WeOne discussmajor motivation for studying the quantum computation of continuous functions is that many scientific problems have mathematical formulations in terms of continuous quantities. Examples include evaluating [[Path integral formulation|path integrals]], solving [[differential equation]]s using the [[Feynman–Kac formula]], and [[numerical optimization]] of continuous functions. A second motivation is to explore and understand the ways in which quantum computers can be more capable or powerful than classical ones. By computational complexity (complexity for brevity) is meant the '''minimal''' computational resources needed to solve a problem. Two of the most important resources for quantum computing are [[qubit]]s and queries. Classical complexity has been extensively studied in [[information-based complexity]]. The classical complexity of many continuous problems is known. Therefore, when the quantum complexity of these problems is obtained, the question as to whether quantum computers are more powerful than classical can be answered. Furthermore, it can be established how much more powerful. In contrast, the complexity of discrete problems is typically unknown; one has to settle for the complexity hierarchy. For example, the classical complexity of integer factorization is unknown.▼
One major motivation for studying the quantum computation of continuous functions is that many scientific problems have mathematical formulations in terms of continuous quantities. Examples of such formulations are
** [[Path Integral Formulation|Path integration]]
** [[Feynman–Kac]] path integration
** [[Schrödinger equation]]
* In their standard monograph Nielsen and Chuang state "Of particular interest is a decisive answer to the problem whether quantum computers are more powerful than classical computers." To answer this question one must know the classical and quantum computational complexities
▲We discuss the second motivation. By computational complexity (complexity for brevity) is meant the '''minimal''' computational resources needed to solve a problem. Two of the most important resources for quantum computing are [[qubit]]s and queries. Classical complexity has been extensively studied in [[information-based complexity]]. The classical complexity of many continuous problems is known. Therefore, when the quantum complexity of these problems is obtained, the question as to whether quantum computers are more powerful than classical can be answered. Furthermore, it can be established how much more powerful. In contrast, the complexity of discrete problems is typically unknown; one has to settle for the complexity hierarchy. For example, the classical complexity of integer factorization is unknown.
