pith. machine review for the scientific record. sign in

arxiv: 2604.19832 · v1 · submitted 2026-04-20 · 🪐 quant-ph · cs.LG

Recognition: unknown

Option Pricing on Noisy Intermediate-Scale Quantum Computers: A Quantum Neural Network Approach

Sebastian Zaj\k{a}c , Rafa{\l} Pracht
Authors on Pith no claims yet

Pith reviewed 2026-05-10 03:55 UTC · model grok-4.3

classification 🪐 quant-ph cs.LG
keywords quantum neural networksoption pricingNISQ hardwareBlack-Scholes-Merton modelquantum machine learningderivative pricingquantum computing applicationsfinancial modeling
0
0 comments X

The pith

Quantum neural networks approximate option prices accurately on today's noisy quantum computers.

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

This paper tests whether quantum neural networks can learn to price options by running directly on real NISQ hardware. The authors build a small 2-qubit circuit and run it on four different quantum processors, finding that the network produces consistent pricing results for the Black-Scholes-Merton model despite device noise. They treat the simple model as a benchmark to check if the geometry of quantum states helps the network capture the pricing function. If the approach holds, it opens a route to quantum methods for more demanding pricing tasks that classical computers handle slowly. The work matters because better pricing tools directly affect risk calculations in trillion-dollar derivatives markets.

Core claim

The paper demonstrates that a fully quantum approach based on quantum neural networks can achieve accurate option pricing approximations on NISQ hardware. By implementing a compact 2-qubit QNN architecture on multiple platforms including IBM Fez, IQM Garnet, IonQ Forte, and Rigetti Ankaa-3, the authors obtain pricing results that remain consistent across devices. This provides empirical evidence that QNNs, by exploiting the geometric structure of Hilbert space, can effectively approximate the option pricing function in the Black-Scholes-Merton framework and constitute a viable method for derivative pricing.

What carries the argument

A compact 2-qubit quantum neural network whose quantum circuit layers map market parameters to option prices through parameterized unitary operations.

If this is right

  • QNN methods can deliver usable pricing results on current quantum processors without waiting for error-corrected hardware.
  • The same circuit approach can be extended to local volatility and stochastic volatility models once hardware improves.
  • Consistent cross-device performance supports using QNNs for real-time risk calculations in derivatives trading.
  • Success on the benchmark model indicates that quantum machine learning can address pricing problems whose classical cost grows rapidly with model complexity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method could be tested next on multi-asset or path-dependent options to see whether qubit count limits block scaling.
  • Integration with classical post-processing might reduce the impact of hardware noise while keeping the quantum advantage in function approximation.
  • If the approach generalizes, trading desks could run pricing updates on cloud quantum access rather than large classical clusters for certain models.

Load-bearing premise

That the structure of quantum states lets the network learn the option pricing function well enough to stay accurate on noisy hardware.

What would settle it

If the same 2-qubit QNN produces pricing errors that exceed classical accuracy benchmarks on all four tested devices for the Black-Scholes-Merton model, the claim of consistent viable approximations would not hold.

Figures

Figures reproduced from arXiv: 2604.19832 by Rafa{\l} Pracht, Sebastian Zaj\k{a}c.

Figure 1
Figure 1. Figure 1: Scatterplots illustrating the dependency of the option price on each input parameter. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The finQbit VQC architecture illustrating the 2-qubit register. [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The finQbit VQC architecture after U4 decomposition gate. [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison between the finQbit predictions and the analytical Black-Scholes prices across the tested [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Repetition Stability on IQM Garnet (R = 25). The dashed lines represent Black-Scholes theoretical values. The distribution of valuation samples across different moneyness levels is shown in [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Point cloud distribution for IQM Garnet. [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Convergence of the option price estimator on IQM Garnet across 500 to 5,000 shots. [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Drift analysis on IonQ Forte (R = 15). The Stability Track ( [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Point cloud distribution on IonQ Forte. The distribution of individual valuation samples is further detailed in the point cloud analysis ( [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Option price convergence trajectory for IonQ Forte. Values reconstructed via Monte Carlo sampling across [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Drift analysis on Rigetti Ankaa-3 (R = 20). The stability analysis (R = 20) for the Rigetti Ankaa-3 system ( [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Point cloud distribution on Rigetti. A distinct systematic bias is visible as moneyness increases, while the [PITH_FULL_IMAGE:figures/full_fig_p014_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Stability and drift analysis on ibm_fez. Dashed lines represent the analytical Black-Scholes solution. (a) Dataset A (2k shots). (b) Dataset B (5k shots) [PITH_FULL_IMAGE:figures/full_fig_p016_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Point cloud distribution of valuations. The vertical shift reveals the hardware’s systematic bias under different [PITH_FULL_IMAGE:figures/full_fig_p016_14.png] view at source ↗
read the original abstract

In a global derivatives market with notional values in the hundreds of trillions of dollars, the accuracy and efficiency of pricing models are of fundamental importance, with direct implications for risk management, capital allocation, and regulatory compliance. In this work, we employ the Black-Scholes-Merton (BSM) framework not as an end in itself, but as a controlled benchmark environment in which to rigorously assess the capabilities of quantum machine learning methods. We propose a fully quantum approach to option pricing based on Quantum Neural Networks (QNNs), and, to the best of our knowledge, present one of the first implementations of such a methodology on currently available quantum hardware. Specifically, we investigate whether QNNs, by exploiting the geometric structure of Hilbert space, can effectively approximate option pricing functions. Our implementation utilizes a compact 2-qubit QNN architecture evaluated across multiple state-of-the-art quantum processors, including IBM Fez, IQM Garnet, IonQ Forte, and Rigetti Ankaa-3. This cross-platform study reveals distinct hardware-dependent performance characteristics while demonstrating that accurate pricing approximations can be achieved consistently across different devices despite the constraints of Noisy Intermediate-Scale Quantum (NISQ) hardware. The results provide empirical evidence that QNN-based approaches constitute a viable framework for derivative pricing. While the analysis is conducted within the BSM setting, the broader significance lies in the potential extension of these methods to more realistic and computationally demanding models, including local volatility, stochastic volatility, and interest rate frameworks commonly used in practice.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript proposes a fully quantum approach to option pricing via compact 2-qubit Quantum Neural Networks (QNNs) within the Black-Scholes-Merton (BSM) framework, treated explicitly as a controlled benchmark rather than a production tool. It reports implementation and evaluation of this architecture on four NISQ processors (IBM Fez, IQM Garnet, IonQ Forte, Rigetti Ankaa-3), claiming that accurate pricing approximations are achieved consistently across devices despite hardware noise, with broader implications for extension to local-volatility or stochastic-volatility models.

Significance. If the reported cross-platform results hold under quantitative scrutiny, the work supplies one of the first empirical demonstrations of QNN-based derivative pricing executed directly on physical NISQ hardware. The multi-vendor evaluation is a concrete strength that addresses hardware dependence, and the explicit framing of BSM as a benchmark avoids overclaiming. This could serve as a reproducible starting point for assessing quantum machine learning feasibility in finance, provided the accuracy claims are backed by error metrics and baselines.

major comments (1)
  1. [Abstract] Abstract and results description: the central claim that 'accurate pricing approximations can be achieved consistently across different devices' is asserted without any reported quantitative error metrics (e.g., mean absolute percentage error against BSM values), training loss curves, hyper-parameter details, data exclusion criteria, or classical baseline comparisons. These omissions make it impossible to assess whether the observed performance substantiates the feasibility conclusion or is consistent with the low soundness noted in the review.
minor comments (1)
  1. The manuscript would benefit from explicit statements of the QNN circuit depth, ansatz parameterization, and optimization method used for the 2-qubit architecture to allow reproduction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive feedback on our manuscript. We address the major comment below and commit to revisions that improve the transparency and substantiation of our results.

read point-by-point responses

  1. Referee: [Abstract] Abstract and results description: the central claim that 'accurate pricing approximations can be achieved consistently across different devices' is asserted without any reported quantitative error metrics (e.g., mean absolute percentage error against BSM values), training loss curves, hyper-parameter details, data exclusion criteria, or classical baseline comparisons. These omissions make it impossible to assess whether the observed performance substantiates the feasibility conclusion or is consistent with the low soundness noted in the review.

    Authors: We agree that the abstract and results description would benefit from explicit quantitative support for the performance claims. In the revised manuscript we will add mean absolute percentage error (MAPE) values comparing QNN outputs to BSM benchmarks, training loss curves, hyper-parameter specifications, data exclusion criteria, and classical baseline comparisons. These additions will allow readers to evaluate the accuracy and feasibility of the approach directly. revision: yes

Circularity Check

0 steps flagged

No significant circularity; empirical hardware benchmark is self-contained

full rationale

The paper treats BSM explicitly as an external benchmark for testing a 2-qubit QNN on physical NISQ devices (IBM, IQM, IonQ, Rigetti). Reported accuracies arise from direct circuit execution and comparison to known closed-form BSM values, with no equations, fitted parameters, or self-citations that reduce the central claim to its own inputs by construction. The derivation chain consists of standard QNN ansatz + hardware runs + classical post-processing against an independent analytical reference; this is a normal experimental demonstration and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard quantum circuit execution and variational training procedures; the only non-standard element is the assumption that a small QNN can approximate the pricing map.

free parameters (1)
  • QNN variational parameters
    Angles and gate parameters adjusted during training to match Black-Scholes outputs
axioms (1)
  • domain assumption Quantum circuits can approximate continuous functions via Hilbert-space geometry
    Invoked to justify why a 2-qubit QNN should be able to learn the pricing function

pith-pipeline@v0.9.0 · 5584 in / 1306 out tokens · 71054 ms · 2026-05-10T03:55:47.125344+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

23 extracted references · 21 canonical work pages

  1. [1]

    Robert C

    doi:10.1086/260062. Robert C. Merton. Theory of rational option pricing.Bell Journal of Economics and Management Science, 4(1): 141–183,

    work page doi:10.1086/260062

  2. [2]

    Emanuel Derman and Iraj Kani

    doi:10.2307/3003143. Emanuel Derman and Iraj Kani. Riding on a smile.Risk, 7:32–39,

    work page doi:10.2307/3003143

  3. [3]

    Alan Brace, Dariusz Gatarek, and Marek Musiela

    doi:10.1016/0304-405X(76)90022-2. Alan Brace, Dariusz Gatarek, and Marek Musiela. The market model of interest rate dynamics.Mathematical Finance, 7(2):127–155,

    work page doi:10.1016/0304-405x(76)90022-2

  4. [4]

    URLhttps://doi.org/10.1111/1467-9965.00028

    doi:10.1111/1467-9965.00028. URLhttps://doi.org/10.1111/1467-9965.00028. Rafał Pracht and Dariusz G ˛ atarek. Quantum Binomial Tree, An Effective Method for Probability Distribution Loading for Derivative Pricing.Wilmott, 2025(139),

    work page doi:10.1111/1467-9965.00028 2025

  5. [5]

    doi:10.54946/wilm.12175

    ISSN 1541-8286. doi:10.54946/wilm.12175. URL http://dx.doi.org/10.54946/wilm.12175. Maria Schuld and Nathan Killoran. Quantum machine learning in feature hilbert spaces.Physical Review Letters, 122 (4):040504,

    work page doi:10.54946/wilm.12175

  6. [6]

    Quantum machine learning in feature Hilbert spaces

    doi:10.1103/PhysRevLett.122.040504. Maria Schuld. Quantum machine learning models are kernel methods.Physical Review A, 103(3):032430,

    work page doi:10.1103/physrevlett.122.040504

  7. [8]

    Edward Farhi and Hartmut Neven

    doi:10.1038/s41467-024-46345-x. Edward Farhi and Hartmut Neven. Classification with quantum neural networks on near term processors.arXiv preprint arXiv:1802.06002,

    work page doi:10.1038/s41467-024-46345-x

  8. [9]

    Schuman, Shruti R

    ISSN 2662-8457. doi:10.1038/s43588- 021-00084-1. URLhttp://dx.doi.org/10.1038/s43588-021-00084-1. Dylan Herman, Cody Googin, Xiaoyuan Mickelin, et al. Quantum computing for finance.Nature Reviews Physics, 5: 450–465,

    work page doi:10.1038/s43588-

  9. [10]

    Quantum computing for fi- nance.Nature Reviews Physics, 5(8):450–465, 2023

    doi:10.1038/s42254-023-00603-1. John Preskill. Quantum computing in the NISQ era and beyond.Quantum, 2:79,

    work page doi:10.1038/s42254-023-00603-1

  10. [11]

    Quantum Computing in the NISQ era and beyond,

    doi:10.22331/q-2018-08-06-79. Frank Arute, Kunal Arya, Ryan Babbush, et al. Quantum supremacy using a programmable superconducting processor. Nature, 574(7779):505–510,

    work page Pith review doi:10.22331/q-2018-08-06-79 2018

  11. [12]

    Laufer and Maxine Wolfe

    doi:10.1111/j.1540- 6261.1994.tb00081.x. Johannes Ruf and Weiguan Wang. Machine learning in option pricing and hedging: a survey.Journal of Computational Finance, 24(1):1–55,

    work page doi:10.1111/j.1540- 1994

  12. [13]

    Andreas Weigand

    doi:10.21314/JCF.2020.375. Andreas Weigand. Machine learning in asset pricing: A survey of recent literature.Working Paper,

    work page doi:10.21314/jcf.2020.375 2020

  13. [14]

    The Review of Financial Studies , volume =

    doi:10.1093/rfs/hhaa009. Serena Della Corte, Laurens Van Mieghem, Antonis Papapantoleon, and Jonas Papazoglou-Hennig. Machine learning for option pricing: an empirical investigation of network architectures,

    work page doi:10.1093/rfs/hhaa009

  14. [15]

    David Anderson and Urban Ulrych

    URL https://arxiv.org/abs/ 2307.07657. David Anderson and Urban Ulrych. Accelerated american option pricing with deep neural networks,

    work page arXiv

  15. [16]

    Information fusion58, 82–115 (2020), https://doi.org/10.1016/j.inffus.2019.12.012

    doi:10.1016/j.inffus.2019.12.012. Tianqi Chen and Carlos Guestrin. XGBoost: A scalable tree boosting system. InProceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 785–794,

    work page doi:10.1016/j.inffus.2019.12.012 2019

  16. [17]

    doi:10.1145/2939672.2939785. John C. Hull.Options, Futures, and Other Derivatives. Pearson, New York, 11th edition,

    work page doi:10.1145/2939672.2939785

  17. [18]

    Mitarai, M

    ISSN 2469-9934. doi:10.1103/physreva.98.032309. URL http://dx.doi.org/10.1103/PhysRevA.98.032309. Jacob L. Cybulski and Sebastian Zaj ˛ ac. Design considerations for denoising quantum time series autoencoder. In Computational Science – ICCS 2024, pages 252–267. Springer Nature Switzerland,

    work page doi:10.1103/physreva.98.032309 2024

  18. [19]

    Fast semi-iterative finite ele- ment Poisson solvers for tensor core GPUs based on prehandling

    doi:10.1007/978-3-031- 63778-0_18. Maria Schuld, Ryan Sweke, and Johannes Jakob Meyer. Effect of data encoding on the expressive power of variational quantum-machine-learning models.Physical Review A, 103(3), March

    work page doi:10.1007/978-3-031-

  19. [20]

    doi:10.1103/physreva.103.032430

    ISSN 2469-9934. doi:10.1103/physreva.103.032430. URLhttp://dx.doi.org/10.1103/PhysRevA.103.032430. Adrián Pérez-Salinas, Alba Cervera-Lierta, Elies Gil-Fuster, and José I. Latorre. Data re-uploading for a universal quantum classifier.Quantum, 4:226, February

    work page doi:10.1103/physreva.103.032430

  20. [21]

    Data re-uploading for a universal quantum classifier,

    ISSN 2521-327X. doi:10.22331/q-2020-02-06-226. URL http://dx.doi.org/10.22331/q-2020-02-06-226. Jarrod R McClean, Sergio Boixo, Vadim N Smelyanskiy, Ryan Babbush, and Hartmut Neven. Barren plateaus in quantum neural network training landscapes.Nature communications, 9(1):1–6,

    work page doi:10.22331/q-2020-02-06-226 2020

  21. [22]

    com/qpu/garnet/

    URL https://www.iqmacademy. com/qpu/garnet/. Accessed: 2025-11-03. Askar Oralkhan and Temirlan Zhaxalykov. Investigation of hardware architecture effects on quantum algorithm performance: A comparative hardware study,

    2025

  22. [23]

    IonQ, Inc

    URLhttps://arxiv.org/abs/2601.05286. IonQ, Inc. Ionq aria and forte quantum systems,

    work page arXiv

  23. [24]

    Accessed: 2025-11-03

    URL https://www.ionq.com/quantum-systems/aria. Accessed: 2025-11-03. Rigetti Computing. Rigetti fourth-generation quantum processing units and qcs architecture. https://qcs.rigetti. com/systems,

    2025