Majorana-Pauli stabilizer codes and duality webs of fermionic topological phases

Abstract: 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 $\mathbb Z_2$ topological order in $(2{+}1)$ dimensions, using $\mathbb Z_8$ 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 bosonic or fermion-parity symmetries. This web connects bosonic topological orders, symmetry-enriched topological phases, and both bosonic and fermionic symmetry-protected topological phases, all within a common stabilizer description. We further show that the construction extends to all Abelian fermionic topological orders with gapped boundaries and to all supercohomology fermionic SPT phases in $(2{+}1)$ dimensions. Going beyond Majorana operators, we introduce fermionic versions of the clock and shift operators and use them to construct an exact bosonization map for $\mathbb Z_D^F$ symmetries for $D$ even. Using this, we realize a stabilizer model for a nontrivial $\mathbb Z_8^F$ 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.

• The paper introduces a unified framework for fermionic topological phases by constructing Majorana-Pauli stabilizer codes that combine generalized Pauli and Majorana operators.
• The paper presents explicit lattice realizations for fermionic toric codes and supercohomology SPT phases, detailing advanced anyon condensation and braiding statistics.
• The paper extends the framework to incorporate higher-order fermionic symmetries using clock and shift operators, enabling exact bosonization maps for Z_D symmetries.

Majorana-Pauli Stabilizer Codes: A Unified Framework for Fermionic Topological Phases

Introduction

This paper systematically constructs Majorana-Pauli stabilizer codes, a class of exactly solvable fermionic lattice models combining generalized Pauli operators and Majorana operators. In contrast to prior approaches restricted to bosonic or Majorana frameworks, this hybrid formalism realizes all Abelian fermionic topological phases in $(2+1)$D admitting gapped boundaries and comprehensively incorporates both anyon condensation and symmetry gauging. The authors present explicit stabilizer realizations for the fermionic toric code, supercohomology fermionic SPT phases, and their associated duality webs, demonstrating algebraic control over anyons, string operators, fusion, and braiding statistics. Additionally, the framework is extended to construct stabilizer codes for symmetry groups with higher-order fermion parity by developing fermionic clock and shift operators, facilitating exact bosonization maps for $Z_D$ symmetries.

Majorana-Pauli Stabilizer Realization of the Fermionic Toric Code

The fermionic toric code is an intrinsically fermionic $Z_2$ topological order in $(2+1)$D, distinct from its bosonic counterpart due to its dependence on the presence of physical fermions. The authors construct an exactly solvable stabilizer model for this phase using $Z_8$ Pauli operators situated on edges and Majorana operators on plaquettes. Key features include:

• The stabilizer algebra derives all anyon types, including a physical transparent fermion, boson $b$, and spin-$1/8$ excitation $d$.
• String, fusion, and braiding operators are explicitly formulated from stabilizer algebra (e.g., $d^2 = by$, mutual braiding between $b$ and $Z_D$0 yields a $Z_D$1 phase).
• Topological ground-state degeneracy on the torus matches theoretical expectations with four sectors, while fermion-parity-resolved labels are organized into even/odd sectors.
• The lattice realization is achieved by a systematic condensation of bosons within the parent $Z_D$2 toric code, binding emergent fermions to physical ones, and enforcing condensation via local stabilizer hopping terms.

This construction yields a stabilizer code whose algebraic structure replicates all aspects of the fermionic toric code anyon theory, including fractional statistics otherwise inaccessible to conventional Majorana-only codes.

Duality Webs and Symmetry Gauging

The authors elucidate a duality web relating bosonic topological orders, fermionic SPT/SET phases, and their shadows via sequences of anyon condensation and gauging:

• Condensing specific bosons in $Z_D$3 toric code yields the fermionic toric code.
• Gauging bosonic or fermion-parity symmetries produces symmetry-enriched versions or bosonic SPTs.
• The web connects phases such as the $Z_D$4 fermionic SPT, $Z_D$5-enriched toric code, and $Z_D$6 Chern-Simons theory, each admitting explicit stabilizer realizations.

Symmetry fractionalization is addressed via boundary effective symmetry operators, confirming nontrivial projective quantum numbers for anyons and nontrivial associator data for defect fusion. The bosonization map extends these constructions, rigorously relating fermionic and bosonic stabilizer codes and implementing symmetry gauging directly at the lattice level.

Fermionic Clock and Shift Operators: Exact Bosonization for Higher-Order Symmetries

To generalize beyond $Z_D$7 fermion parity, the paper introduces fermionic clock and shift operators compatible with cyclic symmetry groups $Z_D$8 ($Z_D$9 even). The construction:

• Defines hybrid operators on local Hilbert spaces combining $Z_2$0 qudit and complex fermion degrees of freedom.
• The algebra incorporates the identification of central elements of $Z_2$1 clock variables with fermion parity, facilitating condensation of higher-order fermionic anyons.
• These operators are used to construct exact bosonization maps for $Z_2$2 symmetries, enabling stabilizer descriptions of SPTs with larger symmetry groups and phases without free-fermion analogs.

The explicit D=8 case illustrates stabilizer construction for a nontrivial $Z_2$3 fermionic SPT, extracting supercohomology data via truncated boundary symmetry operators.

General Stabilizer Construction for Abelian Fermionic Phases

The condensation framework is generalized to all Abelian fermionic twisted gauge theories by specifying K-matrix structure and anyon condensation operations within parent bosonic toric code stacks plus transparent fermions. Lattice stabilizer models are constructed from Pauli and Majorana (or clock/shift) operators, enforcing mutual commutativity through careful cancellation of Pauli and Majorana phases. The method yields:

• All Abelian fermionic twisted quantum doubles with single odd diagonal K-matrix entries.
• Stackable subsystem codes, potentially enabling subsystem code realization for chiral Abelian fermionic phases.

For general SPTs with non-split cyclic fermionic symmetry ($Z_2$4), the framework establishes that only condensations associated with these symmetry extensions must be constructed explicitly, with other symmetry factors handled via stacking and stable transformations.

Discussion and Implications

Majorana-Pauli stabilizer codes provide a unified lattice algebra for all Abelian fermionic phases with gapped boundaries, extending the scope of quantum error correction, exactly solvable models, and anyonic physics. The implications are multifold:

• The algebraic structure allows systematic classification, manipulation, and simulation of fermionic phases of matter, including those lacking free-fermion realizations.
• The duality web approach enables transparent analysis of phase relations via condensation and gauging, with potential for mixed-state topological order under subsystem code decoherence channels.
• Extensions to polynomial algebraic descriptions and fermionic tensor network representations are proposed, enabling efficient encoding and manipulation of stabilizer generators, excitation spectra, and ground state waves.
• The constructions open prospects for translation invariant codes, fermionic subsystem codes, and the simulation of chiral and invertible fermionic phases in quantum computation architectures.

Future directions include stabilizer construction for symmetry-enriched fermionic topological phases, systematic algebraic classification, and practical realization of quantum codes on arbitrary surfaces or with native fermionic degrees of freedom.

Conclusion

The Majorana-Pauli stabilizer codes established in this work unify disparate approaches to fermionic topological phases, encompassing toric codes, supercohomology SPTs, and their bosonic shadows within a tractable algebraic lattice framework. By leveraging hybrid Pauli and Majorana algebra, higher-order clock/shift operators, and rigorous condensation/gauging procedures, these codes bridge fermionic quantum many-body physics and quantum error correction. The framework enables both theoretical exploration and practical realization of a broad class of fermionic Abelian phases, with direct implications for quantum computing, simulation, and classification of topological matter.