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AIP conference proceedings

Efficient quasi-Monte Carlo sampling for quantum random walks

Emanouil I. Atanassov, Mariya K. Durchova

Published Jan 1, 2020• 2 citations
Publisher

Abstract

Quantum random walks are the natural counterpart of the classical random walk, arising when quantum effects are taken into consideration. Many Monte Carlo methods use the classical Brownian motion as a building block. Although it appears naturally in various problems of computational physics, it is also used extensively in the stochastic models used for pricing of financial assets. When quantum random walks are used instead, more complex behaviors can be observed. However, the simulation of a quantum random walk requires more computational resources. By using low-discrepancy sequences instead of pseudorandom numbers, one may hope to decrease the statistical error of the simulation. Another approach for improved efficiency is to use the power of computational accelerators, which are suitable for problems, where a well-ordered sequence of computations is to be performed repeatedly. In our work we concentrated on using GPGPU computing in order to speed-up the operations involved in the generation of the low-discrepancy sequence as well as the subsequent sampling of the quantum random walks. In this work we explain our sampling algorithm and demonstrate its efficiency on model problems from the domain of option pricing.

DOI
10.1063/5.0034824

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Quantum random walks are the natural counterpart of the classical random walk, arising when quantum effects are taken into consideration. Many Monte Carlo methods use the classical Brownian motion as a building block. Although it appears naturally in various problems of computational physics, it is also used extensively in the stochastic models used for pricing of financial assets. When quantum random walks are used instead, more complex behaviors can be observed. However, the simulation of a quantum random walk requires more computational resources. By using low-discrepancy sequences instead of pseudorandom numbers, one may hope to decrease the statistical error of the simulation. Another approach for improved efficiency is to use the power of computational accelerators, which are suitable for problems, where a well-ordered sequence of computations is to be performed repeatedly. In our work we concentrated on using GPGPU computing in order to speed-up the operations involved in the generation of the low-discrepancy sequence as well as the subsequent sampling of the quantum random walks. In this work we explain our sampling algorithm and demonstrate its efficiency on model problems from the domain of option pricing.
Numerical integration with Monte Carlo methods is a central computational task in many scientific and industrial applications, including financial derivative pricing and risk management. Classical Monte Carlo algorithms are computationally demanding: achieving an accuracy εεε typically requires a number of function evaluations scaling as O(1/ε2)O(1/ε^2)O(1/ε2). Quantum-accelerated Monte Carlo methods based on quantum amplitude estimation can in principle quadratically improve this dependence. However, \textit{quasi}-Monte Carlo methods have not been explored in the quantum context. In this work, we introduce a quantum quasi-Monte Carlo algorithm that combines low-discrepancy nets with quantum amplitude estimation. The proposed method prepares the quasi-random point set coherently in superposition. The method does not yield an asymptotic improvement over classical quasi-Monte Carlo, since the total error separates into a discretization error, determined by the finite net, and a quantum estimation error. Instead, we explore a pre-asymptotic advantage window: for a target accuracy that would classically require 2q2^q2q low discrepancy points, one can prepare a higher-resolution net of size 2Q2^Q2Q, with Q>qQ>qQ>q, in superposition and reach the same accuracy using significantly fewer function queries. This window can be controlled by tuning the circuit resolution and amplitude-estimation parameters, making the approach relevant for practical regimes where the number of queries is finite rather than asymptotically large.
As one of the major methods for the simulation of option pricing, Monte Carlo method assumes random fluctuations in the distribution of asset prices. Under certain uncertainties process, different evolution paths could be simulated so as to finally yield the expectation value of the asset price, which requires a lot of simulations to ensure the accuracy based on huge and expensive calculations. In order to solve the above computational problem, quantum Monte Carlo (QMC) has been established and applied in the relevant systems such as European call options. In this work, both MC and QM methods are adopted to simulate European call options. Based on the preparation of quantum states in QMC algorithm and the construction of quantum circuits by simulating a quantum hardware environment on a traditional computer, the amplitude estimation (AE) algorithm is found to play a secondary role in accelerating the pricing of European options. More importantly, the payoff function and the time required for the simulation in QMC method show some improvements than those in MC method.
Abstract Quantum computers have the potential to provide quadratic speedup for Monte Carlo methods currently used in various classical applications. In this work, we examine the advantage of quantum computers for financial option pricing with the Monte Carlo method. Systematic and statistical errors are handled in a joint framework, and a relationship to quantum gate error is established. New metrics are introduced for the assessment of quantum advantage based on sample count and optimized error handling. We implement and analyze a Fourier series based approach and demonstrate its benefit over the more traditional rescaling method in function approximation. Our numerical calculations reveal the unpredictable nature of systematic errors, making consistent quantum advantage difficult with current quantum hardware. Our results indicate that very low noise levels, a two-qubit gate error rate below 10−6, are necessary for the quantum method to outperform the classical one, but a low number of logical qubits (ca. 20) may be sufficient to see quantum advantage already.
Inspired by recent progress in quantum algorithms for ordinary and partial differential equations, we study quantum algorithms for stochastic differential equations (SDEs). Firstly we provide a quantum algorithm that gives a quadratic speed-up for multilevel Monte Carlo methods in a general setting. As applications, we apply it to compute expectation values determined by classical solutions of SDEs, with improved dependence on precision. We demonstrate the use of this algorithm in a variety of applications arising in mathematical finance, such as the Black-Scholes and Local Volatility models, and Greeks. We also provide a quantum algorithm based on sublinear binomial sampling for the binomial option pricing model with the same improvement.
Abstract Financial derivative pricing is a significant challenge in finance, involving the valuation of instruments like options based on underlying assets. While some cases have simple solutions, many require complex classical computational methods like Monte Carlo simulations and numerical techniques. However, as derivative complexities increase, these methods face limitations in computational power. Cases involving Non-Vanilla Basket pricing, American Options, and derivative portfolio risk analysis need extensive computations in higher-dimensional spaces, posing challenges for classical computers. Quantum computing presents a promising avenue by harnessing quantum superposition and entanglement, allowing the handling of high-dimensional spaces effectively. In this paper, we introduce a self-contained and all-encompassing quantum algorithm that operates without reliance on oracles or presumptions. More specifically, we develop an effective stochastic method for simulating exponentially many potential asset paths in quantum parallel, leading to a highly accurate final distribution of stock prices. Furthermore, we demonstrate how this algorithm can be extended to price more complex options and analyze risk within derivative portfolios.
Monte Carlo methods use random sampling to estimate numerical quantities which are hard to compute deterministically. One important example is the use in statistical physics of rapidly mixing Markov chains to approximately compute partition functions. In this work, we describe a quantum algorithm which can accelerate Monte Carlo methods in a very general setting. The algorithm estimates the expected output value of an arbitrary randomized or quantum subroutine with bounded variance, achieving a near-quadratic speedup over the best possible classical algorithm. Combining the algorithm with the use of quantum walks gives a quantum speedup of the fastest known classical algorithms with rigorous performance bounds for computing partition functions, which use multiple-stage Markov chain Monte Carlo techniques. The quantum algorithm can also be used to estimate the total variation distance between probability distributions efficiently.