Learning Gaussian Operations and the Matchgate Hierarchy

TL;DR

This paper extends efficient learning methods from Clifford operations to fermionic Gaussian operations and introduces the Matchgate Hierarchy, an infinite family of gates with similar learnability properties.

Contribution

It generalizes the efficient learning framework to fermionic Gaussian operations and introduces the Matchgate Hierarchy, expanding the class of quantum processes that can be efficiently learned.

Findings

Fermionic Gaussian operations can be efficiently learned from black-box implementations.

The Matchgate Hierarchy is introduced as an infinite family of gates.

Clifford Hierarchy is contained within the Matchgate Hierarchy, enabling efficient learning at all levels.

Abstract

Learning an unknown quantum process is a central task for validation of the functioning of near-term devices. The task is generally hard, requiring exponentially many measurements if no prior assumptions are made on the process. However, an interesting feature of the classically-simulable Clifford group is that unknown Clifford operations may be efficiently determined from a black-box implementation. We extend this result to the important class of fermionic Gaussian operations. These operations have received much attention due to their close links to fermionic linear optics. We then introduce an infinite family of unitary gates, called the Matchgate Hierarchy, with a similar structure to the Clifford Hierarchy. We show that the Clifford Hierarchy is contained within the Matchgate Hierarchy and how operations at any level of the hierarchy can be efficiently learned.