Connectivity constrains quantum codes

Abstract

Quantum error correcting codes are a scheme through which a set of measurements is used to correct for decoherence in a quantum system. Due to experimental limitations, it is natural to require that each of these measurements only involve a constant number of qubits. This requirement motivates the class of quantum low-density parity-check (LDPC) codes, which also limits the number of measurement outcomes a qubit can affect. Seminal results have shown that quantum LDPC codes implemented through local interactions in D-dimensional Euclidean space obey strong restrictions on their code dimension k, distance d, and their ability to implement fault-tolerant operations. However, we lack an understanding of what limits quantum LDPC codes that do not have an explicit embedding in RD. The need for a more general understanding of these limitations is highlighted by recent breakthroughs in the construction of LDPC codes that eschew locality, and yet witness tradeoffs between code parameters. In this work we prove bounds applicable to any quantum LDPC code. Our main results are a) a bound on the distance, b) a bound on the code dimension and c) limitations on certain fault-tolerant gates that can be applied to quantum LDPC codes. All three of these bounds are cast as a function of the graph separator of the connectivity graph representation of the quantum code. We find that unless the connectivity graph contains an expander, the code is notably limited. This implies a necessary, but not sufficient, condition to construct good codes. This is the first bound that studies the limitations of quantum LDPC codes that does not rely on geometric locality. As an application, we present bounds on quantum LDPC codes associated with local graphs in D-dimensional hyperbolic space, and local graph on g-genus surfaces.