A Threshold for Quantum Advantage in Derivative Pricing
1Goldman, Sachs & Co., New York, NY
2University of Maryland, College Park, MD
3IBM Quantum, IBM Research – Zurich
| Published: | 2021-06-01, volume 5, page 463 |
| Eprint: | arXiv:2012.03819v3 |
| Doi: | https://doi.org/10.22331/q-2021-06-01-463 |
| Citation: | Quantum 5, 463 (2021). |
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Abstract
We give an upper bound on the resources required for valuable quantum advantage in pricing derivatives. To do so, we give the first complete resource estimates for useful quantum derivative pricing, using autocallable and Target Accrual Redemption Forward (TARF) derivatives as benchmark use cases. We uncover blocking challenges in known approaches and introduce a new method for quantum derivative pricing – the $\textit{re-parameterization method}$ – that avoids them. This method combines pre-trained variational circuits with fault-tolerant quantum computing to dramatically reduce resource requirements. We find that the benchmark use cases we examine require 8k logical qubits and a T-depth of 54 million. We estimate that quantum advantage would require executing this program at the order of a second. While the resource requirements given here are out of reach of current systems, we hope they will provide a roadmap for further improvements in algorithms, implementations, and planned hardware architectures.
Featured image: Loading the probability distribution for derivative pricing with the re-parameterization method takes variationally-trained standard normal gaussians and distributes the probability mass appropriately by re-parameterizing the basis states corresponding to asset prices.
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[1] A. Prabha, S. Savard, and H. Wickramarachi, Deriving the Economic Impact of Derivatives, Tech. Rep. (Milken Institute, 2013).
https://milkeninstitute.org/sites/default/files/reports-pdf/Derivatives-Report.pdf
[2] F. Black and M. Scholes, ``The pricing of options and corporate liabilities,'' Journal of Political Economy 81, 637 (1973).
https://doi.org/10.1086/260062
[3] A. 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
[4] G. Brassard, P. Hoyer, M. Mosca, and A. Tapp, ``Quantum Amplitude Amplification and Estimation,'' Contemporary Mathematics 305 (2002), 10.1090/conm/305/05215.
https://doi.org/10.1090/conm/305/05215
[5] 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
[6] 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
[7] A. Carrera Vazquez and S. Woerner, ``Efficient state preparation for quantum amplitude estimation,'' Physical Review Applied 15 (2021), 10.1103/physrevapplied.15.034027.
https://doi.org/10.1103/physrevapplied.15.034027
[8] D. J. Egger, R. G. Gutierrez, J. C. Mestre, and S. Woerner, ``Credit risk analysis using quantum computers,'' IEEE Transactions on Computers (2020a), 10.1109/TC.2020.3038063.
https://doi.org/10.1109/TC.2020.3038063
[9] S. Woerner and D. J. Egger, ``Quantum risk analysis,'' npj Quantum Information 5 (2019), 10.1038/s41534-019-0130-6.
https://doi.org/10.1038/s41534-019-0130-6
[10] L. Grover and T. Rudolph, ``Creating superpositions that correspond to efficiently integrable probability distributions,'' (2002), arXiv:quant-ph/0208112.
arXiv:quant-ph/0208112
[11] R. Y. Rubinstein, Simulation and the Monte Carlo Method, Wiley Series in Probability and Statistics (Wiley, 1981).
https://doi.org/10.1002/9780470316511
[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] S. Aaronson and P. Rall, ``Quantum approximate counting, simplified,'' Symposium on Simplicity in Algorithms , 24–32 (2020).
https://doi.org/10.1137/1.9781611976014.5
[14] D. Grinko, J. Gacon, C. Zoufal, and S. Woerner, ``Iterative quantum amplitude estimation,'' npj Quantum Information 7 (2021), 10.1038/s41534-021-00379-1.
https://doi.org/10.1038/s41534-021-00379-1
[15] K. Nakaji, ``Faster Amplitude Estimation,'' (2020), arXiv:2003.02417 [quant-ph].
arXiv:2003.02417
[16] T. Tanaka, Y. Suzuki, S. Uno, R. Raymond, T. Onodera, and N. Yamamoto, ``Amplitude estimation via maximum likelihood on noisy quantum computer,'' (2020), arXiv:2006.16223 [quant-ph].
arXiv:2006.16223
[17] T. Giurgica-Tiron, I. Kerenidis, F. Labib, A. Prakash, and W. Zeng, ``Low depth algorithms for quantum amplitude estimation,'' (2020), arXiv:2012.03348 [quant-ph].
arXiv:2012.03348
[18] M. A. Nielsen and I. L. Chuang, Cambridge University Press (2010) p. 702.
https://doi.org/10.1017/CBO9780511976667
[19] A. Bouland, W. van Dam, H. Joorati, I. Kerenidis, and A. Prakash, ``Prospects and challenges of quantum finance,'' (2020), arXiv:2011.06492 [q-fin.CP].
arXiv:2011.06492
[20] M. Plesch and Č. Brukner, ``Quantum-state preparation with universal gate decompositions,'' Physical Review A 83, 10.1103/physreva.83.032302.
https://doi.org/10.1103/physreva.83.032302
[21] C. Zoufal, A. Lucchi, and S. Woerner, ``Quantum generative adversarial networks for learning and loading random distributions,'' npj Quantum Information 5, 1 (2019).
https://doi.org/10.1038/s41534-019-0223-2
[22] A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O’Brien, ``A variational eigenvalue solver on a photonic quantum processor,'' Nature Communications 5 (2014), 10.1038/ncomms5213.
https://doi.org/10.1038/ncomms5213
[23] P. J. Ollitrault, G. Mazzola, and I. Tavernelli, ``Nonadiabatic molecular quantum dynamics with quantum computers,'' Physical Review Letters 125 (2020), 10.1103/physrevlett.125.260511.
https://doi.org/10.1103/physrevlett.125.260511
[24] C. M. Dawson and M. A. Nielsen, ``The solovay-kitaev algorithm,'' Quantum Info. Comput. 6, 81–95 (2006), arXiv:quant-ph/0505030.
arXiv:quant-ph/0505030
[25] P. Selinger, ``Efficient clifford+t approximation of single-qubit operators,'' Quantum Info. Comput. 15, 159–180 (2015), arXiv:1212.6253 [quant-ph].
arXiv:1212.6253
[26] R. Babbush, J. R. McClean, M. Newman, C. Gidney, S. Boixo, and H. Neven, ``Focus beyond quadratic speedups for error-corrected quantum advantage,'' PRX Quantum 2 (2021), 10.1103/prxquantum.2.010103.
https://doi.org/10.1103/prxquantum.2.010103
[27] A. G. Fowler and C. Gidney, ``Low overhead quantum computation using lattice surgery,'' arXiv:1808.06709 (2018).
arXiv:1808.06709
[28] C. Gidney and M. Ekerå, ``How to factor 2048 bit rsa integers in 8 hours using 20 million noisy qubits,'' Quantum 5, 433 (2021).
https://doi.org/10.22331/q-2021-04-15-433
[29] D. J. Egger, C. Gambella, J. Marecek, S. McFaddin, M. Mevissen, R. Raymond, A. Simonetto, S. Woerner, and E. Yndurain, ``Quantum computing for finance: State-of-the-art and future prospects,'' IEEE Transactions on Quantum Engineering 1, 1–24 (2020b).
https://doi.org/10.1109/tqe.2020.3030314
[30] S. Herbert, ``The problem with grover-rudolph state preparation for quantum monte-carlo,'' (2021), arXiv:2101.02240 [quant-ph].
arXiv:2101.02240
[31] K. Kaneko, K. Miyamoto, N. Takeda, and K. Yoshino, ``Quantum pricing with a smile: Implementation of local volatility model on quantum computer,'' (2020), arXiv:2007.01467 [quant-ph].
arXiv:2007.01467
[32] P. Selinger, ``Quantum circuits of t-depth one,'' Physical Review A 87 (2013), 10.1103/physreva.87.042302.
https://doi.org/10.1103/physreva.87.042302
[33] T. Häner, M. Roetteler, and K. M. Svore, ``Optimizing quantum circuits for arithmetic,'' (2018), arXiv:1805.12445 [quant-ph].
arXiv:1805.12445
[34] 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), arXiv:quant-ph/0406142.
arXiv:quant-ph/0406142
[35] D. Maslov and M. Saeedi, ``Reversible circuit optimization via leaving the boolean domain,'' IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 30, 806 (2011).
https://doi.org/10.1109/TCAD.2011.2105555
[36] Y. Takahashi, S. Tani, and N. Kunihiro, ``Quantum addition circuits and unbounded fan-out,'' (2009), arXiv:0910.2530 [quant-ph].
arXiv:0910.2530
[37] E. Muñoz Coreas and H. Thapliyal, ``T-count and qubit optimized quantum circuit design of the non-restoring square root algorithm,'' J. Emerg. Technol. Comput. Syst. 14 (2018), 10.1145/3264816.
https://doi.org/10.1145/3264816
[38] N. J. Ross and P. Selinger, ``Optimal ancilla-free clifford+t approximation of z-rotations,'' Quantum Info. Comput. 16, 901 (2016), arXiv:1403.2975 [quant-ph].
arXiv:1403.2975
[39] A. Bocharov, M. Roetteler, and K. M. Svore, ``Efficient synthesis of universal repeat-until-success quantum circuits,'' Physical Review Letters 114 (2015), 10.1103/physrevlett.114.080502.
https://doi.org/10.1103/physrevlett.114.080502
[40] T. Kim and B. Choi, ``Efficient decomposition methods for controlled-r n using a single ancillary qubit,'' Scientific Reports 8 (2018), 10.1038/s41598-018-23764-x.
https://doi.org/10.1038/s41598-018-23764-x
[41] C. Zalka, ``Simulating quantum systems on a quantum computer,'' Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313 (1998).
https://doi.org/10.1098/rspa.1998.0162
[42] S. Wiesner, ``Simulations of many-body quantum systems by a quantum computer,'' (1996), arXiv:quant-ph/9603028 [quant-ph].
arXiv:quant-ph/9603028
[43] P. K. Barkoutsos, J. F. Gonthier, I. Sokolov, N. Moll, G. Salis, A. Fuhrer, M. Ganzhorn, D. J. Egger, M. Troyer, A. Mezzacapo, S. Filipp, and I. Tavernelli, ``Quantum algorithms for electronic structure calculations: Particle-hole hamiltonian and optimized wave-function expansions,'' Phys. Rev. A 98, 022322 (2018).
https://doi.org/10.1103/PhysRevA.98.022322
[44] J. Stokes, J. Izaac, N. Killoran, and G. Carleo, ``Quantum natural gradient,'' Quantum 4, 269 (2020).
https://doi.org/10.22331/q-2020-05-25-269
[45] S. McArdle, T. Jones, S. Endo, Y. Li, S. C. Benjamin, and X. Yuan, ``Variational ansatz-based quantum simulation of imaginary time evolution,'' npj Quantum Information 5 (2019), 10.1038/s41534-019-0187-2.
https://doi.org/10.1038/s41534-019-0187-2
Cited by
[1] Ethan Wang, "Efficient Cauchy distribution based quantum state preparation by using the comparison algorithm", AIP Advances 11 10, 105307 (2021).
[2] Kenji Kubo, Koichi Miyamoto, Kosuke Mitarai, and Keisuke Fujii, "Pricing Multiasset Derivatives by Variational Quantum Algorithms", IEEE Transactions on Quantum Engineering 4, 1 (2023).
[3] Amit Saha, Anupam Chattopadhyay, and Amlan Chakrabarti, "Robust Quantum Arithmetic Operations with Intermediate Qutrits in the NISQ-era", International Journal of Theoretical Physics 62 4, 92 (2023).
[4] Koichi Miyamoto and Kenji Kubo, "Pricing Multi-Asset Derivatives by Finite-Difference Method on a Quantum Computer", IEEE Transactions on Quantum Engineering 3, 1 (2022).
[5] 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).
[6] Steven Herbert, "Quantum Monte Carlo Integration: The Full Advantage in Minimal Circuit Depth", Quantum 6, 823 (2022).
[7] Sebastian Senge, Tim Leonhardt, and Kinan Halabi, "Quantum Incubation Journey: Theory Founded Use Case and Technology Selection", Digitale Welt 5 2, 54 (2021).
[8] Roman Rietsche, Christian Dremel, Samuel Bosch, Léa Steinacker, Miriam Meckel, and Jan-Marco Leimeister, "Quantum computing", Electronic Markets 32 4, 2525 (2022).
[9] Steven Herbert, "Quantum computing for data-centric engineering and science", Data-Centric Engineering 3, e36 (2022).
[10] Erik J. Gustafson, Henry Lamm, and Judah Unmuth-Yockey, "Quantum mean estimation for lattice field theory", Physical Review D 107 11, 114511 (2023).
[11] Nils Quetschlich, Lukas Burgholzer, and Robert Wille, 2023 IEEE International Conference on Quantum Software (QSW) 43 (2023) ISBN:979-8-3503-0479-4.
[12] Koichi Miyamoto, "Quantum algorithm for calculating risk contributions in a credit portfolio", EPJ Quantum Technology 9 1, 32 (2022).
[13] 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).
[14] 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).
[15] Zoltán Udvarnoki, Gábor Fáth, and Norbert Fogarasi, "Quantum advantage of Monte Carlo option pricing", Journal of Physics Communications 7 5, 055001 (2023).
[16] Ethan T. Wang, "Efficient Quantum State Preparation for the Cauchy Distribution Based on Piecewise Arithmetic", IEEE Transactions on Quantum Engineering 3, 1 (2022).
[17] Paula García-Molina, Javier Rodríguez-Mediavilla, and Juan José García-Ripoll, "Quantum Fourier analysis for multivariate functions and applications to a class of Schrödinger-type partial differential equations", Physical Review A 105 1, 012433 (2022).
[18] Koichi Miyamoto, "Quantum algorithms for numerical differentiation of expected values with respect to parameters", Quantum Information Processing 21 3, 109 (2022).
[19] Kazuya Kaneko, Koichi Miyamoto, Naoyuki Takeda, and Kazuyoshi Yoshino, "Quantum pricing with a smile: implementation of local volatility model on quantum computer", EPJ Quantum Technology 9 1, 7 (2022).
[20] Eugene Msizi Buthelezi and Phocenah Nyatanga, "Threshold of the CAPB That Can Be Attributed to Fiscal Consolidation Episodes in South Africa", Economies 11 6, 152 (2023).
[21] Mark-Oliver Wolf, Roman Horsky, and Jonas Koppe, "A Quantum Algorithm for Pricing Asian Options on Valuation Trees", Risks 10 12, 221 (2022).
[22] Koichi Miyamoto, "Bermudan option pricing by quantum amplitude estimation and Chebyshev interpolation", EPJ Quantum Technology 9 1, 3 (2022).
[23] Conrad J. Haupt and Daniel J. Egger, "Leakage in restless quantum gate calibration", Physical Review A 108 2, 022614 (2023).
[24] 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).
[25] 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).
[26] Caroline Tornow, Naoki Kanazawa, William E. Shanks, and Daniel J. Egger, "Minimum Quantum Run-Time Characterization and Calibration via Restless Measurements with Dynamic Repetition Rates", Physical Review Applied 17 6, 064061 (2022).
[27] Carsten Blank and Francesco Petruccione, "1. Quantum Applications - Fachbeitrag: Vielversprechend: Monte-Carlo-ähnliche Methoden auf dem Quantencomputer ", Digitale Welt 5 4, 40 (2021).
[28] Alexandru Paler, Oumarou Oumarou, and Robert Basmadjian, "On the Realistic Worst-Case Analysis of Quantum Arithmetic Circuits", IEEE Transactions on Quantum Engineering 3, 1 (2022).
[29] Giorgio Tosti Balducci, Boyang Chen, Matthias Möller, Marc Gerritsma, and Roeland De Breuker, "Review and perspectives in quantum computing for partial differential equations in structural mechanics", Frontiers in Mechanical Engineering 8, 914241 (2022).
[30] Lester Ingber and A. Kwilinski, "Hybrid classical-quantum computing: Applications to statistical mechanics of financial markets", E3S Web of Conferences 307, 04001 (2021).
[31] Gabriel Marin-Sanchez, Javier Gonzalez-Conde, and Mikel Sanz, "Quantum algorithms for approximate function loading", Physical Review Research 5 3, 033114 (2023).
[32] 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).
[33] Annie E. Paine, Vincent E. Elfving, and Oleksandr Kyriienko, "Quantum Quantile Mechanics: Solving Stochastic Differential Equations for Generating Time‐Series", Advanced Quantum Technologies 2300065 (2023).
[34] Felix Fuchs and Blanka Horvath, "A Hybrid Quantum Wasserstein GAN with Applications to Option Pricing", SSRN Electronic Journal (2023).
[35] Andrés Gómez, Álvaro Leitao, Alberto Manzano, Daniele Musso, María R. Nogueiras, Gustavo Ordóñez, and Carlos Vázquez, "A Survey on Quantum Computational Finance for Derivatives Pricing and VaR", Archives of Computational Methods in Engineering 29 6, 4137 (2022).
[36] Natalie Klco, Alessandro Roggero, and Martin J Savage, "Standard model physics and the digital quantum revolution: thoughts about the interface", Reports on Progress in Physics 85 6, 064301 (2022).
[37] Guglielmo Mazzola, "Sampling, rates, and reaction currents through reverse stochastic quantization on quantum computers", Physical Review A 104 2, 022431 (2021).
[38] Dylan Herman, Cody Googin, Xiaoyuan Liu, Yue Sun, Alexey Galda, Ilya Safro, Marco Pistoia, and Yuri Alexeev, "Quantum computing for finance", Nature Reviews Physics 5 8, 450 (2023).
[39] Michele Vischi, Luca Ferialdi, Andrea Trombettoni, and Angelo Bassi, "Possible limits on superconducting quantum computers from spontaneous wave-function collapse models", Physical Review B 106 17, 174506 (2022).
[40] Vladimir Skavysh, Sofia Priazhkina, Diego Guala, and Thomas R. Bromley, "Quantum monte carlo for economics: Stress testing and macroeconomic deep learning", Journal of Economic Dynamics and Control 153, 104680 (2023).
[41] Zsolt Tabi, Bence Bako, Daniel T. R. Nagy, Peter Vaderna, Zsofia Kallus, Peter Haga, and Zoltan Zimboras, 2022 IEEE/ACM 7th Symposium on Edge Computing (SEC) 468 (2022) ISBN:978-1-6654-8611-8.
[42] Luca Cappelli, Francesco Tacchino, Giuseppe Murante, Stefano Borgani, and Ivano Tavernelli, "From Vlasov-Poisson to Schrödinger-Poisson: dark matter simulation with a quantum variational time evolution algorithm", arXiv:2307.06032, (2023).
[43] Javier Gonzalez-Conde, Ángel Rodríguez-Rozas, Enrique Solano, and Mikel Sanz, "Simulating option price dynamics with exponential quantum speedup", arXiv:2101.04023, (2021).
[44] Andriy Miranskyy, Lei Zhang, and Javad Doliskani, "On Testing and Debugging Quantum Software", arXiv:2103.09172, (2021).
[45] Steven Herbert, "No quantum speedup with Grover-Rudolph state preparation for quantum Monte Carlo integration", Physical Review E 103 6, 063302 (2021).
[46] Prasanth Shyamsundar, "Non-Boolean Quantum Amplitude Amplification and Quantum Mean Estimation", arXiv:2102.04975, (2021).
[47] Giuseppe Scriva, Nikita Astrakhantsev, Sebastiano Pilati, and Guglielmo Mazzola, "Challenges of variational quantum optimization with measurement shot noise", arXiv:2308.00044, (2023).
[48] 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).
[49] 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).
[50] Alexandru Paler, Oumarou Oumarou, and Robert Basmadjian, "On the realistic worst case analysis of quantum arithmetic circuits", arXiv:2101.04764, (2021).
[51] Lei Zhang, Andriy Miranskyy, Walid Rjaibi, Greg Stager, Michael Gray, and John Peck, "Making existing software quantum safe: a case study on IBM Db2", arXiv:2110.08661, (2021).
The above citations are from Crossref's cited-by service (last updated successfully 2023-09-21 14:24:47) and SAO/NASA ADS (last updated successfully 2023-09-21 14:24:48). The list may be incomplete as not all publishers provide suitable and complete citation data.
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