  In our previous work, we defined a quantum algorithmic technique known as the
Generalised Phase Kick-Back, or $GPK$, and analysed its applications in
generalising some classical quantum problems, such as the Deutsch-Jozsa problem
or the Bernstein-Vazirani problem. We also proved that using this technique we
can solve Simon's problem in a more efficient manner. In this paper we continue
analysing the potential of this technique, defining the concept of
$\mathbf{y}$-balanced functions and solving a new problem, which further
generalises the generalised Deutsch-Jozsa problem (the fully balanced image
problem). This problem also underlines the relation between quantum computation
and Boolean function theory, and, in particular, the Walsh and Fourier-Hadamard
transforms. We finish our discussion by solving the generalised version of
Simon's problem using the $GPK$ algorithm, while analysing the efficiency of
this new solution.
