  This study investigates the frame potential and expressiveness of commutative
quantum circuits. Based on the Fourier series representation of these circuits,
we express quantum expectation and pairwise fidelity as characteristic
functions of random variables, and expressiveness as the recurrence probability
of a random walk on a lattice. A central outcome of our work includes formulas
to approximate the frame potential and expressiveness for any commutative
quantum circuit, underpinned by convergence theorems in probability theory. We
identify the lattice volume of the random walk as means to approximate
expressiveness based on circuit architecture. In the specific case of
commutative circuits involving Pauli-$Z$ rotations, we provide theoretical
results relating expressiveness and circuit structure. Our probabilistic
representation also provide means for bounding and approximately calculating
the frame potential of a circuit through sampling methods.
