  The quantum switch is a quantum computational primitive that provides
computational advantage by applying operations in a superposition of orders. In
particular, it can reduce the number of gate queries required for solving
promise problems where the goal is to discriminate between a set of properties
of a given set of unitary gates. In this work, we use Complex Hadamard matrices
to introduce more general promise problems, which reduce to the known Fourier
and Hadamard promise problems as limiting cases. Our generalization loosens the
restrictions on the size of the matrices, number of gates and dimension of the
quantum systems, providing more parameters to explore. In addition, it leads to
the conclusion that a continuous variable system is necessary to implement the
most general promise problem. In the finite dimensional case, the family of
matrices is restricted to the so-called Butson-Hadamard type, and the
complexity of the matrix enters as a constraint. We introduce the ``query per
gate'' parameter and use it to prove that the quantum switch provides
computational advantage for both the continuous and discrete cases. Our results
should inspire implementations of promise problems using the quantum switch
where parameters and therefore experimental setups can be chosen much more
freely.
