[Paper Review] Exploiting symmetry in variational quantum machine learning

Johannes Jakob Meyer, Marian Mularski+5 more|arXiv (Cornell University)|May 12, 2022

Quantum Computing Algorithms and Architecture9 citations

TL;DR

This paper introduces a framework to exploit data symmetries in variational quantum machine learning by constructing equivariant quantum gatesets through unitary group representations. By symmetrizing standard gates using representation theory, the method enables invariant predictions under symmetry transformations, significantly improving generalization in quantum learning models, as demonstrated on tic-tac-toe and autonomous driving classification tasks with up to 9 qubits.

ABSTRACT

Variational quantum machine learning is an extensively studied application of near-term quantum computers. The success of variational quantum learning models crucially depends on finding a suitable parametrization of the model that encodes an inductive bias relevant to the learning task. However, precious little is known about guiding principles for the construction of suitable parametrizations. In this work, we holistically explore when and how symmetries of the learning problem can be exploited to construct quantum learning models with outcomes invariant under the symmetry of the learning task. Building on tools from representation theory, we show how a standard gateset can be transformed into an equivariant gateset that respects the symmetries of the problem at hand through a process of gate symmetrization. We benchmark the proposed methods on two toy problems that feature a non-trivial symmetry and observe a substantial increase in generalization performance. As our tools can also be applied in a straightforward way to other variational problems with symmetric structure, we show how equivariant gatesets can be used in variational quantum eigensolvers.

Motivation & Objective

• To develop a systematic method for incorporating known data symmetries into variational quantum machine learning models.
• To address the lack of principled design guidelines for quantum model parametrizations in NISQ devices.
• To demonstrate that symmetry-aware quantum architectures improve generalization and training efficiency.
• To extend the applicability of equivariant quantum circuits beyond machine learning to variational quantum eigensolvers.
• To provide a general blueprint for constructing invariant quantum models using representation theory and gate symmetrization.

Proposed method

• Uses unitary representations of symmetry groups (e.g., O(3)) to embed classical data symmetries into the quantum Hilbert space.
• Applies gate symmetrization to transform a standard gateset into an equivariant gateset that commutes with the symmetry group action.
• Employs representation theory to ensure that the resulting ansatz circuits are equivariant, preserving symmetry structure across the quantum circuit.
• Constructs variational quantum models with data re-uploading that enforce invariance under symmetry transformations.
• Validates the approach using numerical experiments on symmetric learning tasks and variational quantum eigensolvers.
• Extends the framework to ground state problems by leveraging conserved quantities (e.g., particle number, spin) as symmetries in Hamiltonians.

Experimental results

Research questions

• RQ1 How can continuous and discrete data symmetries be systematically embedded into the Hilbert space of a quantum computer?
• RQ2 What is the correct procedure to transform a standard quantum gateset into an equivariant one that respects a given symmetry group?
• RQ3 To what extent does using equivariant gatesets improve generalization in variational quantum machine learning models?
• RQ4 Can equivariant quantum circuits also enhance performance in variational quantum eigensolvers with conserved quantities?
• RQ5 How do equivariant architectures compare to non-equivariant ones in terms of convergence and barren plateau mitigation?

Key findings

• The proposed equivariant gateset construction leads to a substantial increase in generalization performance on symmetric learning tasks such as tic-tac-toe classification.
• Even randomly initialized equivariant architectures outperform their non-invariant counterparts, indicating robustness and inherent inductive bias.
• The method successfully improves convergence and energy estimation in variational quantum eigensolvers for models like the transverse-field Ising and Heisenberg Hamiltonians.
• Equivariant ansätze help alleviate the barren plateau problem by restricting the parameter space to physically relevant subspaces.
• The framework is general and applicable beyond machine learning, as demonstrated on ground state problems with geometric and global symmetries.
• The authors confirm that the symmetry representation on the Hilbert space is sufficient for constructing equivariant gatesets, enabling broad applicability.