[Paper Review] Theory for Equivariant Quantum Neural Networks

Kate Nguyen, Louis Schatzki+6 more|arXiv (Cornell University)|Oct 16, 2022

Quantum Computing Algorithms and Architecture172 references26 citations

TL;DR

The paper develops a comprehensive theoretical framework for designing equivariant quantum neural networks (EQNNs) that respect symmetry groups, introduces methods to construct and parametrize equivariant layers (including non-unitary channels), and demonstrates SU(2)-equivariant QCNNs with improved performance on a quantum phase classification task.

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 whose action 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 (EQNN) 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 (QCNN) 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.

Motivation & Objective

• Identify the role of symmetries in quantum machine learning and motivate the use of equivariant models to improve trainability and generalization.
• Develop a general theoretical framework for EQNNs that accommodates unitary and non-unitary channels for any relevant symmetry group.
• Provide practical methods to construct and parametrize EQNN layers and analyze their trade-offs.
• Demonstrate an explicit group-equivariant QCNN (SU(2)) and benchmark it against a symmetry-agnostic counterpart on a quantum phase classification task.

Proposed method

• Interpret EQNN layers as a generalized Fourier space action via group representations and their commutants.
• Count free parameters of equivariant unitaries and CPTP channels using isotypic decompositions and Choi representations.
• Present three construction approaches: nullspace of matrix equations, twirling over the group, and Choi operator parametrization.
• Introduce intermediate representations as hyperparameters and discuss how changing representations alters accessible information.
• Classify equivariant layers as standard, embedding, or pooling based on input/output representation sizes.
• Demonstrate how to generalize QCNNs to group-equivariant QCNNs and provide SU(2)-equivariant QCNN architecture.

Experimental results

Research questions

• RQ1 How can EQNNs be systematically built for essentially any symmetry group?
• RQ2 What are the exact parameter counts and structural constraints for equivariant channels and unitaries?
• RQ3 How do intermediate representations influence the expressiveness and trainability of EQNNs?
• RQ4 Can group-equivariant QCNNs outperform symmetry-agnostic variants in quantum phase classification tasks?

Key findings

• EQNN layers act as a generalized Fourier transform on the data, with nontrivial action only on multiplicity spaces in the isotypic decomposition.
• The number of free parameters in (G, Rin, Rout)-equivariant CPTP channels is given by a representation-dependent count, enabling parameter-efficiency via symmetry constraints.
• Three constructive methods (nullspace, twirling, Choi representation) enable efficient parametrization of equivariant layers, even for exponentially large or continuous groups.
• Intermediary representations (Rin, R1, ..., Rout) serve as hyperparameters that can alter the model’s access to information and its processing capabilities.
• A SU(2)-equivariant QCNN shows improved performance over a symmetry-agnostic QCNN on a bond-alternating Heisenberg model phase classification task.