Connectivity constrains quantum codes

TL;DR

This paper establishes fundamental bounds on quantum LDPC codes based on the connectivity graph's properties, revealing limitations and necessary conditions for constructing high-performance quantum error-correcting codes.

Contribution

It introduces a unified framework linking graph separator properties to bounds on quantum LDPC code parameters and fault-tolerant gates, advancing understanding of their limitations.

Findings

Bounds on code distance and dimension as functions of graph separators

Limitations on fault-tolerant gates for quantum LDPC codes

Necessity of expander graphs for good quantum LDPC codes

Abstract

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 of syndrome bits, and each syndrome bit only relies on some constant number of qubits. Constructing quantum LDPC codes is challenging. It is an open problem to understand if there exist good quantum LDPC codes, i.e. with constant rate and relative distance. Furthermore, techniques to perform fault-tolerant gates are poorly understood. We present a unified way to address these problems. 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 severely 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 locality. As an application, we present novel bounds on quantum LDPC codes associated with local graphs in $D$-dimensional hyperbolic space.