Quantum weight enumerators and tensor networks

TL;DR

This paper introduces tensor enumerators as an extension of quantum weight enumerators, enabling more efficient analysis of tensor network-based quantum codes and establishing a framework for quantum MacWilliams identities.

Contribution

It extends quantum weight enumerators to tensor enumerators and demonstrates their compatibility with tensor network trace operations, improving analysis of quantum codes.

Findings

Tensor enumerators generalize quantum weight enumerators.

Trace operations are compatible between tensor networks and tensor enumerators.

Framework for quantum MacWilliams identities includes tensor enumerators.

Abstract

We examine the use of weight enumerators for analyzing tensor network constructions, and specifically the quantum lego framework recently introduced. We extend the notion of quantum weight enumerators to so-called tensor enumerators, and prove that the trace operation on tensor networks is compatible with a trace operation on tensor enumerators. This allows us to compute quantum weight enumerators of larger codes such as the ones constructed through tensor network methods more efficiently. We also provide a general framework for quantum MacWilliams identities that includes tensor enumerators as a special case.