I would like to study circuits with only one possible outcome. Quantum phase estimation, Bernstein-Vazirani, and in part Deutsch-Jozsa (for constant functions) come to mind - do you know any other important algorithms with only one possible outcome? Or, even better, do you know any criteria to build a circuit in such a way such that it can have only one possible outcome?
I think the algorithm class you are referring to is EQP (Exact Quantum Polynomial-Time).
There is a famous breakthrough paper by Ambainis that shows for a superlinear advantage between exact quantum algorithms over their classical counterparts (classical deterministic algorithms) for the query complexity of a total Boolean function. Later Ambainis, Iraids and Smotrovs in another paper developed the optimal exact quantum algorithms to determine whether an $n$-bit string has Hamming weight exactly $k$, and to determine whether an $n$-bit string has Hamming weight at least $k$.