Coupled-Layer Construction of Quantum Product Codes

TL;DR

This paper introduces a coupled-layer construction method for quantum product codes, clarifying their assembly and connecting them to concatenated codes and topological phases.

Contribution

It provides an intuitive coupled-layer framework for quantum product codes, linking tensor and balanced products to physical mechanisms and topological phases.

Findings

Tensor and balanced product codes admit a coupled-layer construction.

Tensor product codes can be derived by gauging logicals in concatenated codes.

The framework unifies known mechanisms for higher-dimensional topological phases.

Abstract

Product codes are a class of quantum error correcting codes built from two or more constituent codes. They have recently gained prominence for a breakthrough yielding quantum low-density parity-check (qLDPC) codes with favorable scaling of both code distance and encoding rate. However, despite its powerful algebraic formulation, the physical mechanism for assembling a general product code from its constituents remains unclear. In this letter, we show that the tensor and balanced product codes admit an intuitive coupled-layer construction by taking a stack of one code and condensing a set of excitations in the pattern given by the checks of the other code. We also make a connection to concatenated codes by showing that the tensor product code can be obtained by gauging large-weight logicals in concatenated codes, making them qLDPC. Our framework accommodates both classical or quantum CSS input codes, unifies known physical mechanisms for constructing higher dimensional topological phases via anyon condensation, and naturally extends to non-topological codes.