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

TL;DR

This paper introduces an advanced quantum algorithm framework leveraging SU(d) symmetry, demonstrating potential for quantum speedup in machine learning tasks that are hard for classical algorithms.

Contribution

It develops the PQC+ model, extending permutational quantum computing to achieve potential quantum advantage in symmetry-based machine learning tasks.

Findings

PQC+ can solve certain problems efficiently that are hard for classical algorithms.

PQC+ extends the capabilities of permutational quantum computing.

The framework is applicable to physical systems with SU(d) symmetries.

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) -- 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+.