Quantum Lego: Building Quantum Error Correction Codes from Tensor Networks

TL;DR

This paper presents a tensor network framework for constructing and analyzing complex quantum error correction codes from simpler components, enabling new insights into code properties and decoding circuits.

Contribution

It introduces a modular tensor network approach to build and study quantum error correction codes, including non-stabilizer codes, with graphical intuition and operator flow analysis.

Findings

Constructed various codes including toric, holographic, and 3D stabilizer codes.

Demonstrated the equivalence of surface and Bacon-Shor codes via tensor network dualization.

Provided a method to derive decoding and encoding circuits from tensor network contractions.

Abstract

We introduce a flexible and graphically intuitive framework that constructs complex quantum error correction codes from simple codes or states, generalizing code concatenation. More specifically, we represent the complex code constructions as tensor networks built from the tensors of simple codes or states in a modular fashion. Using a set of local moves known as operator pushing, one can derive properties of the more complex codes, such as transversal non-Clifford gates, by tracing the flow of operators in the network. The framework endows a network geometry to any code it builds and is valid for constructing stabilizer codes as well as non-stabilizer codes over qubits and qudits. For a contractible tensor network, the sequence of contractions also constructs a decoding/encoding circuit. To highlight the framework's range of capabilities and to provide a tutorial, we lay out some examples where we glue together simple stabilizer codes to construct non-trivial codes. These examples include the toric code and its variants, a holographic code with transversal non-Clifford operators, a 3d stabilizer code, and other stabilizer codes with interesting properties. Surprisingly, we find that the surface code is equivalent to the 2d Bacon-Shor code after "dualizing" its tensor network encoding map.