The search for leakage-free entangling Fibonacci braiding gates

It is an open question if there are leakage-free entangling Fibonacci braiding gates. In this article, we give a construction of a large family of leakage-free braiding gates which are then proved to be non-entangling. We also conducted brute-force numerical searches for braids with a word-length up to seven and found no leakage-free entangling gates. These suggest the negative for the conjecture. On the other hand, we provide a much simpler protocol to generate approximately leakage-free entangling Fibonacci braiding gates than existing algorithms in the literature.