Fragmented Topological Excitations in Generalized Hypergraph Product Codes

TL;DR

This paper explores a family of generalized hypergraph product codes, revealing novel fracton topological excitations and properties like non-monotonic ground state degeneracy, advancing understanding of topological quantum codes and fracton orders.

Contribution

It introduces a new class of exactly solvable spin models called orthoplex models, uncovering fragmented topological excitations and complex ground state behaviors in generalized hypergraph product codes.

Findings

Discovery of fragmented topological excitations in 4D models

Identification of non-monotonic ground state degeneracy in 3D models

Revealing the topological nature of isolated point excitations

Abstract

Product code construction is a powerful tool for constructing quantum stabilizer codes, which serve as a promising paradigm for realizing fault-tolerant quantum computation. Furthermore, the natural mapping between stabilizer codes and the ground states of exactly solvable spin models also motivates the exploration of many-body orders in the stabilizer codes. In this work, we investigate the fracton topological orders in a family of codes obtained by a recently proposed general construction. More specifically, this code family can be regarded as a class of generalized hypergraph product (HGP) codes. We term the corresponding exactly solvable spin models \textit{orthoplex models}, based on the geometry of the stabilizers. In the 3D orthoplex model, we identify a series of intriguing properties within this model family, including non-monotonic ground state degeneracy (GSD) as a function of system size and non-Abelian lattice defects. Most remarkably, in 4D we discover \textit{fragmented topological excitations}: while such excitations manifest as discrete, isolated points in real space, their projections onto lower-dimensional subsystems form connected objects such as loops, revealing the intrinsic topological nature of these excitations. Therefore, fragmented excitations constitute an intriguing intermediate class between point-like and spatially extended topological excitations. In addition, these rich features establish the generalized HGP codes as a versatile and analytically tractable platform for studying the physics of fracton orders.