Is there a complexity class “BQP without error”?
Yes, it's called EQP (exact quantum polynomial) and is listed in the complexity zoo:


  The same as BQP, except that the quantum algorithm must return the correct answer with probability 1, and run in polynomial time with probability 1. Unlike bounded-error quantum computing, there is no theory of universal QTMs for exact quantum computing models. In the original definition in [BV97], each language in EQP is computed by a single QTM, equivalently to a uniform family of quantum circuits with a finite gate set K whose amplitudes can be computed in polynomial time. See EQP_K. However, some results require an infinite gate set. The official definition here is that the gate set should be finite.