Expressiveness of Commutative Quantum Circuits: A Probabilistic Approach

TL;DR

This paper explores the expressiveness of commutative quantum circuits using a probabilistic framework, providing formulas and bounds for their expressiveness and frame potential based on random walk models.

Contribution

It introduces a novel probabilistic approach to quantify quantum circuit expressiveness and provides formulas for approximation and bounds, especially for circuits with Pauli-Z rotations.

Findings

Formulas to approximate frame potential and expressiveness.

Expressiveness linked to lattice volume of a random walk.

Theoretical relations between expressiveness and circuit structure.

Abstract

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.