Limitations on Transversal Gates for Hypergraph Product Codes

In a fault tolerant quantum computer, quantum codes are expected to serve the conflicting purposes of protecting quantum information while also allowing that information to be manipulated by fault-tolerant gates. We introduce a new technique for placing limitations on such gates, and apply this technique to a class of quantum codes known as hypergraph product codes contained within the vertical sector. These codes are constructed from input which is a pair of classical linear codes, and generalize the Kitaev surface code which is the hypergraph product of classical repetition codes. We provide a necessary condition on these input codes, under which the resulting hypergraph product code has transversal gates limited to the Clifford group. We conjecture that this condition is satisfied by all $[n,k,d]$ Gallagher codes with $d\ge 3$ and $k\le n/2$ . This work is a generalization of an argument due to Bravyi and König, and we also conjecture this is a refinement of the recent notion of disjointness due to Jochym-O’Connor et al.