Low-Overhead Entangling Gates from Generalised Dehn Twists

We generalise the implementation of logical quantum gates via Dehn twists from topological codes to the hypergraph and balanced products of cyclic codes. These generalised Dehn twists implement logical entangling gates with no additional qubit overhead and $\mathcal{O}(d)$ time overhead. Due to having more logical degrees of freedom in the codes, there is a richer structure of attainable logical gates compared to those for topological codes. To illustrate the scheme, we focus on families of hypergraph and balanced product codes that scale as $\[[18q^2,8,2q\]]_{q\in \mathbb{N}}$ and $\[[18q,8,\leq 2q\]]_{q\in \mathbb{N}}$ respectively. For distance 6 to 12 hypergraph product codes, we find that the set of twists and fold-transversal gates generate the full logical Clifford group. For the balanced product code, we show that Dehn twists apply to codes in this family with odd $q$. We also show that the $[[90,8,10]]$ bivariate bicycle code is a member of the balanced product code family that saturates the distance bound. We also find balanced product codes that saturate the bound up to $q\leq8$ through a numerical search.