Expressiveness of Commutative Quantum Circuits: A Probabilistic Approach

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 we characterize 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 the 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 provides means for bounding and approximately calculating the frame potential of a circuit through sampling methods.