Short Communication: A Quantum Algorithm for Linear PDEs Arising in Finance

Fontanela, Filipe; Jacquier, Antoine; Oumgari, Mugad

doi:10.1137/21m1397878

Cited by 16 publications

(18 citation statements)

References 21 publications

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“…Refs. [13,14,22] solve the discretized Schrödinger equation by VQS, which is a variational quantum algorithm for solving ODEs. In the previous studies mentioned above, the time complexity required to solve the BSPDE depends on the grid points only logarithmically.…”

Section: B Related Workmentioning

confidence: 99%

“…In recent years, applications of quantum computers have been discussed in financial engineering. Specifically, the applications include portfolio optimization [1-3], risk measurement [4][5][6][7][8], and derivative pricing [9][10][11][12][13][14][15][16][17][18][19][20][21][22]. Comprehensive reviews of these topics are presented in Refs.…”

Section: Introductionmentioning

confidence: 99%

“…To overcome this difficulty, several methods [13][14][15]22] have been proposed to efficiently solve the BSPDE using quantum computers. However, when solving the discretized BSPDE with these quantum algorithms, the target derivative price is embedded in the amplitude of one basis of the resulting quantum state, so it requires exponentially large computational complexity to extract it as classical information.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Pricing multi-asset derivatives by variational quantum algorithms

Kubo¹,

Miyamoto²,

Mitarai³

et al. 2022

Preprint

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Pricing a multi-asset derivative is an important problem in financial engineering, both theoretically and practically. Although it is suitable to numerically solve partial differential equations to calculate the prices of certain types of derivatives, the computational complexity increases exponentially as the number of underlying assets increases in some classical methods, such as the finite difference method. Therefore, there are efforts to reduce the computational complexity by using quantum computation. However, when solving with naive quantum algorithms, the target derivative price is embedded in the amplitude of one basis of the quantum state, and so an exponential complexity is required to obtain the solution. To avoid the bottleneck, the previous study [Miyamoto and Kubo, IEEE Transactions on Quantum Engineering, 3, 1-25 ( 2022)] utilizes the fact that the present price of a derivative can be obtained by its discounted expected value at any future point in time and shows that the quantum algorithm can reduce the complexity. In this paper, to make the algorithm feasible to run on a small quantum computer, we use variational quantum simulation to solve the Black-Scholes equation and compute the derivative price from the inner product between the solution and a probability distribution. This avoids the measurement bottleneck of the naive approach and would provide quantum speedup even in noisy quantum computers. We also conduct numerical experiments to validate our method. Our method will be an important breakthrough in derivative pricing using small-scale quantum computers.

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Section: B Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Pricing multi-asset derivatives by variational quantum algorithms

Kubo¹,

Miyamoto²,

Mitarai³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Here, we are interested in variational quantum algorithms for solving differential equations 14 , such as the Black–Scholes equation 15 , 16 , the Poisson equation 17 , 18 , and the Helmholtz equation 19 . Specifically, the Poisson equation can be solved efficiently through explicit decomposition of the coefficient matrix derived from finite difference discretization 17 using minimal cost function evaluations 18 and shallower circuit depth compared to other non-variational quantum algorithms 14 , 20 – 22 .…”

Section: Introductionmentioning

confidence: 99%

“…partial differential equations including a time domain. McArdle et al 23 proposed a variational quantum algorithm which simulates the real (imaginary) time evolution of parametrized trial states via forward Euler time-stepping of the Wick rotated Schrödinger equation, thereby solving the Black–Scholes equation, and by extension, the heat equation 15 , 16 . Besides issues of Ansatz selection and quantum complexity, time-stepping based on an explicit Euler method may be unstable, a limiting condition exacerbated by noise.…”

Section: Introductionmentioning

confidence: 99%

Variational quantum evolution equation solver

Leong

Ewe

Koh

2022

Sci Rep

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Variational quantum algorithms offer a promising new paradigm for solving partial differential equations on near-term quantum computers. Here, we propose a variational quantum algorithm for solving a general evolution equation through implicit time-stepping of the Laplacian operator. The use of encoded source states informed by preceding solution vectors results in faster convergence compared to random re-initialization. Through statevector simulations of the heat equation, we demonstrate how the time complexity of our algorithm scales with the Ansatz volume for gradient estimation and how the time-to-solution scales with the diffusion parameter. Our proposed algorithm extends economically to higher-order time-stepping schemes, such as the Crank–Nicolson method. We present a semi-implicit scheme for solving systems of evolution equations with non-linear terms, such as the reaction–diffusion and the incompressible Navier–Stokes equations, and demonstrate its validity by proof-of-concept results.

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Quantum computing for financial risk measurement

Wilkens¹,

Moorhouse

2023

Quantum Inf Process

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Quantum computing allows a significant speed-up over traditional CPU- and GPU-based algorithms when applied to particular mathematical challenges such as optimisation and simulation. Despite promising advances and extensive research in hard- and software developments, currently available quantum systems are still largely limited in their capability. In line with this, practical applications in quantitative finance are still in their infancy. This paper analyses requirements and concrete approaches for the application to risk management in a financial institution. On the examples of Value-at-Risk for market risk and Potential Future Exposure for counterparty credit risk, the main contribution lies in going beyond textbook illustrations and instead exploring must-have model features and their quantum implementations. While conceptual solutions and small-scale circuits are feasible at this stage, the leap needed for real-life applications is still significant. In order to build a usable risk measurement system, the hardware capacity—measured in number of qubits—would need to increase by several magnitudes from their current value of about $$10^2$$ 10 2 . Quantum noise poses an additional challenge, and research into its control and mitigation would need to advance in order to render risk measurement applications deployable in practice. Overall, given the maturity of established classical simulation-based approaches that allow risk computations in reasonable time and with sufficient accuracy, the business case for a move to quantum solutions is not very strong at this point.

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Short Communication: A Quantum Algorithm for Linear PDEs Arising in Finance

Cited by 16 publications

References 21 publications

Pricing multi-asset derivatives by variational quantum algorithms

Pricing multi-asset derivatives by variational quantum algorithms

Variational quantum evolution equation solver

Quantum computing for financial risk measurement

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