{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,19]],"date-time":"2026-06-19T00:43:30Z","timestamp":1781829810113,"version":"3.54.5"},"reference-count":33,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2024,4,30]],"date-time":"2024-04-30T00:00:00Z","timestamp":1714435200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"<jats:p>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 <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mo>&amp;#x223C;<\/mml:mo><mml:mn>16<\/mml:mn><\/mml:math>x and the number of logical qubits by <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mo>&amp;#x223C;<\/mml:mo><mml:mn>4<\/mml:mn><\/mml:math>x. Additionally, we estimate that the logical clock rate needed for quantum advantage is also reduced by a factor of <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mo>&amp;#x223C;<\/mml:mo><mml:mn>5<\/mml:mn><\/mml:math>x. Overall, we find that quantum advantage will require <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mn>4.7<\/mml:mn><\/mml:math>k logical qubits, and quantum devices that can execute <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msup><mml:mn>10<\/mml:mn><mml:mn>9<\/mml:mn><\/mml:msup><\/mml:math> T-gates at a rate of <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mn>45<\/mml:mn><\/mml:math>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.<\/jats:p>","DOI":"10.22331\/q-2024-04-30-1322","type":"journal-article","created":{"date-parts":[[2024,4,30]],"date-time":"2024-04-30T14:48:49Z","timestamp":1714488529000},"page":"1322","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":18,"title":["Derivative Pricing using Quantum Signal Processing"],"prefix":"10.22331","volume":"8","author":[{"given":"Nikitas","family":"Stamatopoulos","sequence":"first","affiliation":[{"name":"Goldman Sachs, New York, NY"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"William J.","family":"Zeng","sequence":"additional","affiliation":[{"name":"Goldman Sachs, New York, NY"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"9598","published-online":{"date-parts":[[2024,4,30]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"G. Brassard, P. Hoyer, M. Mosca, and A. Tapp, ``Quantum Amplitude Amplification and Estimation,&apos;&apos; Contemporary Mathematics 305 (2002).","DOI":"10.1090\/conm\/305\/05215"},{"key":"1","doi-asserted-by":"publisher","unstructured":"A. Montanaro, `` Quantum speedup of Monte Carlo methods,&apos;&apos; Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 471 (2015).","DOI":"10.1098\/rspa.2015.0301"},{"key":"2","doi-asserted-by":"publisher","unstructured":"P. Rebentrost, B. Gupt, and T. R. Bromley, ``Quantum computational finance: Monte Carlo pricing of financial derivatives,&apos;&apos; Phys. Rev. A 98, 022321 (2018).","DOI":"10.1103\/PhysRevA.98.022321"},{"key":"3","doi-asserted-by":"publisher","unstructured":"S. Woerner and D. J. Egger, ``Quantum risk analysis,&apos;&apos; npj Quantum Information 5 (2019).","DOI":"10.1038\/s41534-019-0130-6"},{"key":"4","doi-asserted-by":"publisher","unstructured":"N. Stamatopoulos, D. J. Egger, Y. Sun, C. Zoufal, R. Iten, N. Shen, and S. Woerner, ``Option Pricing using Quantum Computers,&apos;&apos; Quantum 4, 291 (2020).","DOI":"10.22331\/q-2020-07-06-291"},{"key":"5","doi-asserted-by":"publisher","unstructured":"J. a. F. Doriguello, A. Luongo, J. Bao, P. Rebentrost, and M. Santha, ``Quantum Algorithm for Stochastic Optimal Stopping Problems with Applications in Finance,&apos;&apos; in 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022), Leibniz International Proceedings in Informatics (LIPIcs), Vol. 232 (Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik, Dagstuhl, Germany, 2022) pp. 2:1\u20132:24.","DOI":"10.4230\/LIPIcs.TQC.2022.2"},{"key":"6","doi-asserted-by":"publisher","unstructured":"S. Herbert, ``Quantum Monte Carlo Integration: The Full Advantage in Minimal Circuit Depth,&apos;&apos; Quantum 6, 823 (2022).","DOI":"10.22331\/q-2022-09-29-823"},{"key":"7","doi-asserted-by":"publisher","unstructured":"S. Chakrabarti, R. Krishnakumar, G. Mazzola, N. Stamatopoulos, S. Woerner, and W. J. Zeng, ``A Threshold for Quantum Advantage in Derivative Pricing,&apos;&apos; Quantum 5, 463 (2021).","DOI":"10.22331\/q-2021-06-01-463"},{"key":"8","unstructured":"``Using Q# to estimate resources needed for quantum advantage in derivative pricing,&apos;&apos; Accessed: 2023-06-21."},{"key":"9","doi-asserted-by":"publisher","unstructured":"C. Zoufal, A. Lucchi, and S. Woerner, ``Quantum Generative Adversarial Networks for learning and loading random distributions,&apos;&apos; npj Quantum Information 5 (2019).","DOI":"10.1038\/s41534-019-0223-2"},{"key":"10","doi-asserted-by":"publisher","unstructured":"N. Stamatopoulos, G. Mazzola, S. Woerner, and W. J. Zeng, ``Towards Quantum Advantage in Financial Market Risk using Quantum Gradient Algorithms,&apos;&apos; Quantum 6, 770 (2022).","DOI":"10.22331\/q-2022-07-20-770"},{"key":"11","doi-asserted-by":"publisher","unstructured":"Y. Suzuki, S. Uno, R. Raymond, T. Tanaka, T. Onodera, and N. Yamamoto, ``Amplitude estimation without phase estimation,&apos;&apos; Quantum Information Processing 19, 75 (2020).","DOI":"10.1007\/s11128-019-2565-2"},{"key":"12","doi-asserted-by":"publisher","unstructured":"D. Grinko, J. Gacon, C. Zoufal, and S. Woerner, ``Iterative quantum amplitude estimation,&apos;&apos; npj Quantum Information 7 (2021).","DOI":"10.1038\/s41534-021-00379-1"},{"key":"13","doi-asserted-by":"publisher","unstructured":"A. Gily\u00e9n, Y. Su, G. H. Low, and N. Wiebe, ``Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics,&apos;&apos; in Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (2019) pp. 193\u2013204.","DOI":"10.1145\/3313276.3316366"},{"key":"14","doi-asserted-by":"publisher","unstructured":"J. M. Martyn, Z. M. Rossi, A. K. Tan, and I. L. Chuang, ``Grand Unification of Quantum Algorithms,&apos;&apos; PRX Quantum 2 (2021).","DOI":"10.1103\/prxquantum.2.040203"},{"key":"15","doi-asserted-by":"publisher","unstructured":"G. H. Low and I. L. Chuang, ``Optimal Hamiltonian Simulation by Quantum Signal Processing,&apos;&apos; Phys. Rev. Lett. 118, 010501 (2017).","DOI":"10.1103\/PhysRevLett.118.010501"},{"key":"16","doi-asserted-by":"publisher","unstructured":"J. M. Martyn, Y. Liu, Z. E. Chin, and I. L. Chuang, ``Efficient fully-coherent quantum signal processing algorithms for real-time dynamics simulation,&apos;&apos; The Journal of Chemical Physics 158, 024106 (2023).","DOI":"10.1063\/5.0124385"},{"key":"17","doi-asserted-by":"publisher","unstructured":"L. Lin and Y. Tong, ``Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems,&apos;&apos; Quantum 4, 361 (2020).","DOI":"10.22331\/q-2020-11-11-361"},{"key":"18","doi-asserted-by":"publisher","unstructured":"P. Rall and B. Fuller, ``Amplitude Estimation from Quantum Signal Processing,&apos;&apos; Quantum 7, 937 (2023).","DOI":"10.22331\/q-2023-03-02-937"},{"key":"19","doi-asserted-by":"publisher","unstructured":"S. McArdle, A. Gily\u00e9n, and M. Berta, ``Quantum state preparation without coherent arithmetic,&apos;&apos; arXiv preprint arXiv:2210.14892 (2022).","DOI":"10.48550\/ARXIV.2210.14892"},{"key":"20","doi-asserted-by":"publisher","unstructured":"J. Hull, Options, futures, and other derivatives, 6th ed. (Pearson Prentice Hall, Upper Saddle River, NJ [u.a.], 2006).","DOI":"10.1007\/978-1-4419-9230-7_2"},{"key":"21","doi-asserted-by":"publisher","unstructured":"J. Haah, ``Product Decomposition of Periodic Functions in Quantum Signal Processing,&apos;&apos; Quantum 3, 190 (2019).","DOI":"10.22331\/q-2019-10-07-190"},{"key":"22","unstructured":"R. Chao, D. Ding, A. Gilyen, C. Huang, and M. Szegedy, ``Finding Angles for Quantum Signal Processing with Machine Precision,&apos;&apos; arXiv preprint arXiv:2003.02831 (2020), arXiv:2003.02831 [quant-ph]."},{"key":"23","doi-asserted-by":"publisher","unstructured":"Y. Dong, X. Meng, K. B. Whaley, and L. Lin, ``Efficient phase-factor evaluation in quantum signal processing,&apos;&apos; Physical Review A 103, 042419 (2021).","DOI":"10.1103\/physreva.103.042419"},{"key":"24","unstructured":"Microsoft, Q# Language Specification (2020)."},{"key":"25","doi-asserted-by":"crossref","unstructured":"T. G. Draper, S. A. Kutin, E. M. Rains, and K. M. Svore, ``A logarithmic-depth quantum carry-lookahead adder,&apos;&apos; Quantum Information and Computation 6, 351 (2006).","DOI":"10.26421\/QIC6.4-5-4"},{"key":"26","doi-asserted-by":"publisher","unstructured":"T. H\u00e4ner, M. Roetteler, and K. M. Svore, ``Optimizing quantum circuits for arithmetic,&apos;&apos; arXiv preprint arXiv:1805.12445 (2018).","DOI":"10.48550\/ARXIV.1805.12445"},{"key":"27","doi-asserted-by":"publisher","unstructured":"P. Selinger, ``Quantum circuits of T-depth one,&apos;&apos; Physical Review A 87 (2013).","DOI":"10.1103\/physreva.87.042302"},{"key":"28","doi-asserted-by":"crossref","unstructured":"N. J. Ross and P. Selinger, ``Optimal Ancilla-Free Clifford+T Approximation of z-Rotations,&apos;&apos; Quantum Info. Comput. 16, 901 (2016).","DOI":"10.26421\/QIC16.11-12-1"},{"key":"29","unstructured":"`` QSPPACK,&apos;&apos; Accessed: 2023-06-21."},{"key":"30","unstructured":"`` pyqsp,&apos;&apos; Accessed: 2023-06-21."},{"key":"31","doi-asserted-by":"publisher","unstructured":"Z. M. Rossi and I. L. Chuang, `` Multivariable quantum signal processing (M-QSP): prophecies of the two-headed oracle,&apos;&apos; Quantum 6, 811 (2022).","DOI":"10.22331\/q-2022-09-20-811"},{"key":"32","doi-asserted-by":"publisher","unstructured":"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,&apos;&apos; PRX Quantum 3 (2022).","DOI":"10.1103\/prxquantum.3.040305"}],"container-title":["Quantum"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/quantum-journal.org\/papers\/q-2024-04-30-1322\/pdf\/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2024,4,30]],"date-time":"2024-04-30T14:50:18Z","timestamp":1714488618000},"score":1,"resource":{"primary":{"URL":"https:\/\/quantum-journal.org\/papers\/q-2024-04-30-1322\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,4,30]]},"references-count":33,"URL":"https:\/\/doi.org\/10.22331\/q-2024-04-30-1322","archive":["CLOCKSS"],"relation":{},"ISSN":["2521-327X"],"issn-type":[{"value":"2521-327X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,4,30]]},"article-number":"1322"}}