Topological quantum compilation of two-qubit gates

TL;DR

This paper explores how to topologically compile two-qubit gates using Fibonacci anyons, aiming to generate nearly leakage-free CNOT-like gates and identify which two-qubit operations can be naturally realized through braiding.

Contribution

It demonstrates the generation of specific two-qubit gates via Fibonacci anyon braiding, including a numerically exact implementation of a SWAP gate with minimal braiding elements.

Findings

Most generated classes are near the edges of the Weyl chamber.

Identified that many classes lie between identity and CNOT, and between DCNOT and SWAP.

Achieved a nine-element braiding sequence for an exact SWAP gate.

Abstract

We investigate the topological quantum compilation of two-qubit operations within a system of Fibonacci anyons. Our primary goal is to generate gates that are approximately leakage-free and equivalent to the controlled-NOT (CNOT) gate up to single-qubit operations. These gates belong to the local equivalence class [CNOT]. Additionally, we explore which local equivalence classes of two-qubit operations can be naturally generated by braiding Fibonacci anyons. We discovered that most of the generated classes are located near the edges of the Weyl chamber representation of two-qubit gates, specifically between the local equivalence classes of the identity [1] and [CNOT], and between those of the double-controlled-NOT [DCNOT] and [SWAP]. Furthermore, we found a numerically exact implementation of a local equivalent of the SWAP gate using a sequence of only nine elements from the Fibonacci braiding gate set.