Quantum Algorithms and Complexity for Continuous Problems

TL;DR

This paper explores the potential advantages of quantum algorithms over classical methods for solving continuous problems in science and engineering, analyzing their complexity and speedup in high-dimensional and differential equation contexts.

Contribution

It provides an overview of quantum complexity results for various continuous problems and discusses the potential quantum speedups compared to classical algorithms.

Findings

Quantum algorithms can offer speedups for high-dimensional integration.

Quantum complexity bounds are established for differential equations.

Potential quantum advantages in simulating quantum systems are discussed.

Abstract

Most continuous mathematical formulations arising in science and engineering can only be solved numerically and therefore approximately. We shall always assume that we're dealing with a numerical approximation to the solution. There are two major motivations for studying quantum algorithms and complexity for continuous problems. 1. Are quantum computers more powerful than classical computers for important scientific problems? How much more powerful? 2. Many important scientific and engineering problems have continuous formulations. To answer the first question we must know the classical computational complexity of the problem. Knowing the classical complexity of a continuous problem we obtain the quantum computation speedup if we know the quantum complexity. If we know an upper bound on the quantum complexity through the cost of a particular quantum algorithm then we can obtain a lower bound on the quantum speedup. Regarding the second motivation, in this article we'll report on high-dimensional integration, path integration, Feynman path integration, the smallest eigenvalue of a differential equation, approximation, partial differential equations, ordinary differential equations and gradient estimation. We'll also briefly report on the simulation of quantum systems on a quantum computer.