Limitations on Transversal Gates for Hypergraph Product Codes

Abstract

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 <inline-formula> <tex-math notation="LaTeX">$[n,k,d]$ </tex-math></inline-formula> Gallagher codes with <inline-formula> <tex-math notation="LaTeX">$d\ge 3$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$k\le n/2$ </tex-math></inline-formula>. This work is a generalization of an argument due to Bravyi and K&#x00F6;nig, and we also conjecture this is a refinement of the recent notion of disjointness due to Jochym-O&#x2019;Connor <i>et al.</i>