  In this paper, the Quantum Approximate Optimization Algorithm (QAOA) is
analyzed by leveraging symmetries inherent in problem Hamiltonians. We focus on
the generalized formulation of optimization problems defined on the sets of
$n$-element $d$-ary strings. Our main contribution encompasses dimension
reductions for the originally proposed QAOA. These reductions retain the same
problem Hamiltonian as the original QAOA but differ in terms of their mixer
Hamiltonian, and initial state. The vast QAOA space has a daunting dimension of
exponential scaling in $n$, where certain reduced QAOA spaces exhibit
dimensions governed by polynomial functions. This phenomenon is illustrated in
this paper, by providing partitions corresponding to polynomial dimensions of
the corresponding subspaces. As a result, each reduced QAOA partition
encapsulates unique classical solutions absent in others, allowing us to
establish a lower bound on the number of solutions to the initial optimization
problem. Our novel approach opens promising practical advantages in
accelerating the algorithm. Restricting the algorithm to Hilbert spaces of
smaller dimension may lead to significant acceleration of both quantum and
classical simulation of circuits and serve as a tool to cope with barren
plateaus problem.
