Efficient state preparation for multivariate Monte Carlo simulation

TL;DR

This paper introduces a quantum algorithm for multivariate state preparation that significantly reduces the gate complexity from exponential to linear in the number of variables, enabling more efficient quantum Monte Carlo simulations.

Contribution

The paper presents a novel quantum algorithm utilizing multivariable quantum signal processing to efficiently prepare multivariate states with linear gate complexity.

Findings

Reduces gate complexity from exponential to linear in variables D

Uses multivariable quantum signal processing for polynomial transformations

Enables efficient quantum Monte Carlo simulations in finance

Abstract

Quantum state preparation is a task to prepare a state with a specific function encoded in the amplitude, which is an essential subroutine in many quantum algorithms. In this paper, we focus on multivariate state preparation, as it is an important extension for many application areas. Specifically in finance, multivariate state preparation is required for multivariate Monte Carlo simulation, which is used for important numerical tasks such as risk aggregation and multi-asset derivative pricing. Using existing methods, multivariate quantum state preparation requires the number of gates exponential in the number of variables $D$. For this task, we propose a quantum algorithm that only requires the number of gates linear in $D$. Our algorithm utilizes multivariable quantum signal processing (M-QSP), a technique to perform the multivariate polynomial transformation of matrix elements. Using easily prepared block-encodings corresponding to each variable, we apply the M-QSP to construct the target function. In this way, our algorithm prepares the target state efficiently for functions achievable with M-QSP.