Local tensor-network codes

TL;DR

This paper introduces tensor-network codes for topological stabilizer codes, providing methods for code construction, distance calculation, and error correction improvements, with applications to surface and colour codes.

Contribution

It demonstrates how to represent topological codes as tensor-network codes and introduces an efficient distance calculation method that enhances error correction performance.

Findings

Modified surface code with fewer logical operators

Tensor-network distance calculator provides detailed code info

Error correction success improved by up to 2%

Abstract

Tensor-network codes enable the construction of large stabilizer codes out of tensors describing smaller stabilizer codes. An application of tensor-network codes was an efficient and exact decoder for holographic codes. Here, we show how to write some topological codes, including the surface code and colour code, as simple tensor-network codes. We also show how to calculate distances of stabilizer codes by contracting a tensor network. The algorithm actually gives more information, including a histogram of all logical coset weights. We prove that this method is efficient in the case of holographic codes. Using our tensor-network distance calculator, we find a modification of the rotated surface code that has the same distance but fewer minimum-weight logical operators by injecting the non-CSS five-qubit code tensor into the tensor network. This corresponds to an improvement in successful error correction of up to 2% against depolarizing noise (in the perfect-measurement setting), but comes at the cost of introducing four higher-weight stabilizers. Our general construction lets us pick a network geometry (e.g., a Euclidean lattice in the case of the surface code), and, using only a small set of seed codes (constituent tensors), build extensive codes with the potential for optimisation.