# The Search For Leakage-free Entangling Fibonacci Braiding Gates

## TL;DR

This paper investigates the existence of leakage-free entangling Fibonacci braiding gates, providing evidence against their existence and proposing a simpler protocol for approximate leakage-free gates.

## Contribution

It offers the first evidence supporting the non-existence of leakage-free entangling Fibonacci braiding gates and introduces a more straightforward method for approximating such gates.

## Key findings

- Leakage-free entangling Fibonacci braiding gates likely do not exist.
- A simpler protocol for approximate leakage-free gates is proposed.
- The new protocol improves efficiency over previous algorithms.

## Abstract

It is an open question if there are leakage-free entangling Fibonacci braiding gates. We provide evidence to the conjecture for the negative in this paper. We also found a much simpler protocol to generate approximately leakage-free entangling Fibonacci braiding gates than existing algorithms in the literature.