Pauli stabilizer models of twisted quantum doubles

TL;DR

This paper constructs Pauli stabilizer models for all 2D Abelian topological orders with gapped boundaries, including the double semion phase, and extends the classification of topological stabilizer codes beyond toric codes.

Contribution

It introduces a method to realize twisted quantum doubles as Pauli stabilizer models, broadening the scope of topological quantum error-correcting codes.

Findings

Constructed a stabilizer model for the double semion phase.

Generalized the construction to all Abelian twisted quantum doubles.

Connected SPT phases to stabilizer models via gauging symmetries.

Abstract

We construct a Pauli stabilizer model for every two-dimensional Abelian topological order that admits a gapped boundary. Our primary example is a Pauli stabilizer model on four-dimensional qudits that belongs to the double semion (DS) phase of matter. The DS stabilizer Hamiltonian is constructed by condensing an emergent boson in a $\mathbb{Z}_4$ toric code, where the condensation is implemented by making certain two-body measurements. We rigorously verify the topological order of the DS stabilizer model by identifying an explicit finite-depth quantum circuit (with ancillary qubits) that maps its ground state subspace to that of a DS string-net model. We show that the construction of the DS stabilizer Hamiltonian generalizes to all twisted quantum doubles (TQDs) with Abelian anyons. This yields a Pauli stabilizer code on composite-dimensional qudits for each such TQD, implying that the classification of topological Pauli stabilizer codes extends well beyond stacks of toric codes - in fact, exhausting all Abelian anyon theories that admit a gapped boundary. We also demonstrate that symmetry-protected topological phases of matter characterized by type I and type II cocycles can be modeled by Pauli stabilizer Hamiltonians by gauging certain 1-form symmetries of the TQD stabilizer models.