In class, someone asked about languages known to be in BQP that are not known to be in NP. There is at least one such language, known as the Recursive Fourier Sampling problem. It is described in this paper: *Quantum complexity theory* by Ethan Bernstein and Umesh Vazirani, ACM Symposium on Theory of Computing, 1993. [Full version: PDF] Section 8.4 of this paper describes the problem, but it is definitely not light reading! Before tackling this paper, I would recommend reading Scott Aaronson’s *Scientific American* article: *The Limits of Quantum Computers*.