Continuous Quantum Computation Two major motivations for studying continuous quantum computation are:
- Many scientific problems have continuous mathematical formulations. Exaples of such formulations are
- 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 computationsl complexities
In the section An Example: Path Integration solving a continuous problem on a quantum computer will be discussed. Here the second motivation will be amplified. 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 qubits and queries. Classical complexity has been extensively studied in informations-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 integrer factorization is unknown.