Meet-in-the-Middle Compilation of the [CNOT] Class with Fibonacci Anyons

@misc{fedin2026meetinthemiddle,
  title = {{Meet-in-the-Middle Compilation of the $[\mathrmCNOT]$ Class with Fibonacci Anyons}},
  author = {Matvei Fedin and Andrey Morozov},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.37552},
  eprint = {2610.37552},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.37552},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Meet-in-the-Middle Compilation of the [CNOT] Class with Fibonacci Anyons
AU  - Matvei Fedin
AU  - Andrey Morozov
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.37552
UR  - https://arxiv.org/abs/2610.37552
AB  - Compilation of two-qubit Fibonacci anyon gates is complicated by leakage (ℓ) from the four-dimensional computational sector to the fifth state of the fusion space. We construct a GPU pipeline to find low-leakage representatives of the [CNOT] class on different subsets of the braid group. We exhaustively explore the full positive five-generator alphabet up to L=21 with minimal leakage ℓ∼10^−3 and error d_[CNOT]∼ 10^−8 on it. To reach larger lengths, the search space is successively narrowed to three-generator, boundary-σ_3, and palindromic braids, and by considering only the Garside left normal form. These optimizations allow us to reach L=72. For L=69, a braid with ℓ∼10^−10 and d_[CNOT]∼ 10^−10 is found. In addition to directly enumerating optimal braids, we proved that ℓ∈ℤ[φ], and any nonzero leakage obeys an exponential lower bound in the braid length, and obtained the limit laws F_3,∞(ℓ)=ℓ^2 and F_5,∞(ℓ)=ℓ^4 for our sets. Using asymptotics for the distribution function F, we showed that for sufficiently long braids, the asymptotics of exhaustive search are better than the asymptotics of random sampling for finding low-leakage braids.
ER  -