Two of the motivations for studying continuous problems are:
1. Many scientific problems have continuous formulations. Examples are path integration, Feynman-Kac path integration, and the Schrodinger equation.
2. The classical complexity of many continuous problems is known. Therefore when we obtain the quantum complexity of these problems we know if quantum computers are more powerful than classical computers and also how much faster.
Why can we obtain the classical and quantum complexity of continuous problems but have to settle for the complexity hierarchy for combinatorial problems such as 3-SAT? We'll discuss the key difference between continuous and combinatorial problems.
There is much good news to report on algorithms and quantum speedups for continuous problems. Successes include path integration, Feynman-Kac path integration, high dimensional integration, high dimensional approximation, smallest eigenvalue of a Hermitean matrix, and the Sturm-Liouville eigenvalue problem (in progress). Typical results are the following: On a quantum computer the qubit complexity of path integration is of order 1/E where E is the error threshold. The query complexity is of order 1/E for a number of problems. For these problems there is exponential improvement of the query complexity over the classical worst case and polynomial improvement over the classical randomized (Monte Carlo) case.
Current and future research includes the following: Can we improve 1/E to log(1/E) qubits for path integration? Can we improve 1/E to log(1/E) queries for important problems? Study faulty tolerant computing for continuous problems.