A variational quantum algorithm for the Feynman-Kac formula

Alghassi, Hedayat; Deshmukh, Amol; Ibrahim, Noelle; Robles, Nicolas; Woerner, Stefan; Zoufal, Christa

doi:10.22331/q-2022-06-07-730

Cited by 18 publications

(31 citation statements)

References 61 publications

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“…Such decomposition can be obtained in a similar way to Ref. [22,41] and is shown in Appendix B. F can be represented as a sum of O(d 2 n 4 ) unitaries each of which requires at most O(n 2 ) gates to be implemented. G for typical boundary conditions discussed in Sec.…”

Section: Proposed Methodsmentioning

confidence: 99%

“…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%

“…Our algorithm has the following three parts; embedding the probability distribution of the underlying asset prices into the quantum state, solving the BSPDE with boundary conditions, and calculating the inner product. For the first part, we can use the quantum generative algorithms [33][34][35][36][37] or variational quantum simulation (VQS) for the Fokker-Planck equations [22,[38][39][40][41], which describe the time evolution of the probability density functions of the stochastic processes. For the second part, we discretize the BSPDE using the FDM and solve it using VQS.…”

Section: Introductionmentioning

confidence: 99%

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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: Proposed Methodsmentioning

confidence: 99%

Section: B Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Pricing multi-asset derivatives by variational quantum algorithms

Kubo¹,

Miyamoto²,

Mitarai³

et al. 2022

Preprint

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A novel method for response probability density of nonlinear stochastic dynamic systems

Wang,

Jiang,

Hong

et al. 2024

Nonlinear Dyn

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This paper presents a novel method for analyzing high-dimensional nonlinear stochastic dynamic systems. In particular, we attempt to obtain the solution of the Fokker–Planck–Kolmogorov (FPK) equation governing the response probability density of the system without using the FPK equation directly. The method consists of several important components including the radial basis function neural networks (RBFNN), Feynman–Kac formula and the short-time Gaussian property of the response process. In the area of solving partial differential equations (PDEs) using neural networks, known as physics-informed neural network (PINN), the proposed method presents an effective alternative for obtaining solutions of PDEs without directly dealing with the equation, thus avoids evaluating the derivatives of the equation. This approach has a potential to make the neural network-based solution more efficient and accurate. Several highly challenging examples of nonlinear stochastic systems are presented in the paper to illustrate the effectiveness of the proposed method in comparison to the equation-based RBFNN approach.

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Extracting a function encoded in amplitudes of a quantum state by tensor network and orthogonal function expansion

Miyamoto

Ueda

2023

Quantum Inf Process

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There are quantum algorithms for finding a function f satisfying a set of conditions, such as solving partial differential equations, and these achieve exponential quantum speedup compared to existing classical methods, especially when the number d of the variables of f is large. In general, however, these algorithms output the quantum state which encodes f in the amplitudes, and reading out the values of f as classical data from such a state can be so time-consuming that the quantum speedup is ruined. In this study, we propose a general method for this function readout task. Based on the function approximation by a combination of tensor network and orthogonal function expansion, we present a quantum circuit and its optimization procedure to obtain an approximating function of f that has a polynomial number of degrees of freedom with respect to d and is efficiently evaluable on a classical computer. We also conducted a numerical experiment to approximate a finance-motivated function to demonstrate that our method works.

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A variational quantum algorithm for the Feynman-Kac formula

Cited by 18 publications

References 61 publications

Pricing multi-asset derivatives by variational quantum algorithms

Pricing multi-asset derivatives by variational quantum algorithms

A novel method for response probability density of nonlinear stochastic dynamic systems

Extracting a function encoded in amplitudes of a quantum state by tensor network and orthogonal function expansion

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