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arXiv:2601.00064 · Journal of High Energy Physics

Pauli stabilizer formalism for topological quantum field theories and generalized statistics

Yitao Feng, Hanyu Xue, Ryohei Kobayashi, Po-Shen Hsin, Yu-An Chen

quant-phcond-mat.str-elhep-thmath.QASubmitted Dec 31, 2025• Announced Jan 2, 2026• 1 citation
v1 · Dec 31, 2025v2 · Jan 5, 2026v3 · Jun 24, 2026
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Abstract

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 the lattice 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 spacetime dimensions. We develop a lattice description of the resulting extended excitations and systematically determine their generalized statistics. Our main example is the $(4+1)$D \emph{fermionic-loop toric code}, obtained by condensing the $e^2 m^2$-loop in the $(4+1)$D $\mathbb{Z}_4$ 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 $H^{5}(B^{2}G, U(1))$. % Beyond $(4+1)$D, the fermionic-loop toric codes form a family of $\mathbb{Z}_2$ topological orders in arbitrary dimensions featuring fermionic loop excitations, realized as explicit Pauli stabilizer codes using $\mathbb{Z}_4$ 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 generalized 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 $\mathbb{Z}_2$ topological orders: the \emph{fermionic-membrane toric code} and the \emph{fermionic-volume toric code}.

Comments
48 pages, 2 figures

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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 the lattice 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 spacetime dimensions. We develop a lattice description of the resulting extended excitations and systematically determine their generalized statistics.
Our main example is the (4+1)(4+1)(4+1)D \emph{fermionic-loop toric code}, obtained by condensing the e2m2e^2 m^2e2m2-loop in the (4+1)(4+1)(4+1)D Z4\mathbb{Z}_4Z4​ toric code. We show that the loop excitation exhibits fermionic loop statistics: the 24-step loop-flipping process yields a phase of −1-1−1. Our Pauli stabilizer models realize all twisted 2-form gauge theories in (4+1)(4+1)(4+1)D, the higher-form Dijkgraaf-Witten TQFT classified by H5(B2G,U(1))H^{5}(B^{2}G, U(1))H5(B2G,U(1)). % Beyond (4+1)(4+1)(4+1)D, the fermionic-loop toric codes form a family of Z2\mathbb{Z}_2Z2​ topological orders in arbitrary dimensions featuring fermionic loop excitations, realized as explicit Pauli stabilizer codes using Z4\mathbb{Z}_4Z4​ 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 generalized statistics. For example, we propose anyonic membrane statistics in (6+1)(6+1)(6+1)D, as well as fermionic membrane and volume statistics in arbitrary dimensions. We construct new families of Z2\mathbb{Z}_2Z2​ topological orders: the \emph{fermionic-membrane toric code} and the \emph{fermionic-volume toric code}.
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\mathbb Z_2Z2​ topological order in (2+1)(2{+}1)(2+1) dimensions, using Z8\mathbb Z_8Z8​ 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\mathbb Z_D^FZDF​ symmetries for DDD even. Using this, we realize a stabilizer model for a nontrivial Z8F\mathbb Z_8^FZ8F​ 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 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.
We construct Pauli topological subsystem codes characterized by arbitrary two-dimensional Abelian anyon theories–this includes anyon theories with degenerate braiding relations and those without a gapped boundary to the vacuum. Our work both extends the classification of two-dimensional Pauli topological subsystem codes to systems of composite-dimensional qudits and establishes that the classification is at least as rich as that of Abelian anyon theories. We exemplify the construction with topological subsystem codes defined on four-dimensional qudits based on the Z4(1) anyon theory with degenerate braiding relations and the chiral semion theory–both of which cannot be captured by topological stabilizer codes. The construction proceeds by "gauging out" certain anyon types of a topological stabilizer code. This amounts to defining a gauge group generated by the stabilizer group of the topological stabilizer code and a set of anyonic string operators for the anyon types that are gauged out. The resulting topological subsystem code is characterized by an anyon theory containing a proper subset of the anyons of the topological stabilizer code. We thereby show that every Abelian anyon theory is a subtheory of a stack of toric codes and a certain family of twisted quantum doubles that generalize the double semion anyon theory. We further prove a number of general statements about the logical operators of translation invariant topological subsystem codes and define their associated anyon theories in terms of higher-form symmetries.
The statistics of particles and extended excitations, such as loops and membranes, are fundamental to modern condensed matter physics, high-energy physics, and quantum information science, yet a comprehensive lattice-level framework for computing them remains elusive. In this work, we develop a universal microscopic method to determine the generalized statistics of Abelian excitations on lattices of arbitrary dimension and demonstrate it by deriving the statistics of particles, loops, and membranes in up to three spatial dimensions. Our approach constructs a sequence of local unitary operators whose many-body Berry phase encodes the desired statistical invariant. The required sequence is generated automatically from the Smith normal form of locality constraints and therefore needs no extra physical input. We prove that the resulting invariants are quantized, provide an algorithm that computes them efficiently, and show how they unify familiar braiding and fusion data of particles while also uncovering new self- and mutual statistics of loop and membrane excitations. We further demonstrate that each statistical invariant corresponds to an ’t Hooft anomaly of a generalized symmetry; we show that a nontrivial invariant both (i) obstructs gauging that symmetry and (ii) forbids any short-range-entangled (symmetry-preserving) ground state. This establishes a precise connection between microscopic lattice anomalies and many-body dynamics, providing a generalization of the Lieb-Schultz-Mattis theorem that constrains a wide class of quantum lattice systems.
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.
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.