rho=0 restriction.

assumptions (6)
  • domain assumption Central finite-difference discretization (3.4)-(3.5) converges to the Black-Scholes solution on the truncated domain with Dirichlet or mixed boundaries.
    The error analysis in Theorem 5.1 assumes the ODE (3.4) approximates the PDE (3.3) to the desired accuracy; no discrete maximum principle or convergence proof is given here.
  • domain assumption The Schrodingerisation framework of prior work (warped phase transform, Hermitian dilation, p-recovery) applies to the ODE system (3.4).
    The paper cites [10] Theorem 5.1 for the recovery condition p* greater than lambda_max(C1) T and uses the method as a black box.
  • standard math The Bell-basis circuit identity W_j(gamma tau, lambda) exactly implements exp(i gamma tau (e^{i lambda} s_-^j + e^{-i lambda} s_+^j)).
    Taken from [8]; used in Eq. (3.16) to build V1 and V2.
  • standard math First-order Lie-Trotter-Suzuki splitting error bounds from [6,18] apply with the stated commutator scalings.
    Used in the proof of Lemma 5.1 for the operator-norm error between UBS and VBS.
  • domain assumption For the d-dimensional case, correlations are set to zero, rho_mn = 0 for m unequal to n, so the cross-derivative terms are discarded.
    Stated in Section 4.1; Example 6.2 nevertheless uses rho=0.6 outside this assumption.
  • standard math Truncation in p and Fourier grid Np produce discretization error O(pi Lp / Np + e^{-pi Lp}).
    Used in Theorem 5.1; imported from Fourier analysis of the warped-phase initial data.

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Cite this review

Pith. "Pith review of Quantum Circuits for the Black-Scholes equations via Schr\"{o}dingerisation." pith.science (2026). https://pith.science/paper/TRTQXTHL

@misc{pith2026250504304,
  author       = {Pith},
  title        = {Pith review of: Quantum Circuits for the Black-Scholes equations via Schr\"odingerisation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TRTQXTHL}},
  note         = {Machine review of arXiv:2505.04304}
}
read the original abstract

In this paper, we construct quantum circuits for the Black-Scholes equations, a cornerstone of financial modeling, based on a quantum algorithm that overcome the cure of high dimensionality. Our approach leverages the Schr\"odingerisation technique, which converts linear partial and ordinary differential equations with non-unitary dynamics into a system evolved by unitary dynamics. This is achieved through a warped phase transformation that lifts the problem into a higher-dimensional space, enabling the simulation of the Black-Scholes equation on a quantum computer. We will conduct a thorough complexity analysis to highlight the quantum advantages of our approach compared to existing algorithms. The effectiveness of our quantum circuit is substantiated through extensive numerical experiments.

Figures

Figures reproduced from arXiv: 2505.04304 by the authors.

Figure 3.1
Figure 3.1. Quantum circuit for Wj (γτ, λ) q1 X • • • • X • Rx q2 W1(γ1τ, 0) · · · Wnx−1(γ1τ, 0) Wnx (γ1τ, 0) Ph(γ1τ ) Ph(τ r/Lp) • . . . · · · qnx • qnx+1 • [PITH_FULL_IMAGE:figures/full_fig_p009_3_1.png] view at source ↗
Figure 3.2
Figure 3.2. Quantum circuit for V˜ 1(τ ) q1 X • • • X Ry q2 W1(γ2τ, − π 2 ) Wnx−1(γ2τ, − π 2 ) Wnx (γ2τ, − π 2 ) • . . . · · · qnx • qnx+1 • [PITH_FULL_IMAGE:figures/full_fig_p009_3_2.png] view at source ↗
Figure 3.3
Figure 3.3. Quantum circuit for V˜ 2(τ ) q / nx+1 V˜ 1(τ ) V˜ 2 1 (τ ) · · · V˜ 2 np−1 1 (τ ) V˜ −2 np−1 1 (τ ) V˜ 2(τ ) k0 • · · · k1 • · · · . . . knp−1 · · · • [PITH_FULL_IMAGE:figures/full_fig_p009_3_3.png] view at source ↗
Figures from the paper (12 more)
Figure 3.4
Figure 3.4. Figure 3.4: Quantum circuit for V˜BS For time T = Nτ , one can obtain |vˆ(T)⟩ := V N BS|vˆ(0)⟩, which means the operator VBS(τ ) is applied N times. We can perform an inverse quantum Fourier transform on it to retrieve |v(T)⟩. Then a measurement (the projection P := I ⊗nx+1 ⊗ |p…
Figure 3.5
Figure 3.5. Figure 3.5: Quantum circuit for the Schrodingerisation method, where the measurement requires ¨ only projection onto P > 0 and QFT (IQFT ) denotes the (inverse) quantum Fourier transform. 4 The d-dimension Black-Scholes equation Denote s1, s2, · · · , sd as the prices of d risky…
Figure 4.1
Figure 4.1. Figure 4.1: Quantum circuit for V˜ 1(τ ) q 1 / nx V˜ 2(τ ) V2((r − σ 2 1 2 )τ ) q 2 / nx V2((r − σ 2 2 2 )τ ) . . . q j / nx := V2((r − σ 2 j 2 )τ ) . . . q d / nx V2((r − σ 2 d 2 )τ ) [PITH_FULL_IMAGE:figures/full_fig_p013_4_1.png]
Figure 4.2
Figure 4.2. Figure 4.2: Quantum circuit for V˜ 2(τ ) q / dnx V˜ 1(τ ) V˜ 2 1 (τ ) · · · V˜ 2 np−1 1 (τ ) V˜ −2 np−1 1 (τ ) V˜ 2(τ ) k0 • · · · k1 • · · · . . . knp−1 · · · • [PITH_FULL_IMAGE:figures/full_fig_p013_4_2.png]
Figure 4.3
Figure 4.3. Figure 4.3: Quantum circuit for V˜ BS Lemma 5.1. For the Schrodinger equation ¨ d|u(t)⟩ dt = iHBS|u(t)⟩, with the Hamiltonian HBS as specified in Equation (3.10), the time evolution operator UBS(τ ) = exp(iHBSτ ) over a time step τ can be effectively approximated by the unitary …
Figure 6.1
Figure 6.1. Figure 6.1: Figures of the numerical solution uh and error when nx = 6, np = 4. CPU runtime: 00:20:17. Maximum RAM occupancy: 5.47 GB 0 10 20 30 40 50 60 70 Asset Price (US Dollar $) 0 10 20 30 40 Option Price (US Dollar $) Schrodingerisation Exact Initial 10 20 30 40 50 60 Asse…
Figure 6.2
Figure 6.2. Figure 6.2: Figures of the numerical solution uh and error when nx = 6, np = 5. CPU runtime: 01:11:15. Maximum RAM occupancy: 6.09 GB 20 [PITH_FULL_IMAGE:figures/full_fig_p020_6_2.png]
Figure 6.3
Figure 6.3. Figure 6.3: Figures of the numerical solution uh and error when nx = 6, np = 6. CPU runtime: 03:37:11. Maximum RAM occupancy: 7.12 GB 0 10 20 30 40 50 60 70 Asset Price (US Dollar $) 0 10 20 30 40 Option Price (US Dollar $) Schrodingerisation Exact Initial 10 20 30 40 50 60 Asse…
Figure 6.4
Figure 6.4. Figure 6.4: Figures of the numerical solution uh and error when nx = 5, 6 and np = 6. CPU runtime: 00:18:08. Maximum RAM occupancy: 5.36 GB To be clarified, the classical error in the figures comes from the directly solving of ODE (3.4) with matrix exponential. The point number …
Figure 6.5
Figure 6.5. Figure 6.5: Figures of the numerical solution u ∗ h and error when nx = 6, np = 6. (∆p, ∆x) ( Lp 2 7 , Lx 2 6 ) order ( Lp 2 8 , Lx 2 7 ) order ( Lp 2 9 , Lx 2 8 ) order ∥uh − u∥L2 6.698e-01 - 3.271e-01 1.03 1.776e-01 0.89 ∥u ∗ h − u∥L2 2.925e-01 - 7.540e-02 1.96 2.060e-02 1.88 …
Figure 6.6
Figure 6.6. Figure 6.6: The solutions of the 2d Black Sholes equation with two initial values . [PITH_FULL_IMAGE:figures/full_fig_p023_6_6.png]
Figure 6.7
Figure 6.7. Figure 6.7: The solutions of 2d Black Sholes equation (upper right). [PITH_FULL_IMAGE:figures/full_fig_p023_6_7.png]

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