Expressiveness of Commutative Quantum Circuits: A Probabilistic Approach

Jorge M. Ramirez; Elaine Wong; Caio Alves; Sarah Chehade; Ryan Bennink

doi:10.1109/TQE.2024.3488518

Expressiveness of Commutative Quantum Circuits: A Probabilistic Approach

Jorge M. Ramirez, Elaine Wong, Caio Alves, Sarah Chehade, Ryan Bennink

Research output: Contribution to journal › Article › peer-review

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 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.

Original language	English
Article number	3104015
Journal	IEEE Transactions on Quantum Engineering
Volume	5
DOIs	https://doi.org/10.1109/TQE.2024.3488518
State	Published - 2024

Funding

This work was supported in part by the Department of Energy Advanced Scientific Computing Research Accelerated Research in Quantum Computing Program under field work proposal 3ERKJ354 and in part by UT-Battelle, LLC under Contract DE-AC05-00OR22725 with the U.S. Department of Energy. This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).

Keywords

Commutative quantum circuit
expressiveness
frame potential

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10.1109/TQE.2024.3488518

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title = "Expressiveness of Commutative Quantum Circuits: A Probabilistic Approach",

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 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.",

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PY - 2024

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AB - 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.

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JO - IEEE Transactions on Quantum Engineering

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Expressiveness of Commutative Quantum Circuits: A Probabilistic Approach

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