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

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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.

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.

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[1] Javier Gonzalez-Conde, Ángel Rodríguez-Rozas, Enrique Solano, and Mikel Sanz, "Pricing Financial Derivatives with Exponential Quantum Speedup", arXiv:2101.04023.

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

[3] Alexandru Paler, Oumarou Oumarou, and Robert Basmadjian, "On the realistic worst case analysis of quantum arithmetic circuits", arXiv:2101.04764.

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

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

[6] Paula García-Molina, Javier Rodríguez-Mediavilla, and Juan José García-Ripoll, "Solving partial differential equations in quantum computers", arXiv:2104.02668.

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