A Threshold for Quantum Advantage in Derivative Pricing

Shouvanik Chakrabarti1,2, Rajiv Krishnakumar1, Guglielmo Mazzola3, Nikitas Stamatopoulos1, Stefan Woerner3, and William J. Zeng1

1Goldman, Sachs & Co., New York, NY
2University of Maryland, College Park, MD
3IBM Quantum, IBM Research – Zurich

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


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.

Derivative contracts are ubiquitous in finance with various uses from hedging risk to speculation, and currently have an estimated global gross market value in the tens of trillions of dollars. The goal of derivative pricing is to determine the value of entering a derivative contract today, given the uncertainty about future values of the underlying assets. In many cases, the pricing of derivative contracts uses Monte Carlo methods which consume significant computational resources for financial institutions and therefore, finding a quantum advantage for this application would be very valuable to the financial sector as a whole. This work gives the first detailed resource estimates of the required conditions for quantum advantage in derivative pricing. In doing so, it also introduces a new method for loading stochastic processes into a quantum computer, the re-parameterization method, which overcomes blocking challenges in previous approaches. We find that the benchmark use cases we examine require 8k logical qubits and a T-depth of 54 million and 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.

► BibTeX data

► References

[1] A. Prabha, S. Savard, and H. Wickramarachi, Deriving the Economic Impact of Derivatives, Tech. Rep. (Milken Institute, 2013).

[2] F. Black and M. Scholes, ``The pricing of options and corporate liabilities,'' Journal of Political Economy 81, 637 (1973).

[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.

[4] G. Brassard, P. Hoyer, M. Mosca, and A. Tapp, ``Quantum Amplitude Amplification and Estimation,'' Contemporary Mathematics 305 (2002), 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).

[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).

[7] A. Carrera Vazquez and S. Woerner, ``Efficient state preparation for quantum amplitude estimation,'' Physical Review Applied 15 (2021), 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.

[9] S. Woerner and D. J. Egger, ``Quantum risk analysis,'' npj Quantum Information 5 (2019), 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.

[11] R. Y. Rubinstein, Simulation and the Monte Carlo Method, Wiley Series in Probability and Statistics (Wiley, 1981).

[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).

[13] S. Aaronson and P. Rall, ``Quantum approximate counting, simplified,'' Symposium on Simplicity in Algorithms , 24–32 (2020).

[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.

[15] K. Nakaji, ``Faster Amplitude Estimation,'' (2020), arXiv:2003.02417 [quant-ph].

[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].

[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].

[18] M. A. Nielsen and I. L. Chuang, Cambridge University Press (2010) p. 702.

[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].

[20] M. Plesch and Č. Brukner, ``Quantum-state preparation with universal gate decompositions,'' Physical Review A 83, 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).

[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.

[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.

[24] C. M. Dawson and M. A. Nielsen, ``The solovay-kitaev algorithm,'' Quantum Info. Comput. 6, 81–95 (2006), 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].

[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.

[27] A. G. Fowler and C. Gidney, ``Low overhead quantum computation using lattice surgery,'' arXiv:1808.06709 (2018).

[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).

[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).

[30] S. Herbert, ``The problem with grover-rudolph state preparation for quantum monte-carlo,'' (2021), arXiv:2101.02240 [quant-ph].

[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].

[32] P. Selinger, ``Quantum circuits of t-depth one,'' Physical Review A 87 (2013), 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].

[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.

[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).

[36] Y. Takahashi, S. Tani, and N. Kunihiro, ``Quantum addition circuits and unbounded fan-out,'' (2009), arXiv:0910.2530 [quant-ph].

[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.

[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].

[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.

[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.

[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).

[42] S. Wiesner, ``Simulations of many-body quantum systems by a quantum computer,'' (1996), arXiv:quant-ph/​9603028 [quant-ph].

[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).

[44] J. Stokes, J. Izaac, N. Killoran, and G. Carleo, ``Quantum natural gradient,'' Quantum 4, 269 (2020).

[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.

Cited by

[1] Ethan Wang, "Efficient Cauchy distribution based quantum state preparation by using the comparison algorithm", AIP Advances 11 10, 105307 (2021).

[2] 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).

[3] 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).

[4] 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).

[5] Steven Herbert, "Quantum Monte Carlo Integration: The Full Advantage in Minimal Circuit Depth", Quantum 6, 823 (2022).

[6] 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).

[7] Roman Rietsche, Christian Dremel, Samuel Bosch, Léa Steinacker, Miriam Meckel, and Jan-Marco Leimeister, "Quantum computing", Electronic Markets (2022).

[8] 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).

[9] Carsten Blank and Francesco Petruccione, "1. Quantum Applications - Fachbeitrag: Vielversprechend: Monte-Carlo-ähnliche Methoden auf dem Quantencomputer ", Digitale Welt 5 4, 40 (2021).

[10] Alexandru Paler, Oumarou Oumarou, and Robert Basmadjian, "On the Realistic Worst-Case Analysis of Quantum Arithmetic Circuits", arXiv:2101.04764, IEEE Transactions on Quantum Engineering 3, 1 (2022).

[11] 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).

[12] Lester Ingber and A. Kwilinski, "Hybrid classical-quantum computing: Applications to statistical mechanics of financial markets", E3S Web of Conferences 307, 04001 (2021).

[13] 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).

[14] Ethan T. Wang, "Efficient Quantum State Preparation for the Cauchy Distribution Based on Piecewise Arithmetic", IEEE Transactions on Quantum Engineering 3, 1 (2022).

[15] 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).

[16] Koichi Miyamoto, "Quantum algorithms for numerical differentiation of expected values with respect to parameters", Quantum Information Processing 21 3, 109 (2022).

[17] 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).

[18] 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).

[19] 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).

[20] Mark-Oliver Wolf, Roman Horsky, and Jonas Koppe, "A Quantum Algorithm for Pricing Asian Options on Valuation Trees", Risks 10 12, 221 (2022).

[21] Guglielmo Mazzola, "Sampling, rates, and reaction currents through reverse stochastic quantization on quantum computers", Physical Review A 104 2, 022431 (2021).

[22] 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).

[23] Koichi Miyamoto, "Bermudan option pricing by quantum amplitude estimation and Chebyshev interpolation", EPJ Quantum Technology 9 1, 3 (2022).

[24] Javier Gonzalez-Conde, Ángel Rodríguez-Rozas, Enrique Solano, and Mikel Sanz, "Simulating option price dynamics with exponential quantum speedup", arXiv:2101.04023.

[25] Prasanth Shyamsundar, "Non-Boolean Quantum Amplitude Amplification and Quantum Mean Estimation", arXiv:2102.04975.

[26] Andriy Miranskyy, Lei Zhang, and Javad Doliskani, "On Testing and Debugging Quantum Software", arXiv:2103.09172.

[27] Steven Herbert, "No quantum speedup with Grover-Rudolph state preparation for quantum Monte Carlo integration", Physical Review E 103 6, 063302 (2021).

[28] M. C. Braun, T. Decker, N. Hegemann, and S. F. Kerstan, "Error Resilient Quantum Amplitude Estimation from Parallel Quantum Phase Estimation", arXiv:2204.01337.

[29] Lei Zhang, Andriy Miranskyy, Walid Rjaibi, Greg Stager, Michael Gray, and John Peck, "Making Existing Software Quantum Safe: Lessons Learned", arXiv:2110.08661.

The above citations are from Crossref's cited-by service (last updated successfully 2022-11-30 08:51:53) and SAO/NASA ADS (last updated successfully 2022-11-30 08:51:54). The list may be incomplete as not all publishers provide suitable and complete citation data.

1 thought on “A Threshold for Quantum Advantage in Derivative Pricing

  1. Pingback: Goldman Sachs and AWS examine efficient ways to load data into quantum computers | AWS Quantum Technologies Blog