Equivalence Classes of Quantum Error-Correcting Codes

TL;DR

This paper introduces canonical forms and prime decompositions for quantum error-correcting codes using ZX calculus, providing a systematic way to classify and analyze code equivalence classes.

Contribution

It presents the first canonical forms for CSS codes and states, introduces prime code diagrams, and proves the uniqueness of prime decompositions for Clifford codes.

Findings

Canonical forms for CSS codes and states are established.

Prime decompositions of Clifford codes are proven to be unique.

Equivalence classes of ZX diagrams are tabulated under local and output permutations.

Abstract

Quantum error-correcting codes (QECC's) are needed to combat the inherent noise affecting quantum processes. Using ZX calculus, we represent QECC's in a form called a ZX diagram, consisting of a tensor network. In this paper, we present canonical forms for CSS codes and CSS states (which are CSS codes with 0 inputs), and we show the resulting canonical forms for the toric code and certain surface codes. Next, we introduce the notion of prime code diagrams, ZX diagrams of codes that have a single connected component with the property that no sequence of rewrite rules can split such a diagram into two connected components. We also show the Fundamental Theorem of Clifford Codes, proving the existence and uniqueness of the prime decomposition of Clifford codes. Next, we tabulate equivalence classes of ZX diagrams under a different definition of equivalence that allows output permutations and any local operations on the outputs. Possible representatives of these equivalence classes are analyzed. This work expands on previous works in exploring the canonical forms of QECC's in their ZX diagram representations.