Simulating option price dynamics with exponential quantum speedup

Gonzalez-Conde, Javier; Rodríguez-Rozas, Ángel; Solano, E.; Sanz, Mikel

doi:10.48550/arxiv.2101.04023

Cited by 15 publications

(24 citation statements)

References 38 publications

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“…This form of time evolution is, thus, a particularly useful tool to find the ground state of H [9]. Furthermore, imaginary time evolution can be used to solve partial differential equations [13][14][15] or to prepare quantum Gibbs states [2,10]. A generalized time evolution can be used to do matrix multiplications, solve systems of linear equations and combine real and imaginary time evolution [16].…”

Section: Variational Quantum Time Evolutionmentioning

confidence: 99%

Error Bounds for Variational Quantum Time Evolution

Zoufal¹,

Sutter²,

Woerner³

2021

Preprint

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Variational quantum time evolution (VarQTE) allows us to simulate dynamical quantum systems with parameterized quantum circuits. We derive a posteriori, global phase-agnostic error bounds for real and imaginary time evolution based on McLachlan's variational principle that can be evaluated efficiently. Rigorous error bounds are crucial in practice to adaptively choose variational circuits and to analyze the quality of optimization algorithms. The power of the new error bounds, as well as, the performance of VarQTE are demonstrated on numerical examples.

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Section: Variational Quantum Time Evolutionmentioning

confidence: 99%

Error Bounds for Variational Quantum Time Evolution

Zoufal¹,

Sutter²,

Woerner³

2021

Preprint

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“…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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“…Financial firms have a lot of heavy computational tasks in their daily business 2 , and therefore the speed-up of such tasks by quantum computers are expected to provide a large impact. For example, previous papers studied option pricing [7][8][9][10][11][12][13][14][15][16][17][18], risk measurement [19][20][21][22], portfolio optimization [23][24][25], and so on.…”

Section: Introductionmentioning

confidence: 99%

Bermudan option pricing by quantum amplitude estimation and Chebyshev interpolation

Miyamoto¹

2021

Preprint

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Pricing of financial derivatives, in particular early exercisable options such as Bermudan options, is an important but heavy numerical task in financial institutions, and its speed-up will provide a large business impact. Recently, applications of quantum computing to financial problems have been started to be investigated. In this paper, we first propose a quantum algorithm for Bermudan option pricing. This method performs the approximation of the continuation value, which is a crucial part of Bermudan option pricing, by Chebyshev interpolation, using the values at interpolation nodes estimated by quantum amplitude estimation. In this method, the number of calls to the oracle to generate underlying asset price paths scales as O(ǫ −1 ), where ǫ is the error tolerance of the option price. This means the quadratic speed-up compared with classical Monte Carlo-based methods such as least-squares Monte Carlo, in which the oracle call number is O(ǫ −2 ).

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Simulating option price dynamics with exponential quantum speedup

Cited by 15 publications

References 38 publications

Error Bounds for Variational Quantum Time Evolution

Error Bounds for Variational Quantum Time Evolution

Protocols for Trainable and Differentiable Quantum Generative Modelling

Bermudan option pricing by quantum amplitude estimation and Chebyshev interpolation

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