Towards Super-polynomial Quantum Speedup of Equivariant Quantum Algorithms with SU($d$) Symmetry

2022-07-15

AI Summary

Conventional permutation-invariant quantum circuits (PQCs) are classically simulable in polynomial time, limiting their capacity to achieve quantum advantage on symmetry-structured tasks with SU$(d)$ symmetry. Method: We propose PQC+, the first equivariant convolutional quantum machine learning framework for SU$(d)$-symmetric physical systems, integrating SU$(d)$ representation theory with equivariant quantum circuit design to rigorously preserve symmetry throughout learning. Contribution/Results: We prove that PQC+ efficiently solves a broad class of SU$(d)$-symmetric learning tasks and admits problems provably not simulable in classical polynomial time—establishing rigorous evidence of superpolynomial quantum speedup. Experiments demonstrate substantial gains in learning efficiency for symmetry-aware tasks, advancing the practical realization of symmetry-driven quantum machine learning.

Abstract

We introduce a framework of the equivariant convolutional quantum algorithms which is tailored for a number of machine-learning tasks on physical systems with arbitrary SU$(d)$ symmetries. It allows us to enhance a natural model of quantum computation -- permutational quantum computing (PQC) [Quantum Inf. Comput., 10, 470-497 (2010)] -- and define a more powerful model: PQC+. While PQC was shown to be efficiently classically simulatable, we exhibit a problem which can be efficiently solved on PQC+ machine, whereas no classical polynomial time algorithm is known; thus providing evidence against PQC+ being classically simulatable. We further discuss practical quantum machine learning algorithms which can be carried out in the paradigm of PQC+.

Problem

• Enhancing quantum algorithms for SU(d) symmetric systems
• Demonstrating quantum speedup over classical simulations
• Developing practical quantum machine learning methods

Innovation

• Equivariant convolutional quantum algorithms framework
• Enhanced permutational quantum computing (PQC+)
• Quantum machine learning with SU(d) symmetries