Option Pricing using Quantum Computers
1Quantitative Research, JPMorgan Chase & Co., New York, NY, 10017
2IBM Quantum, IBM Research – Zurich
3ETH Zurich
| Published: | 2020-07-06, volume 4, page 291 |
| Eprint: | arXiv:1905.02666v5 |
| Doi: | https://doi.org/10.22331/q-2020-07-06-291 |
| Citation: | Quantum 4, 291 (2020). |
Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.
Abstract
We present a methodology to price options and portfolios of options on a gate-based quantum computer using amplitude estimation, an algorithm which provides a quadratic speedup compared to classical Monte Carlo methods. The options that we cover include vanilla options, multi-asset options and path-dependent options such as barrier options. 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.
Featured image: Probability distribution represented on a quantum register and the payoff function for a European call option, inputs to the algorithm enabling its pricing on a quantum computer.
► BibTeX data
► References
[1] John C. 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
[2] Fischer Black and Myron Scholes, ``The pricing of options and corporate liabilities,'' Journal of Political Economy 81, 637–654 (1973).
https://doi.org/10.1086/260062
[3] Bruno Dupire, ``Pricing with a smile,'' Risk Magazine , 18–20 (1994).
[4] Phelim P. Boyle, ``Options: A Monte Carlo approach,'' Journal of Financial Economics 4, 323–338 (1977).
https://doi.org/10.1016/0304-405X(77)90005-8
[5] Paul Glasserman, Monte Carlo Methods in Financial Engineering (Springer-Verlag New York, 2003) p. 596.
https://doi.org/10.1007/978-0-387-21617-1
[6] Michael A. Nielsen and Isaac L. Chuang, Cambridge University Press (2010) p. 702.
https://doi.org/10.1017/CBO9780511976667
[7] Abhinav Kandala, Antonio Mezzacapo, Kristan Temme, Maika Takita, Markus Brink, Jerry M. Chow, and Jay M. Gambetta, ``Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets,'' Nature 549, 242 (2017).
https://doi.org/10.1038/nature23879
[8] Abhinav Kandala, Kristan Temme, Antonio D. Corcoles, Antonio Mezzacapo, Jerry M. Chow, and Jay M. Gambetta, ``Error mitigation extends the computational reach of a noisy quantum processor,'' Nature 567, 491–495 (2019).
https://doi.org/10.1038/s41586-019-1040-7
[9] N. Moll, P. Barkoutsos, L. S. Bishop, J. M. Chow, A. Cross, D. J. Egger, S. Filipp, A. Fuhrer, J. M. Gambetta, M. Ganzhorn, A. Kandala, A. Mezzacapo, P. Müller, W. Riess, G. Salis, J. Smolin, I. Tavernelli, and K. Temme, ``Quantum optimization using variational algorithms on near-term quantum devices,'' Quantum Science and Technology 3, 030503 (2018).
https://doi.org/10.1088/2058-9565/aab822
[10] 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).
https://doi.org/10.1103/PhysRevApplied.11.044092
[11] Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd, ``Quantum algorithm for linear systems of equations,'' Phys. Rev. Lett. 103, 150502 (2009).
https://doi.org/10.1103/PhysRevLett.103.150502
[12] Seth Lloyd, Masoud Mohseni, and Patrick Rebentrost, ``Quantum principal component analysis,'' Nature Physics 10, 631–633 (2014).
https://doi.org/10.1038/nphys3029
[13] Jacob Biamonte, Peter Wittek, Nicola Pancotti, Patrick Rebentrost, Nathan Wiebe, and Seth Lloyd, ``Quantum machine learning,'' Nature 549, 195–202 (2017).
https://doi.org/10.1038/nature23474
[14] Vojtech Havlicek, Antonio D. Corcoles, Kristan Temme, Aram W. Harrow, Abhinav Kandala, Jerry M. Chow, and Jay M. Gambetta, ``Supervised learning with quantum-enhanced feature spaces,'' Nature 567, 209 – 212 (2019).
https://doi.org/10.1038/s41586-019-0980-2
[15] Roman Orus, Samuel Mugel, and Enrique Lizaso, ``Quantum computing for finance: Overview and prospects,'' Reviews in Physics 4, 100028 (2019).
https://doi.org/10.1016/j.revip.2019.100028
[16] Patrick Rebentrost and Seth Lloyd, ``Quantum computational finance: quantum algorithm for portfolio optimization,'' (2018), arXiv:1811.03975.
arXiv:1811.03975
[17] Stefan Woerner and Daniel J. Egger, ``Quantum risk analysis,'' npj Quantum Information 5, 15 (2019).
https://doi.org/10.1038/s41534-019-0130-6
[18] Patrick Rebentrost, Brajesh Gupt, and Thomas 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
[19] Christa Zoufal, Aurélien Lucchi, and Stefan Woerner, ``Quantum generative adversarial networks for learning and loading random distributions,'' npj Quantum Information 5, 1–9 (2019).
https://doi.org/10.1038/s41534-019-0223-2
[20] Ana Martin, Bruno Candelas, Angel Rodriguez-Rozas, Jose D. Martin-Guerrero, Xi Chen, Lucas Lamata, Roman Orus, Enrique Solano, and Mikel Sanz, ``Towards pricing financial derivatives with an ibm quantum computer,'' (2019), arXiv:1904.05803.
arXiv:1904.05803
[21] Gilles Brassard, Peter Hoyer, Michele Mosca, and Alain Tapp, ``Quantum Amplitude Amplification and Estimation,'' Contemporary Mathematics 305 (2002), 10.1090/conm/305/05215.
https://doi.org/10.1090/conm/305/05215
[22] Yohichi Suzuki, Shumpei Uno, Rudy Raymond, Tomoki Tanaka, Tamiya Onodera, and Naoki Yamamoto, ``Amplitude estimation without phase estimation,'' Quantum Information Processing 19, 75 (2020).
https://doi.org/10.1007/s11128-019-2565-2
[23] Reuven Y. Rubinstein, Simulation and the Monte Carlo Method, Wiley Series in Probability and Statistics (Wiley, 1981).
https://doi.org/10.1002/9780470316511
[24] Daniel S Abrams and Colin P Williams, ``Fast quantum algorithms for numerical integrals and stochastic processes,'' (1999), arxiv:quant-ph/9908083.
arXiv:quant-ph/9908083
[25] Ashley Montanaro, ``Quantum speedup of monte carlo methods,'' Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 471 (2015), 10.1098/rspa.2015.0301.
https://doi.org/10.1098/rspa.2015.0301
[26] A. Yu. Kitaev, ``Quantum measurements and the Abelian Stabilizer Problem,'' (1995), arXiv:quant-ph/9511026.
arXiv:arXiv:quant-ph/9511026
[27] R. Cleve, A. Ekert, C. Macchiavello, and M. Mosca, ``Quantum algorithms revisited,'' Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 339–354 (1998).
https://doi.org/10.1098/rspa.1998.0164
[28] Paul Glasserman, Philip Heidelberger, and Perwez Shahabuddin, ``Efficient Monte Carlo Methods for Value-at-Risk,'' in Mastering Risk, Vol. 2 (2000) pp. 5–18.
[29] Robert C. Merton, ``Theory of rational option pricing,'' The Bell Journal of Economics and Management Science 4, 141–183 (1973).
https://doi.org/10.2307/3003143
[30] Lov Grover and Terry Rudolph, ``Creating superpositions that correspond to efficiently integrable probability distributions,'' (2002), arXiv:quant-ph/0208112.
arXiv:quant-ph/0208112
[31] Adriano Koshiyama, Nick Firoozye, and Philip Treleaven, ``Generative adversarial networks for financial trading strategies fine-tuning and combination,'' (2019), arXiv:1901.01751.
arXiv:arXiv:1901.01751
[32] Blanka Horvath, Aitor Muguruza, and Mehdi Tomas, ``Deep learning volatility,'' SSRN Electronic Journal (2019), 10.2139/ssrn.3322085.
https://doi.org/10.2139/ssrn.3322085
[33] Martin Plesch and Časlav Brukner, ``Quantum-state preparation with universal gate decompositions,'' Phys. Rev. A 83, 032302 (2011).
https://doi.org/10.1103/PhysRevA.83.032302
[34] Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio, ``Generative adversarial nets,'' in Advances in Neural Information Processing Systems 27, edited by Z. Ghahramani, M. Welling, C. Cortes, N. D. Lawrence, and K. Q. Weinberger (Curran Associates, Inc., 2014) pp. 2672–2680.
arXiv:1406.2661
[35] Adriano Barenco, Charles H. Bennett, Richard Cleve, David P. Divincenzo, Norman Margolus, Peter Shor, Tycho Sleator, John A. Smolin, and Harald Weinfurter, ``Elementary gates for quantum computation,'' Phys. Rev. A 52, 3457–3467 (1995).
https://doi.org/10.1103/PhysRevA.52.3457
[36] Steven A Cuccaro, Thomas G Draper, Samuel A Kutin, and David Petrie Moulton, ``A new quantum ripple-carry addition circuit,'' (2004), arxiv:quant-ph/0410184.
arXiv:quant-ph/0410184
[37] Gadi Aleksandrowicz, Thomas Alexander, Panagiotis Barkoutsos, Luciano Bello, Yael Ben-Haim, David Bucher, Francisco Jose Cabrera-Hernández, Jorge Carballo-Franquis, Adrian Chen, Chun-Fu Chen, Jerry M. Chow, Antonio D. Córcoles-Gonzales, Abigail J. Cross, Andrew Cross, Juan Cruz-Benito, Chris Culver, Salvador De La Puente González, Enrique De La Torre, Delton Ding, Eugene Dumitrescu, Ivan Duran, Pieter Eendebak, Mark Everitt, Ismael Faro Sertage, Albert Frisch, Andreas Fuhrer, Jay Gambetta, Borja Godoy Gago, Juan Gomez-Mosquera, Donny Greenberg, Ikko Hamamura, Vojtech Havlicek, Joe Hellmers, Łukasz Herok, Hiroshi Horii, Shaohan Hu, Takashi Imamichi, Toshinari Itoko, Ali Javadi-Abhari, Naoki Kanazawa, Anton Karazeev, Kevin Krsulich, Peng Liu, Yang Luh, Yunho Maeng, Manoel Marques, Francisco Jose Martín-Fernández, Douglas T. McClure, David McKay, Srujan Meesala, Antonio Mezzacapo, Nikolaj Moll, Diego Moreda Rodríguez, Giacomo Nannicini, Paul Nation, Pauline Ollitrault, Lee James O'Riordan, Hanhee Paik, Jesús Pérez, Anna Phan, Marco Pistoia, Viktor Prutyanov, Max Reuter, Julia Rice, Abdón Rodríguez Davila, Raymond Harry Putra Rudy, Mingi Ryu, Ninad Sathaye, Chris Schnabel, Eddie Schoute, Kanav Setia, Yunong Shi, Adenilton Silva, Yukio Siraichi, Seyon Sivarajah, John A. Smolin, Mathias Soeken, Hitomi Takahashi, Ivano Tavernelli, Charles Taylor, Pete Taylour, Kenso Trabing, Matthew Treinish, Wes Turner, Desiree Vogt-Lee, Christophe Vuillot, Jonathan A. Wildstrom, Jessica Wilson, Erick Winston, Christopher Wood, Stephen Wood, Stefan Wörner, Ismail Yunus Akhalwaya, and Christa Zoufal, ``Qiskit: An open-source framework for quantum computing,'' (2019).
https://doi.org/10.5281/zenodo.2562110
[38] Vlatko Vedral, Adriano Barenco, and Artur Ekert, ``Quantum networks for elementary arithmetic operations,'' Phys. Rev. A 54, 147–153 (1996).
https://doi.org/10.1103/PhysRevA.54.147
[39] Thomas G Draper, ``Addition on a Quantum Computer,'' (2000), arXiv:quant-ph/0008033.
arXiv:quant-ph/0008033
[40] Thomas G Draper, Samuel A Kutin, Eric M Rains, and Krysta M Svore, ``A logarithmic-depth quantum carry-lookahead adder,'' Quantum Information and Computation 6, 351–369 (2006), arXiv:quant-ph/0406142.
arXiv:quant-ph/0406142
[41] Ghasem Tarmast, ``Multivariate Log-Normal Distribution,'' International Statistical Institute Proceedings: 53rd Session, Seoul (2001).
[42] Emmanuel Gobet, ``Advanced monte carlo methods for barrier and related exotic options,'' Handbook of Numerical Analysis, 15, 497 – 528 (2009).
https://doi.org/10.1016/S1570-8659(08)00012-4
[43] Pavel V. Shevchenko and Pierre Del Moral, ``Valuation of barrier options using sequential monte carlo,'' Journal of Computational Finance 20, 107–135 (2014).
https://doi.org/10.2139/ssrn.2529539
[44] M. Žnidarič, O. Giraud, and B. Georgeot, ``Optimal number of controlled-not gates to generate a three-qubit state,'' Physical Review A 77, 032320 (2008).
https://doi.org/10.1103/PhysRevA.77.032320
[45] Vivek V Shende, Stephen S Bullock, and Igor L Markov, ``Synthesis of quantum-logic circuits,'' IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006).
https://doi.org/10.1109/TCAD.2005.855930
[46] Raban Iten, Oliver Reardon-Smith, Luca Mondada, Ethan Redmond, Ravjot Singh Kohli, and Roger Colbeck, ``Introduction to UniversalQCompiler,'' (2019), arXiv:1904.01072.
arXiv:1904.01072
[47] A. Dewes, F. R. Ong, V. Schmitt, R. Lauro, N. Boulant, P. Bertet, D. Vion, and D. Esteve, ``Characterization of a two-transmon processor with individual single-shot qubit readout,'' Phys. Rev. Lett. 108, 057002 (2012).
https://doi.org/10.1103/PhysRevLett.108.057002
[48] Kristan Temme, Sergey Bravyi, and Jay M. Gambetta, ``Error mitigation for short-depth quantum circuits,'' Phys. Rev. Lett. 119, 180509 (2017).
https://doi.org/10.1103/PhysRevLett.119.180509
[49] E. F. Dumitrescu, A. J. McCaskey, G. Hagen, G. R. Jansen, T. D. Morris, T. Papenbrock, R. C. Pooser, D. J. Dean, and P. Lougovski, ``Cloud quantum computing of an atomic nucleus,'' Phys. Rev. Lett. 120, 210501 (2018).
https://doi.org/10.1103/PhysRevLett.120.210501
[50] Mark Broadie and Paul Glasserman, ``Estimating security price derivatives using simulation,'' Management Science 42 (1996), 10.1287/mnsc.42.2.269.
https://doi.org/10.1287/mnsc.42.2.269
[51] Austin G. Fowler, Matteo Mariantoni, John M. Martinis, and Andrew N. Cleland, ``Surface codes: Towards practical large-scale quantum computation,'' Phys. Rev. A 86, 032324 (2012).
https://doi.org/10.1103/PhysRevA.86.032324
[52] Miroslav Dobšíček, Göran Johansson, Vitaly Shumeiko, and Göran Wendin, ``Arbitrary accuracy iterative quantum phase estimation algorithm using a single ancillary qubit: A two-qubit benchmark,'' Phys. Rev. A 76, 030306 (2007).
https://doi.org/10.1103/PhysRevA.76.030306
[53] C. J. O'Loan, ``Iterative phase estimation,'' Journal of Physics A: Mathematical and Theoretical 43 (2010), 10.1088/1751-8113/43/1/015301.
https://doi.org/10.1088/1751-8113/43/1/015301
[54] Krysta M Svore, Matthew B Hastings, and Michael Freedman, ``Faster phase estimation,'' Quantum Information & Computation 14, 306–328 (2014), arXiv:1304.0741.
arXiv:1304.0741
[55] S.S. Bullock and I.L. Markov, ``Smaller circuits for arbitrary n-qubit diagonal computations,'' Quantum Information & Computation 4, 027–047 (2004), arXiv:quant-ph/0303039.
arXiv:quant-ph/0303039
Cited by
[1] 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).
[2] Dmitry Grinko, Julien Gacon, Christa Zoufal, and Stefan Woerner, "Iterative quantum amplitude estimation", npj Quantum Information 7, 52 (2021).
[3] Daniel J. Egger, Jakub Mareček, and Stefan Woerner, "Warm-starting quantum optimization", Quantum 5, 479 (2021).
[4] Ismail Yunus Akhalwaya, Adam Connolly, Roland Guichard, Steven Herbert, Cahit Kargi, Alexandre Krajenbrink, Michael Lubasch, Conor Mc Keever, Julien Sorci, Michael Spranger, and Ifan Williams, "A Modular Engine for Quantum Monte Carlo Integration", arXiv:2308.06081, (2023).
[5] Javier Gonzalez-Conde, Ángel Rodríguez-Rozas, Enrique Solano, and Mikel Sanz, "Efficient Hamiltonian simulation for solving option price dynamics", Physical Review Research 5 4, 043220 (2023).
[6] Samuel Mugel, Carlos Kuchkovsky, Escolástico Sánchez, Samuel Fernández-Lorenzo, Jorge Luis-Hita, Enrique Lizaso, and Román Orús, "Dynamic portfolio optimization with real datasets using quantum processors and quantum-inspired tensor networks", Physical Review Research 4 1, 013006 (2022).
[7] Ikko Hamamura and Takashi Imamichi, "Efficient evaluation of quantum observables using entangled measurements", npj Quantum Information 6, 56 (2020).
[8] Shouvanik Chakrabarti, Rajiv Krishnakumar, Guglielmo Mazzola, Nikitas Stamatopoulos, Stefan Woerner, and William J. Zeng, "A Threshold for Quantum Advantage in Derivative Pricing", Quantum 5, 463 (2021).
[9] Julia E. Rice, Tanvi P. Gujarati, Mario Motta, Tyler Y. Takeshita, Eunseok Lee, Joseph A. Latone, and Jeannette M. Garcia, "Quantum computation of dominant products in lithium-sulfur batteries", Journal of Chemical Physics 154 13, 134115 (2021).
[10] Mudassir Moosa, Thomas W. Watts, Yiyou Chen, Abhijat Sarma, and Peter L. McMahon, "Linear-depth quantum circuits for loading Fourier approximations of arbitrary functions", Quantum Science and Technology 9 1, 015002 (2024).
[11] Pauline J. Ollitrault, Guglielmo Mazzola, and Ivano Tavernelli, "Nonadiabatic Molecular Quantum Dynamics with Quantum Computers", Physical Review Letters 125 26, 260511 (2020).
[12] Steven Herbert, "Quantum Monte Carlo Integration: The Full Advantage in Minimal Circuit Depth", Quantum 6, 823 (2022).
[13] Almudena Carrera Vazquez, Daniel J. Egger, David Ochsner, and Stefan Woerner, "Well-conditioned multi-product formulas for hardware-friendly Hamiltonian simulation", Quantum 7, 1067 (2023).
[14] Dong An, Noah Linden, Jin-Peng Liu, Ashley Montanaro, Changpeng Shao, and Jiasu Wang, "Quantum-accelerated multilevel Monte Carlo methods for stochastic differential equations in mathematical finance", Quantum 5, 481 (2021).
[15] Daniel J. Egger, Ricardo Gacía Gutiérrez, Jordi Cahué Mestre, and Stefan Woerner, "Credit Risk Analysis using Quantum Computers", arXiv:1907.03044, (2019).
[16] 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).
[17] Nikitas Stamatopoulos, Guglielmo Mazzola, Stefan Woerner, and William J. Zeng, "Towards Quantum Advantage in Financial Market Risk using Quantum Gradient Algorithms", Quantum 6, 770 (2022).
[18] Almudena Carrera Vazquez and Stefan Woerner, "Efficient State Preparation for Quantum Amplitude Estimation", Physical Review Applied 15 3, 034027 (2021).
[19] Gabriel Marin-Sanchez, Javier Gonzalez-Conde, and Mikel Sanz, "Quantum algorithms for approximate function loading", Physical Review Research 5 3, 033114 (2023).
[20] Arthur G. Rattew, Yue Sun, Pierre Minssen, and Marco Pistoia, "The Efficient Preparation of Normal Distributions in Quantum Registers", Quantum 5, 609 (2021).
[21] Koichi Miyamoto and Kenji Shiohara, "Reduction of qubits in a quantum algorithm for Monte Carlo simulation by a pseudo-random-number generator", Physical Review A 102 2, 022424 (2020).
[22] Ar A. Melnikov, A. A. Termanova, S. V. Dolgov, F. Neukart, and M. R. Perelshtein, "Quantum state preparation using tensor networks", Quantum Science and Technology 8 3, 035027 (2023).
[23] Javier Gonzalez-Conde, Thomas W. Watts, Pablo Rodriguez-Grasa, and Mikel Sanz, "Efficient quantum amplitude encoding of polynomial functions", Quantum 8, 1297 (2024).
[24] João F. Doriguello, Alessandro Luongo, Jinge Bao, Patrick Rebentrost, and Miklos Santha, "Quantum algorithm for stochastic optimal stopping problems with applications in finance", arXiv:2111.15332, (2021).
[25] Austin Gilliam, Stefan Woerner, and Constantin Gonciulea, "Grover Adaptive Search for Constrained Polynomial Binary Optimization", Quantum 5, 428 (2021).
[26] Philippe Suchsland, Francesco Tacchino, Mark H. Fischer, Titus Neupert, Panagiotis Kl. Barkoutsos, and Ivano Tavernelli, "Algorithmic Error Mitigation Scheme for Current Quantum Processors", Quantum 5, 492 (2021).
[27] Eric G. Brown, Oktay Goktas, and W. K. Tham, "Quantum Amplitude Estimation in the Presence of Noise", arXiv:2006.14145, (2020).
[28] Kirill Plekhanov, Matthias Rosenkranz, Mattia Fiorentini, and Michael Lubasch, "Variational quantum amplitude estimation", Quantum 6, 670 (2022).
[29] Ana Martin, Bruno Candelas, Ángel Rodríguez-Rozas, José D. Martín-Guerrero, Xi Chen, Lucas Lamata, Román Orús, Enrique Solano, and Mikel Sanz, "Toward pricing financial derivatives with an IBM quantum computer", Physical Review Research 3 1, 013167 (2021).
[30] Juan José García-Ripoll, "Quantum-inspired algorithms for multivariate analysis: from interpolation to partial differential equations", Quantum 5, 431 (2021).
[31] 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).
[32] Xi-Ning Zhuang, Zhao-Yun Chen, Yu-Chun Wu, and Guo-Ping Guo, "Quantum computational quantitative trading: high-frequency statistical arbitrage algorithm", New Journal of Physics 24 7, 073036 (2022).
[33] Daniel J. Egger, Chiara Capecci, Bibek Pokharel, Panagiotis Kl. Barkoutsos, Laurin E. Fischer, Leonardo Guidoni, and Ivano Tavernelli, "Pulse variational quantum eigensolver on cross-resonance-based hardware", Physical Review Research 5 3, 033159 (2023).
[34] Bobak Toussi Kiani, Giacomo De Palma, Milad Marvian, Zi-Wen Liu, and Seth Lloyd, "Learning quantum data with the quantum earth mover's distance", Quantum Science and Technology 7 4, 045002 (2022).
[35] Christa Zoufal, Aurélien Lucchi, and Stefan Woerner, "Variational Quantum Boltzmann Machines", arXiv:2006.06004, (2020).
[36] Nikitas Stamatopoulos and William J. Zeng, "Derivative Pricing using Quantum Signal Processing", Quantum 8, 1322 (2024).
[37] Hedayat Alghassi, Amol Deshmukh, Noelle Ibrahim, Nicolas Robles, Stefan Woerner, and Christa Zoufal, "A variational quantum algorithm for the Feynman-Kac formula", Quantum 6, 730 (2022).
[38] Hans Hon Sang Chan, Richard Meister, Tyson Jones, David P. Tew, and Simon C. Benjamin, "Grid-based methods for chemistry simulations on a quantum computer", Science Advances 9 9, eabo7484 (2023).
[39] Zoltán Udvarnoki, Gábor Fáth, and Norbert Fogarasi, "Quantum advantage of Monte Carlo option pricing", Journal of Physics Communications 7 5, 055001 (2023).
[40] Juan José García-Ripoll, "Quantum-inspired algorithms for multivariate analysis: from interpolation to partial differential equations", arXiv:1909.06619, (2019).
[41] Kazuya Kaneko, Koichi Miyamoto, Naoyuki Takeda, and Kazuyoshi Yoshino, "Quantum Pricing with a Smile: Implementation of Local Volatility Model on Quantum Computer", arXiv:2007.01467, (2020).
[42] Adam Holmes and A. Y. Matsuura, "Efficient Quantum Circuits for Accurate State Preparation of Smooth, Differentiable Functions", arXiv:2005.04351, (2020).
[43] Iordanis Kerenidis, Anupam Prakash, and Dániel Szilágyi, "Quantum Algorithms for Portfolio Optimization", arXiv:1908.08040, (2019).
[44] Javier Alcazar, Andrea Cadarso, Amara Katabarwa, Marta Mauri, Borja Peropadre, Guoming Wang, and Yudong Cao, "Quantum algorithm for credit valuation adjustments", New Journal of Physics 24 2, 023036 (2022).
[45] Almudena Carrera Vazquez and Stefan Woerner, "Efficient State Preparation for Quantum Amplitude Estimation", arXiv:2005.07711, (2020).
[46] David Fitzek, Toheed Ghandriz, Leo Laine, Mats Granath, and Anton Frisk Kockum, "Applying quantum approximate optimization to the heterogeneous vehicle routing problem", arXiv:2110.06799, (2021).
[47] Gabriele Agliardi, Michele Grossi, Mathieu Pellen, and Enrico Prati, "Quantum integration of elementary particle processes", Physics Letters B 832, 137228 (2022).
[48] Shumpei Uno, Yohichi Suzuki, Keigo Hisanaga, Rudy Raymond, Tomoki Tanaka, Tamiya Onodera, and Naoki Yamamoto, "Modified Grover operator for quantum amplitude estimation", New Journal of Physics 23 8, 083031 (2021).
[49] Carsten Blank, Daniel K. Park, and Francesco Petruccione, "Quantum-enhanced analysis of discrete stochastic processes", npj Quantum Information 7, 126 (2021).
[50] Julien Zylberman and Fabrice Debbasch, "Efficient quantum state preparation with Walsh series", Physical Review A 109 4, 042401 (2024).
[51] Tudor Giurgica-Tiron, Sonika Johri, Iordanis Kerenidis, Jason Nguyen, Neal Pisenti, Anupam Prakash, Ksenia Sosnova, Ken Wright, and William Zeng, "Low-depth amplitude estimation on a trapped-ion quantum computer", Physical Review Research 4 3, 033034 (2022).
[52] Philippe Suchsland, Panagiotis Kl. Barkoutsos, Ivano Tavernelli, Mark H. Fischer, and Titus Neupert, "Simulating a ring-like Hubbard system with a quantum computer", Physical Review Research 4 1, 013165 (2022).
[53] Alessandro Tammaro, Davide E. Galli, Julia E. Rice, and Mario Motta, "N-Electron Valence Perturbation Theory with Reference Wave Functions from Quantum Computing: Application to the Relative Stability of Hydroxide Anion and Hydroxyl Radical", Journal of Physical Chemistry A 127 3, 817 (2023).
[54] Julien Gacon, Christa Zoufal, and Stefan Woerner, "Quantum-Enhanced Simulation-Based Optimization", arXiv:2005.10780, (2020).
[55] Lee Braine, Daniel J. Egger, Jennifer Glick, and Stefan Woerner, "Quantum Algorithms for Mixed Binary Optimization applied to Transaction Settlement", arXiv:1910.05788, (2019).
[56] Imran Khan Tutul, Sara Karimi, Mohammadreza Soltaninia, and Junpeng Zhan, "Shallow Depth Factoring Based on Quantum Feasibility Labeling and Variational Quantum Search", arXiv:2305.19542, (2023).
[57] Raúl Berganza Gómez, Corey O'Meara, Giorgio Cortiana, Christian B. Mendl, and Juan Bernabé-Moreno, "Towards AutoQML: A Cloud-Based Automated Circuit Architecture Search Framework", arXiv:2202.08024, (2022).
[58] Austin Gilliam, Charlene Venci, Sreraman Muralidharan, Vitaliy Dorum, Eric May, Rajesh Narasimhan, and Constantin Gonciulea, "Foundational Patterns for Efficient Quantum Computing", arXiv:1907.11513, (2019).
[59] Caleb Rotello, Peter Graf, Matthew Reynolds, Cody James Winkleblack, and Wesley Jones, "Calculating the expected value function of a two-stage stochastic optimization program with a quantum algorithm", arXiv:2402.15029, (2024).
[60] Tomoki Tanaka, Shumpei Uno, Tamiya Onodera, Naoki Yamamoto, and Yohichi Suzuki, "Noisy quantum amplitude estimation without noise estimation", Physical Review A 105 1, 012411 (2022).
[61] Israel F. Araujo, Carsten Blank, Ismael C. S. Araújo, and Adenilton J. da Silva, "Low-rank quantum state preparation", arXiv:2111.03132, (2021).
[62] Ethan Wang, "Efficient Cauchy distribution based quantum state preparation by using the comparison algorithm", AIP Advances 11 10, 105307 (2021).
[63] Jonas Koppe and Mark-Oliver Wolf, "Amplitude-based implementation of the unit step function on a quantum computer", Physical Review A 107 2, 022606 (2023).
[64] Kazuya Kaneko, Koichi Miyamoto, Naoyuki Takeda, and Kazuyoshi Yoshino, "Quantum speedup of Monte Carlo integration with respect to the number of dimensions and its application to finance", Quantum Information Processing 20 5, 185 (2021).
[65] Wei Hu, Yang Yang, Weiye Xia, Jiawei Pi, Enyi Huang, Xin-Ding Zhang, and Hua Xu, "Performance of superconducting quantum computing chips under different architecture designs", Quantum Information Processing 21 7, 237 (2022).
[66] Antonio Sannia, Rodrigo Martínez-Peña, Miguel C. Soriano, Gian Luca Giorgi, and Roberta Zambrini, "Dissipation as a resource for Quantum Reservoir Computing", Quantum 8, 1291 (2024).
[67] Cristina Sanz-Fernandez, Rodrigo Hernandez, Christian D. Marciniak, Ivan Pogorelov, Thomas Monz, Francesco Benfenati, Samuel Mugel, and Roman Orus, "Quantum portfolio value forecasting", arXiv:2111.14970, (2021).
[68] M. C. Braun, T. Decker, N. Hegemann, S. F. Kerstan, and C. Schäfer, "A Quantum Algorithm for the Sensitivity Analysis of Business Risks", arXiv:2103.05475, (2021).
[69] Kang-Da Wu, Chengran Yang, Ren-Dong He, Mile Gu, Guo-Yong Xiang, Chuan-Feng Li, Guang-Can Guo, and Thomas J. Elliott, "Implementing quantum dimensionality reduction for non-Markovian stochastic simulation", Nature Communications 14, 2624 (2023).
[70] Prithvi Gundlapalli and Junyi Lee, "Deterministic and Entanglement-Efficient Preparation of Amplitude-Encoded Quantum Registers", Physical Review Applied 18 2, 024013 (2022).
[71] Monit Sharma, Hoong Chuin Lau, and Rudy Raymond, "Quantum-Enhanced Simulation-Based Optimization for Newsvendor Problems", arXiv:2403.17389, (2024).
[72] Nils Quetschlich, Lukas Burgholzer, and Robert Wille, "Towards an Automated Framework for Realizing Quantum Computing Solutions", arXiv:2210.14928, (2022).
[73] D. V. Babukhin and W. V. Pogosov, "The effect of quantum noise on algorithmic perfect quantum state transfer on NISQ processors", Quantum Information Processing 21 1, 7 (2022).
[74] Mark-Oliver Wolf, Tom Ewen, and Ivica Turkalj, "Quantum Architecture Search for Quantum Monte Carlo Integration via Conditional Parameterized Circuits with Application to Finance", arXiv:2304.08793, (2023).
[75] M. C. Braun, T. Decker, N. Hegemann, and S. F. Kerstan, "Error Resilient Quantum Amplitude Estimation from Parallel Quantum Phase Estimation", arXiv:2204.01337, (2022).
[76] Jirawat Tangpanitanon, Jirawat Saiphet, Pantita Palittapongarnpim, Poompong Chaiwongkhot, Pinn Prugsanapan, Nuntanut Raksasri, Wipada Wannasiwaporn, Yarnvith Raksri, Pairash Thajchayapong, and Thiparat Chotibut, "Hybrid Quantum-Classical Algorithms for Loan-Collection Optimization with Loan-Loss Provisions", Physical Review Applied 19 6, 064001 (2023).
[77] Byeongyong Park and Doyeol Ahn, "Reducing CNOT count in quantum Fourier transform for the linear nearest-neighbor architecture", Scientific Reports 13, 8638 (2023).
[78] Yen-Jui Chang, Wei-Ting Wang, Hao-Yuan Chen, Shih-Wei Liao, and Ching-Ray Chang, "A novel approach for quantum financial simulation and quantum state preparation", arXiv:2308.01844, (2023).
[79] Yewei Yuan, Chao Wang, Bei Wang, Zhao-Yun Chen, Meng-Han Dou, Yu-Chun Wu, and Guo-Ping Guo, "An improved QFT-based quantum comparator and extended modular arithmetic using one ancilla qubit", New Journal of Physics 25 10, 103011 (2023).
[80] Alessandro Carbone, Davide Emilio Galli, Mario Motta, and Barbara Jones, "Quantum Circuits for the Preparation of Spin Eigenfunctions on Quantum Computers", Symmetry 14 3, 624 (2022).
[81] Xiaojian Du and Wenyang Qian, "Accelerated quantum circuit Monte Carlo simulation for heavy quark thermalization", Physical Review D 109 7, 076025 (2024).
[82] Francesca Cibrario, Or Samimi Golan, Giacomo Ranieri, Emanuele Dri, Mattia Ippoliti, Ron Cohen, Christian Mattia, Bartolomeo Montrucchio, Amir Naveh, and Davide Corbelletto, "Quantum Amplitude Loading for Rainbow Options Pricing", arXiv:2402.05574, (2024).
[83] Kazuya Kaneko, Koichi Miyamoto, Naoyuki Takeda, and Kazuyoshi Yoshino, "Linear regression by quantum amplitude estimation and its extension to convex optimization", Physical Review A 104 2, 022430 (2021).
[84] S. E. Rasmussen and N. T. Zinner, "Multiqubit state learning with entangling quantum generative adversarial networks", Physical Review A 106 3, 032429 (2022).
[85] Adam Holmes and A. Y. Matsuura, "Entanglement Properties of Quantum Superpositions of Smooth, Differentiable Functions", arXiv:2009.09096, (2020).
[86] Christian Laudagé and Ivica Turkalj, "Quantum Risk Analysis: Beyond (Conditional) Value-at-Risk", arXiv:2211.04456, (2022).
[87] Tom Ewen, "Pricing of European Calls with the Quantum Fourier Transform", arXiv:2404.14115, (2024).
[88] Xavier Vasques, "A new step for computing", arXiv:2108.03997, (2021).
[89] Xavier Vasques, "The data center of tomorrow is made up of heterogeneous accelerators", arXiv:2003.10950, (2020).
[90] Koichi Miyamoto, "Quantum algorithms for numerical differentiation of expected values with respect to parameters", Quantum Information Processing 21 3, 109 (2022).
The above citations are from SAO/NASA ADS (last updated successfully 2024-06-02 09:54:28). The list may be incomplete as not all publishers provide suitable and complete citation data.
Could not fetch Crossref cited-by data during last attempt 2024-06-02 09:54:24: Encountered the unhandled forward link type postedcontent_cite while looking for citations to DOI 10.22331/q-2020-07-06-291.
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.