A variational quantum algorithm for the Feynman-Kac formula

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

doi:10.48550/arxiv.2108.10846

Cited by 6 publications

(16 citation statements)

References 38 publications

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“…In addition, VarQITE has been used to prepare states that do not result from unitary evolution in real time, such as the quantum Gibbs state [352]. Specifically for finance it has been used to simulate the Feynman-Kac partial differential equation [10].…”

Section: Variational Quantum Imaginary Time Evolution (Varqite)mentioning

confidence: 99%

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A Survey of Quantum Computing for Finance

Herman¹,

Googin²,

Liu³

et al. 2022

Preprint

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Quantum computers are expected to surpass the computational capabilities of classical computers during this decade and have transformative impact on numerous industry sectors, particularly finance. In fact, finance is estimated to be the first industry sector to benefit from quantum computing, not only in the medium and long terms, but even in the short term. This survey paper presents a comprehensive summary of the state of the art of quantum computing for financial applications, with particular emphasis on Monte Carlo integration, optimization, and machine learning, showing how these solutions, adapted to work on a quantum computer, can help solve more efficiently and accurately problems such as derivative pricing, risk analysis, portfolio optimization, natural language processing, and fraud detection. We also discuss the feasibility of these algorithms on near-term quantum computers with various hardware implementations and demonstrate how they relate to a wide range of use cases in finance. We hope this article will not only serve as a reference for academic researchers and industry practitioners but also inspire new ideas for future research.

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Section: Variational Quantum Imaginary Time Evolution (Varqite)mentioning

confidence: 99%

“…As mentioned, there exist polynomial speedups provided by quantum computing for common cone programs (Section 6.2). Equation (10) with additional positivity constraints on the allocation vector can be represented as an SOCP and solved with quantum IPMs [193].…”

Section: Combinatorial Formulations the First Combinatorial Formulati...mentioning

confidence: 99%

A Survey of Quantum Computing for Finance

Herman¹,

Googin²,

Liu³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…This can correspond to a Fokker-Planck equation (FPE) or a Kolmogorov backward equation (KBE) [62], written for the time-dependent probability distribution function p(x, t) of the stochastic variable X t . More generally, the evolution can be described by the Feynman-Kac formula [37]. Importantly, once we learn the p(x, t) in the domain of interest t ∈ T , inprinciple we can obtain stochastic trajectories (samples from time-incremented distributions), offering full generative modelling of time-series.…”

Section: Model Differentiation and Constrained Training From Stochast...mentioning

confidence: 99%

“…SDE-based sampling is also motivated by works in the financial sector where Monte-Carlo techniques are used. To date, various quantum protocols for associated PDEs has been considered, in many cases taking the perspective of real and imaginary time evolution [34][35][36][37] or using amplitude amplification for tasks like option pricing [38][39][40][41][42]. More broadly, the area of differential equations with quantum computers has been developing rapidly, starting from fault-tolerant QC oriented [43][44][45][46] to near-term and quantum-inspired protocols [29,30,[47][48][49][50][51].…”

Section: Introductionmentioning

confidence: 99%

Protocols for Trainable and Differentiable Quantum Generative Modelling

Kyriienko¹,

Paine²,

Elfving³

2022

Preprint

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We propose an approach for learning probability distributions as differentiable quantum circuits (DQC) that enable efficient quantum generative modelling (QGM) and synthetic data generation. Contrary to existing QGM approaches, we perform training of a DQCbased model, where data is encoded in a latent space with a phase feature map, followed by a variational quantum circuit. We then map the trained model to the bit basis using a fixed unitary transformation, coinciding with a quantum Fourier transform circuit in the simplest case. This allows fast sampling from parametrized distributions using a single-shot readout. Importantly, latent space training provides models that are automatically differentiable, and we show how samples from solutions of stochastic differential equations (SDEs) can be accessed by solving stationary and time-dependent Fokker-Planck equations with a quantum protocol. Finally, our approach opens a route to multidimensional generative modelling with qubit registers explicitly correlated via a (fixed) entangling layer. In this case quantum computers can offer advantage as efficient samplers, which perform complex inverse transform sampling enabled by the fundamental laws of quantum mechanics. On a technical side the advances are multiple, as we introduce the phase feature map, analyze its properties, and develop frequency-taming techniques that include qubit-wise training and feature map sparsification.

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“…Since large financial institutions perform enormous computational tasks in their daily business 2 , it is naturally expected that quantum computers will tremendously speedup them and make a large impact on the industry. In fact, some recent papers have already discussed applications of quantum algorithms to concrete problems in financial engineering: for example, derivative pricing [4][5][6][7][8][9][10][11][12][13][14][15][16], risk measurement [17][18][19], portfolio optimization [20][21][22], and so on. See [23][24][25] as comprehensive reviews.…”

Section: Introductionmentioning

confidence: 99%

Pricing multi-asset derivatives by finite difference method on a quantum computer

Miyamoto¹,

Kubo²

2021

Preprint

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Following the recent great advance of quantum computing technology, there are growing interests in its applications to industries, including finance. In this paper, we focus on derivative pricing based on solving the Black-Scholes partial differential equation by finite difference method (FDM), which is a suitable approach for some types of derivatives but suffers from the curse of dimensionality, that is, exponential growth of complexity in the case of multiple underlying assets. We propose a quantum algorithm for FDM-based pricing of multi-asset derivative with exponential speedup with respect to dimensionality compared with classical algorithms. The proposed algorithm utilizes the quantum algorithm for solving differential equations, which is based on quantum linear system algorithms. Addressing the specific issue in derivative pricing, that is, extracting the derivative price for the present underlying asset prices from the output state of the quantum algorithm, we present the whole of the calculation process and estimate its complexity. We believe that the proposed method opens the new possibility of accurate and high-speed derivative pricing by quantum computers.

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A variational quantum algorithm for the Feynman-Kac formula

Cited by 6 publications

References 38 publications

A Survey of Quantum Computing for Finance

A Survey of Quantum Computing for Finance

Protocols for Trainable and Differentiable Quantum Generative Modelling

Pricing multi-asset derivatives by finite difference method on a quantum computer

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