A Quantum algorithm for linear PDEs arising in Finance

Fontanela, Filipe; Jacquier, Antoine; Oumgari, Mugad

doi:10.48550/arxiv.1912.02753

Cited by 10 publications

(17 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For the potential case, the left side of the figures are shown with different f (0) and a significant dip is shown in the middle of the analytical curves due to the effect of the harmonic potential centred at x = 1/2 with V max = 32 in the region of 0 ≤ x ≤ 1. For its discretised version on the right side, it shows that the fidelity is roughly 0.91 between |ψ 3 3 in the red dashed line and f (x) with f (0) = −0.19 in the green solid one (p c = 2).…”

Section: B Ml-aided Resultsmentioning

confidence: 99%

“…To overcome the disadvantage in the QP and to enhance the advantage in the CP, a hybrid (quantum-classical) algorithm might make a synergy between the performance of CP and QP. It is still, however, an open question whether this hybrid computing could fundamentally enhance computational power beyond conventional classical computation or not although several positive efforts have been recently tackled on the problems by quantum-enhanced solvers [2][3][4][5][6].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Quantum variational PDE solver with machine learning

Joo,

Moon

2021

Preprint

View full text Add to dashboard Cite

To solve nonlinear partial differential equations (PDEs) is one of the most common but important tasks in not only basic sciences but also many practical industries. We here propose a quantum variational (QuVa) PDE solver with the aid of machine learning (ML) schemes to synergise two emerging technologies in mathematically hard problems. The core quantum processing in this solver is to calculate efficiently the expectation value of specially designed quantum operators. For a large quantum system, we only obtain data from measurements of few control qubits to avoid the exponential cost in the measurements of the whole quantum system and optimise a pathway to find possible solution sets of the desired PDEs using ML techniques. As an example, a few different types of the second-order DEs are examined with randomly chosen samples and a regression method is implemented to chase the best candidates of solution functions with another trial samples. We demonstrated that a three-qubit system successfully follows the pattern of analytical solutions of three different DEs with high fidelity since the variational solutions are given by a necessary condition to obtain the exact solution of the DEs. Thus, we believe that final solution candidate sets are efficiently extracted from the QuVa PDE solver with the support of ML techniques and this algorithm could be beneficial to search for the solutions of complex mathematical problems as well as to find good ansatzs for eigenstates in large quantum systems (e.g., for quantum chemistry).

show abstract

Section: B Ml-aided Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Quantum variational PDE solver with machine learning

Joo,

Moon

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…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

View full text Add to dashboard Cite

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 ).

show abstract

“…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

View full text Add to dashboard Cite

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.

show abstract

A Quantum algorithm for linear PDEs arising in Finance

Cited by 10 publications

References 0 publications

Quantum variational PDE solver with machine learning

Quantum variational PDE solver with machine learning

Bermudan option pricing by quantum amplitude estimation and Chebyshev interpolation

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

Contact Info

Product

Resources

About