Connectivity constrains quantum codes

Quantum low-density parity-check (LDPC) codes are an important class of quantum error correcting codes. In such codes, each qubit only affects a constant number syndrome bits, and bit relies on some qubits. Constructing LDPC is challenging. It open problem to understand if there exist good i.e. with rate relative distance. Furthermore, techniques perform fault-tolerant gates poorly understood. We present unified way address these problems. Our main results a) bound the distance, b) code dimension c) limitations certain that can be applied All three bounds cast as function graph separator connectivity representation code. find unless contains expander, severely limited. This implies necessary, but not sufficient, condition construct first studies does rely locality. As application, we novel associated local graphs in D-dimensional hyperbolic space.