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We put an emphasis on the implementation of the quantum circuits required to build the input states and operators needed by amplitude estimation to price the different option types. Additionally, we show simulation results to highlight how the circuits that we implement price the different option contracts. Finally, we examine the performance of option pricing circuits on quantum hardware using the IBM Q Tokyo quantum device. We employ a simple, yet effective, error mitigation scheme that allows us to significantly reduce the errors arising from noisy two-qubit gates.<\/jats:p>","DOI":"10.22331\/q-2020-07-06-291","type":"journal-article","created":{"date-parts":[[2020,7,6]],"date-time":"2020-07-06T15:49:54Z","timestamp":1594050594000},"page":"291","source":"Crossref","is-referenced-by-count":217,"title":["Option Pricing using Quantum Computers"],"prefix":"10.22331","volume":"4","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0906-5340","authenticated-orcid":false,"given":"Nikitas","family":"Stamatopoulos","sequence":"first","affiliation":[{"name":"Quantitative Research, JPMorgan Chase & Co., New York, NY, 10017"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5523-9807","authenticated-orcid":false,"given":"Daniel J.","family":"Egger","sequence":"additional","affiliation":[{"name":"IBM Quantum, IBM Research \u2013 Zurich"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0756-164X","authenticated-orcid":false,"given":"Yue","family":"Sun","sequence":"additional","affiliation":[{"name":"Quantitative Research, JPMorgan Chase & Co., New York, NY, 10017"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4126-3141","authenticated-orcid":false,"given":"Christa","family":"Zoufal","sequence":"additional","affiliation":[{"name":"IBM Quantum, IBM Research \u2013 Zurich"},{"name":"ETH Zurich"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5332-5093","authenticated-orcid":false,"given":"Raban","family":"Iten","sequence":"additional","affiliation":[{"name":"IBM Quantum, IBM Research \u2013 Zurich"},{"name":"ETH Zurich"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4059-5187","authenticated-orcid":false,"given":"Ning","family":"Shen","sequence":"additional","affiliation":[{"name":"Quantitative Research, JPMorgan Chase & Co., New York, NY, 10017"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5945-4707","authenticated-orcid":false,"given":"Stefan","family":"Woerner","sequence":"additional","affiliation":[{"name":"IBM Quantum, IBM Research \u2013 Zurich"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"9598","published-online":{"date-parts":[[2020,7,6]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"John C. 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Tavernelli, and K. Temme, ``Quantum optimization using variational algorithms on near-term quantum devices,'' Quantum Science and Technology 3, 030503 (2018).","DOI":"10.1088\/2058-9565\/aab822"},{"key":"9","doi-asserted-by":"publisher","unstructured":"M. Ganzhorn, D.J. Egger, P. Barkoutsos, P. Ollitrault, G. Salis, N. Moll, M. Roth, A. Fuhrer, P. Mueller, S. Woerner, I. Tavernelli, and S. Filipp, ``Gate-efficient simulation of molecular eigenstates on a quantum computer,'' Phys. Rev. Applied 11, 044092 (2019).","DOI":"10.1103\/PhysRevApplied.11.044092"},{"key":"10","doi-asserted-by":"publisher","unstructured":"Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd, ``Quantum algorithm for linear systems of equations,'' Phys. Rev. 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