Derivative Pricing using Quantum Signal Processing

Nikitas Stamatopoulos; William J. Zeng

doi:10.22331/q-2024-04-30-1322

Derivative Pricing using Quantum Signal Processing

Nikitas Stamatopoulos and William J. Zeng

Goldman Sachs, New York, NY

Published:	2024-04-30, volume 8, page 1322
Eprint:	arXiv:2307.14310v2
Doi:	https://doi.org/10.22331/q-2024-04-30-1322
Citation:	Quantum 8, 1322 (2024).

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Abstract

Pricing financial derivatives on quantum computers typically includes quantum arithmetic components which contribute heavily to the quantum resources required by the corresponding circuits. In this manuscript, we introduce a method based on Quantum Signal Processing (QSP) to encode financial derivative payoffs directly into quantum amplitudes, alleviating the quantum circuits from the burden of costly quantum arithmetic. Compared to current state-of-the-art approaches in the literature, we find that for derivative contracts of practical interest, the application of QSP significantly reduces the required resources across all metrics considered, most notably the total number of T-gates by $\sim 16$x and the number of logical qubits by $\sim 4$x. Additionally, we estimate that the logical clock rate needed for quantum advantage is also reduced by a factor of $\sim 5$x. Overall, we find that quantum advantage will require $4.7$k logical qubits, and quantum devices that can execute $10^9$ T-gates at a rate of $45$MHz. While in this work we focus specifically on the payoff component of the derivative pricing process where the method we present is most readily applicable, similar techniques can be employed to further reduce the resources in other applications, such as state preparation.

Featured image: Payoff approximation using QSP for an autocallable derivative contract, and the corresponding quantum resources compared to those when quantum arithmetic is employed instead.

► BibTeX data

@article{Stamatopoulos2024derivativepricing,
  doi = {10.22331/q-2024-04-30-1322},
  url = {https://doi.org/10.22331/q-2024-04-30-1322},
  title = {Derivative {P}ricing using {Q}uantum {S}ignal {P}rocessing},
  author = {Stamatopoulos, Nikitas and Zeng, William J.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1322},
  month = apr,
  year = {2024}
}

► References

[1] G. Brassard, P. Hoyer, M. Mosca, and A. Tapp, ``Quantum Amplitude Amplification and Estimation,'' Contemporary Mathematics 305 (2002).
https://doi.org/10.1090/conm/305/05215

[2] A. Montanaro, `` Quantum speedup of Monte Carlo methods,'' Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 471 (2015).
https://doi.org/10.1098/rspa.2015.0301

[3] P. Rebentrost, B. Gupt, and T. R. Bromley, ``Quantum computational finance: Monte Carlo pricing of financial derivatives,'' Phys. Rev. A 98, 022321 (2018).
https://doi.org/10.1103/PhysRevA.98.022321

[4] S. Woerner and D. J. Egger, ``Quantum risk analysis,'' npj Quantum Information 5 (2019).
https://doi.org/10.1038/s41534-019-0130-6

[5] N. Stamatopoulos, D. J. Egger, Y. Sun, C. Zoufal, R. Iten, N. Shen, and S. Woerner, ``Option Pricing using Quantum Computers,'' Quantum 4, 291 (2020).
https://doi.org/10.22331/q-2020-07-06-291

[6] J. a. F. Doriguello, A. Luongo, J. Bao, P. Rebentrost, and M. Santha, ``Quantum Algorithm for Stochastic Optimal Stopping Problems with Applications in Finance,'' in 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022), Leibniz International Proceedings in Informatics (LIPIcs), Vol. 232 (Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 2022) pp. 2:1–2:24.
https://doi.org/10.4230/LIPIcs.TQC.2022.2

[7] S. Herbert, ``Quantum Monte Carlo Integration: The Full Advantage in Minimal Circuit Depth,'' Quantum 6, 823 (2022).
https://doi.org/10.22331/q-2022-09-29-823

[8] S. Chakrabarti, R. Krishnakumar, G. Mazzola, N. Stamatopoulos, S. Woerner, and W. J. Zeng, ``A Threshold for Quantum Advantage in Derivative Pricing,'' Quantum 5, 463 (2021).
https://doi.org/10.22331/q-2021-06-01-463

[9] ``Using Q# to estimate resources needed for quantum advantage in derivative pricing,'' Accessed: 2023-06-21.
https://cloudblogs.microsoft.com/quantum/2022/09/15/using-q-to-estimate-resources-needed-for-quantum-advantage-in-derivative-pricing/

[10] C. Zoufal, A. Lucchi, and S. Woerner, ``Quantum Generative Adversarial Networks for learning and loading random distributions,'' npj Quantum Information 5 (2019).
https://doi.org/10.1038/s41534-019-0223-2

[11] N. Stamatopoulos, G. Mazzola, S. Woerner, and W. J. Zeng, ``Towards Quantum Advantage in Financial Market Risk using Quantum Gradient Algorithms,'' Quantum 6, 770 (2022).
https://doi.org/10.22331/q-2022-07-20-770

[12] Y. Suzuki, S. Uno, R. Raymond, T. Tanaka, T. Onodera, and N. Yamamoto, ``Amplitude estimation without phase estimation,'' Quantum Information Processing 19, 75 (2020).
https://doi.org/10.1007/s11128-019-2565-2

[13] D. Grinko, J. Gacon, C. Zoufal, and S. Woerner, ``Iterative quantum amplitude estimation,'' npj Quantum Information 7 (2021).
https://doi.org/10.1038/s41534-021-00379-1

[14] A. Gilyén, Y. Su, G. H. Low, and N. Wiebe, ``Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics,'' in Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (2019) pp. 193–204.
https://doi.org/10.1145/3313276.3316366

[15] J. M. Martyn, Z. M. Rossi, A. K. Tan, and I. L. Chuang, ``Grand Unification of Quantum Algorithms,'' PRX Quantum 2 (2021).
https://doi.org/10.1103/prxquantum.2.040203

[16] G. H. Low and I. L. Chuang, ``Optimal Hamiltonian Simulation by Quantum Signal Processing,'' Phys. Rev. Lett. 118, 010501 (2017).
https://doi.org/10.1103/PhysRevLett.118.010501

[17] J. M. Martyn, Y. Liu, Z. E. Chin, and I. L. Chuang, ``Efficient fully-coherent quantum signal processing algorithms for real-time dynamics simulation,'' The Journal of Chemical Physics 158, 024106 (2023).
https://doi.org/10.1063/5.0124385

[18] L. Lin and Y. Tong, ``Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems,'' Quantum 4, 361 (2020).
https://doi.org/10.22331/q-2020-11-11-361

[19] P. Rall and B. Fuller, ``Amplitude Estimation from Quantum Signal Processing,'' Quantum 7, 937 (2023).
https://doi.org/10.22331/q-2023-03-02-937

[20] S. McArdle, A. Gilyén, and M. Berta, ``Quantum state preparation without coherent arithmetic,'' arXiv preprint arXiv:2210.14892 (2022).
https://doi.org/10.48550/ARXIV.2210.14892
arXiv:2210.14892

[21] J. Hull, Options, futures, and other derivatives, 6th ed. (Pearson Prentice Hall, Upper Saddle River, NJ [u.a.], 2006).
https://doi.org/10.1007/978-1-4419-9230-7_2

[22] J. Haah, ``Product Decomposition of Periodic Functions in Quantum Signal Processing,'' Quantum 3, 190 (2019).
https://doi.org/10.22331/q-2019-10-07-190

[23] R. Chao, D. Ding, A. Gilyen, C. Huang, and M. Szegedy, ``Finding Angles for Quantum Signal Processing with Machine Precision,'' arXiv preprint arXiv:2003.02831 (2020), arXiv:2003.02831 [quant-ph].
arXiv:2003.02831

[24] Y. Dong, X. Meng, K. B. Whaley, and L. Lin, ``Efficient phase-factor evaluation in quantum signal processing,'' Physical Review A 103, 042419 (2021).
https://doi.org/10.1103/physreva.103.042419

[25] Microsoft, Q# Language Specification (2020).
https://github.com/microsoft/qsharp-language/tree/main/Specifications/Language#q-language

[26] T. G. Draper, S. A. Kutin, E. M. Rains, and K. M. Svore, ``A logarithmic-depth quantum carry-lookahead adder,'' Quantum Information and Computation 6, 351 (2006).
https://dl.acm.org/doi/10.5555/2012086.2012090

[27] T. Häner, M. Roetteler, and K. M. Svore, ``Optimizing quantum circuits for arithmetic,'' arXiv preprint arXiv:1805.12445 (2018).
https://doi.org/10.48550/ARXIV.1805.12445
arXiv:1805.12445

[28] P. Selinger, ``Quantum circuits of T-depth one,'' Physical Review A 87 (2013).
https://doi.org/10.1103/physreva.87.042302

[29] N. J. Ross and P. Selinger, ``Optimal Ancilla-Free Clifford+T Approximation of z-Rotations,'' Quantum Info. Comput. 16, 901 (2016).
https://dl.acm.org/doi/abs/10.5555/3179330.3179331

[30] `` QSPPACK,'' Accessed: 2023-06-21.
https://github.com/qsppack/QSPPACK

[31] `` pyqsp,'' Accessed: 2023-06-21.
https://github.com/ichuang/pyqsp

[32] Z. M. Rossi and I. L. Chuang, `` Multivariable quantum signal processing (M-QSP): prophecies of the two-headed oracle,'' Quantum 6, 811 (2022).
https://doi.org/10.22331/q-2022-09-20-811

[33] Y. Dong, L. Lin, and Y. Tong, ``Ground-State Preparation and Energy Estimation on Early Fault-Tolerant Quantum Computers via Quantum Eigenvalue Transformation of Unitary Matrices,'' PRX Quantum 3 (2022).
https://doi.org/10.1103/prxquantum.3.040305

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[2] Guoming Wang and Angus Kan, "Option pricing under stochastic volatility on a quantum computer", Quantum 8, 1504 (2024).

[3] Manqoba Q. Hlatshwayo, Manav Babel, Dalila Islas-Sanchez, and Konstantinos Georgopoulos, "A Technical Review of Quantum Computing Use Cases for Finance and Economics", Quantum Reports 8 1, 26 (2026).

[4] Aleksandr Chuvakov and Rodion Boryaev, "Factoring Decision Support System Based on Optimized Quantum Algorithms QMC", Informatics and Automation 24 2, 657 (2025).

[5] Hidetaka Manabe and Yuichi Sano, "The State Preparation of Multivariate Normal Distributions using Tree Tensor Network", Quantum 9, 1755 (2025).

[6] Sam McArdle, András Gilyén, and Mario Berta, "Quantum State Preparation without Coherent Arithmetic", Physical Review Letters 136 24, 240603 (2026).

[7] Ugo Fiore, Federica Gioia, and Paolo Zanetti, "A perspective on quantum Fintech", Decisions in Economics and Finance 49 1, 187 (2026).

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[10] Alberto Manzano, Gonzalo Ferro, Álvaro Leitao, Carlos Vázquez, and Andrés Gómez, "Alternative pipeline for option pricing using quantum computers", EPJ Quantum Technology 12 1, 28 (2025).

[11] Christian Laudagé and Ivica Turkalj, "Quantum Risk Analysis: Beyond (Conditional) Value-at-Risk", Quantum Economics and Finance 2 2, 148 (2025).

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[14] Zhu Sun and Bálint Koczor, "Space and time cost of continuous rotations in surface codes", Quantum Science and Technology 11 2, 025046 (2026).

[15] Matthias Rosenkranz, Eric Brunner, Gabriel Marin-Sanchez, Nathan Fitzpatrick, Silas Dilkes, Yao Tang, Yuta Kikuchi, and Marcello Benedetti, "Quantum state preparation for multivariate functions", Quantum 9, 1703 (2025).

[16] Siyi Wang, Xiufan Li, Wei Jie Bryan Lee, Suman Deb, Eugene Lim, and Anupam Chattopadhyay, "A comprehensive study of quantum arithmetic circuits", Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 383 2288, 20230392 (2025).

[17] Zhu Sun, Gregory Boyd, Zhenyu Cai, Hamza Jnane, Bálint Koczor, Richard Meister, Romy Minko, Benjamin Pring, Simon C. Benjamin, and Nikitas Stamatopoulos, "Low-depth phase oracle using a parallel piecewise circuit", Physical Review A 111 6, 062420 (2025).

[18] Yuan Liu, Shraddha Singh, Kevin C. Smith, Eleanor Crane, John M. Martyn, Alec Eickbusch, Alexander Schuckert, Richard D. Li, Jasmine Sinanan-Singh, Micheline B. Soley, Takahiro Tsunoda, Isaac L. Chuang, Nathan Wiebe, and Steven M. Girvin, "Hybrid Oscillator-Qubit Quantum Processors: Instruction Set Architectures, Abstract Machine Models, and Applications", PRX Quantum 7 1, 010201 (2026).

[19] Yichen Xie and Nadav Ben-Ami, 2025 IEEE International Conference on Quantum Computing and Engineering (QCE) 242 (2025) ISBN:979-8-3315-5736-2.

[20] Alexander M. Dalzell, Sam McArdle, Mario Berta, Przemyslaw Bienias, Chi-Fang Chen, András Gilyén, Connor T. Hann, Michael J. Kastoryano, Emil T. Khabiboulline, Aleksander Kubica, Grant Salton, Samson Wang, and Fernando G. S. L. Brandão, "Quantum algorithms: A survey of applications and end-to-end complexities", arXiv:2310.03011, (2023).

[21] Travis L. Scholten, Carl J. Williams, Dustin Moody, Michele Mosca, William Hurley, William J. Zeng, Matthias Troyer, and Jay M. Gambetta, "Assessing the Benefits and Risks of Quantum Computers", arXiv:2401.16317, (2024).

[22] Titos Matsakos and Stuart Nield, "Quantum Monte Carlo simulations for financial risk analytics: scenario generation for equity, rate, and credit risk factors", Quantum 8, 1306 (2024).

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