It has long been known that long-ranged entangled topological phases can be exploited to protect quantum information against unwanted local errors. Indeed, conditions for intrinsic topological order are reminiscent of criteria for faithful quantum error correction. At the same time, the promise of using general topological orders for practical error correction remains largely unfulfilled to date. In this work, we significantly contribute to establishing such a connection by showing that Abelian twisted quantum double models can be used for quantum error correction. By exploiting the group cohomological data sitting at the heart of these lattice models, we transmute the terms of these Hamiltonians into full-rank, pairwise commuting operators, defining commuting stabilizers. The resulting codes are defined by commuting non-Pauli stabilizers, with local systems that can either be qubits or higher dimensional quantum systems. Thus, this work establishes a new connection between condensed matter physics and quantum information theory, and constructs tools to systematically devise new topological quantum error correcting codes beyond toric or surface code models.
It has long been known that long-ranged entangled topological phases can be exploited to protect quantum information against unwanted local errors. Indeed, conditions for intrinsic topological order are reminiscent of criteria for faithful quantum error correction. At the same time, the promise of using general topological orders for practical error correction remains largely unfulfilled to date. In this work, we significantly contribute to establishing such a connection by showing that Abelian twisted quantum double models can be used for quantum error correction. By exploiting the group cohomological data sitting at the heart of these lattice models, we transmute the terms of these Hamiltonians into full-rank, pairwise commuting operators, defining commuting stabilizers. The resulting codes are defined by non-Pauli commuting stabilizers, with local systems that can either be qubits or higher dimensional quantum systems. Thus, this work establishes a new connection between condensed matter physics and quantum information theory, and constructs tools to systematically devise new topological quantum error correcting codes beyond toric or surface code models.
Topological quantum field theory (TQFT) provides a unifying framework for describing topological phases of matter and for constructing quantum error-correcting codes, playing a central role across high-energy physics, condensed matter, and quantum information. A central challenge is to formulate topological order on lattices and to extract the properties of topological excitations from microscopic Hamiltonians. In this work, we construct new classes of lattice gauge theories as Pauli stabilizer models, realizing a wide range of TQFTs in general dimensions. We develop a lattice description of extended excitations and systematically determine their generalized statistics. Our main example is the (4+1)D fermionic-loop toric code, obtained by condensing the
e2m2-loop in the (4+1)D
Z4 toric code. We show that the loop excitation exhibits fermionic loop statistics: the 24-step loop-flipping process yields a phase of
−1. Our Pauli stabilizer models realize all twisted 2-form gauge theories in (4+1)D, the higher-form Dijkgraaf-Witten TQFT classified by
H5(B2G,U(1)). Beyond (4+1)D, the fermionic-loop toric codes form a family of
Z2 topological orders in arbitrary dimensions, realized as explicit Pauli stabilizer codes using
Z4 qudits. Finally, we develop a Pauli-based framework that defines generalized statistics for extended excitations in any dimension, yielding computable lattice unitary processes to detect nontrivial statistics. For example, we propose anyonic membrane statistics in (6+1)D, as well as fermionic membrane and volume statistics in arbitrary dimensions. We construct new families of
Z2 topological orders: the fermionic-membrane toric code and the fermionic-volume toric code. In addition, we demonstrate that
p-dimensional excitations in
2p+2 spatial dimensions can support anyonic
p-brane statistics for only even
p.
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 Z 4 toric code, where the condensation is implemented at the level of the ground states by 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.
Stabilizer codes provide exact lattice realizations of bosonic topological orders. In contrast, systematic stabilizer descriptions of intrinsically fermionic topological phases remain much less developed. In this work, we introduce Majorana-Pauli stabilizer codes, a class of exactly solvable fermionic lattice models whose stabilizers are built from both generalized Pauli operators and Majorana operators. As a main example, we construct an exactly solvable stabilizer realization of the fermionic toric code: an intrinsically fermionic
Z2 topological order in
(2+1) dimensions, using
Z8 Pauli operators coupled to Majorana modes. Within this stabilizer framework, the anyons, string operators, fusion rules, and braiding statistics all follow naturally from the stabilizer algebra. More broadly, we show that the fermionic toric code belongs to a duality web generated by anyon condensation and by gauging or ungauging bosonic or fermionic symmetries. This web connects bosonic topological orders, symmetry-enriched topological (SET) phases, and both bosonic and fermionic symmetry-protected topological (SPT) phases, all within a common stabilizer description. We extend this construction to all Abelian fermionic topological orders with gapped boundaries and to fermionic SPT phases whose fully gauged theories are Abelian. Going beyond Majorana operators, we introduce fermionic versions of the clock and shift operators and use them to construct an exact bosonization map for
ZDF symmetries for
D even. Using this, we realize a stabilizer model for a nontrivial
Z8F fermionic SPT phase with no free-fermion analog. Altogether, these results extend the stabilizer-code paradigm to a broad class of intrinsically fermionic phases bridging fermionic quantum many-body physics to quantum error correction.
We present a versatile and mathematically rigorous technique for bounding recovery thresholds in topological codes subject to noise. Our method captures the effect of applying an arbitrary (possibly non-Pauli) local noise channel to the code state of a broad class of two-dimensional codes, including surface codes, non-Abelian quantum doubles, and string-net codes. In each case, we prove that for noise strengths up to some explicit constant value, any initially encoded logical information can be recovered to high precision, and that the noise-corrupted state exhibits key hallmarks of mixed-state topological order: long-range entanglement and emergent higher-form symmetries. We also describe how these methods can be adapted to higher dimensions and correlated noise models.
This thesis presents a detailed analysis of the quantum double model and its twisted version, focusing on their properties as quantum error-correcting codes. In the case of Kitaev's quantum double model for finite Abelian groups, the error correction process is explicitly described. The number of correctable errors depends on the lattice and the topology of the underlying surface. Although there exists a theoretical maximum number of errors that can be corrected, it is proven that correcting this number of errors is, in general, an NP-complete problem. As an alternative, a polynomial-time correction algorithm is proposed that corrects a number of errors below the theoretical maximum. In regard to the twisted quantum double model, it is studied within the framework of Local Topological Order (LTO). Originally formulated for square lattices, the definition of LTO is extended here to arbitrary two-dimensional lattices, enabling an explicit characterization of the ground states space through invariant spaces of monomial representations. By reformulating the LTO conditions to include such general lattices, it is proven that the twisted model satisfies all four axioms of LTO on any 2D lattice. As a result, the ground states space of the model is shown to define a quantum error-correcting code, highlighting the role of topological order in fault-tolerant quantum computation.