Theory for Equivariant Quantum Neural Networks

Author:

Nguyen Quynh T.12,Schatzki Louis34,Braccia Paolo15ORCID,Ragone Michael16,Coles Patrick J.1,Sauvage Frédéric1,Larocca Martín17,Cerezo M.3

Affiliation:

1. Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

2. School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA

3. Information Sciences, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

4. Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA

5. Dipartimento di Fisica e Astronomia, Università di Firenze, Sesto Fiorentino, Florence 50019, Italy

6. Department of Mathematics, University of California Davis, Davis, California 95616, USA

7. Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

Abstract

Quantum neural network architectures that have little to no inductive biases are known to face trainability and generalization issues. Inspired by a similar problem, recent breakthroughs in machine learning address this challenge by creating models encoding the symmetries of the learning task. This is materialized through the usage of equivariant neural networks the action of which commutes with that of the symmetry. In this work, we import these ideas to the quantum realm by presenting a comprehensive theoretical framework to design equivariant quantum neural networks (EQNNs) for essentially any relevant symmetry group. We develop multiple methods to construct equivariant layers for EQNNs and analyze their advantages and drawbacks. Our methods can find unitary or general equivariant quantum channels efficiently even when the symmetry group is exponentially large or continuous. As a special implementation, we show how standard quantum convolutional neural networks (QCNNs) can be generalized to group-equivariant QCNNs where both the convolution and pooling layers are equivariant to the symmetry group. We then numerically demonstrate the effectiveness of a SU(2)-equivariant QCNN over symmetry-agnostic QCNN on a classification task of phases of matter in the bond-alternating Heisenberg model. Our framework can be readily applied to virtually all areas of quantum machine learning. Lastly, we discuss about how symmetry-informed models such as EQNNs provide hopes to alleviate central challenges such as barren plateaus, poor local minima, and sample complexity. Published by the American Physical Society 2024

Funder

U.S. Department of Energy

Los Alamos National Laboratory

NSF

Office of Science

Office of Advanced Scientific Computing Research

Laboratory Directed Research and Development

National Science Foundation (NSF) Quantum Leap Challenge Institute

Publisher

American Physical Society (APS)

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