Majorana-Pauli Stabilizer Codes

• Majorana-Pauli stabilizer codes are frameworks that encode Majorana zero modes and Ising-anyon fusion channels using commuting even fermionic operators and Pauli measurements.
• They bridge qubit and fermionic models through matching, hybrid, and algebraic constructions, enabling simulation of topological orders such as the toric and fermionic toric codes.
• Classical-code correspondences and parity-preserving Clifford operations provide systematic methods for encoding, logical operations, and error correction in intrinsically fermionic phases.

Searching arXiv for relevant papers on Majorana-Pauli stabilizer codes and closely related Majorana/Ising-anyon stabilizer frameworks. Majorana-Pauli stabilizer codes are stabilizer constructions in which Majorana or Ising-anyon degrees of freedom are represented within a Pauli stabilizer formalism, or more generally by commuting even fermionic operators and hybrid Pauli–Majorana stabilizers. In one important realization, they are qubit stabilizer codes whose defect or subsystem degrees of freedom behave as Majorana zero modes and whose braiding is implemented by Pauli measurements or local unitaries. In another, they are fermionic stabilizer codes built directly from even products of Majorana operators. In a more recent hybrid formulation, the stabilizers combine generalized Pauli operators with Majorana modes to realize intrinsically fermionic topological phases (Wootton, 2015, Dutta, 11 Mar 2025, Sun et al., 23 Jun 2026).

1. Conceptual scope and terminology

The term encompasses several closely related but not identical frameworks. A broad operational definition is a stabilizer or subsystem code in which Majorana operators, Majorana parities, or Ising-anyon fusion channels are the protected degrees of freedom, while the control, measurement, or exact solvability is expressed in Pauli language. In the Pauli-only version, Majorana modes are emergent: they arise as defect or subsystem variables encoded nonlocally in qubit operators. In the fermionic version, the basic operators are Majorana monomials or even Clifford operators. In the hybrid version, generalized Pauli operators on qudits and Majorana operators on fermions appear in the same stabilizer algebra (Wootton, 2015, Bravyi et al., 2010, Sun et al., 23 Jun 2026).

A recurrent source of ambiguity is that “Majorana-Pauli stabilizer code” can denote either a qubit stabilizer code with effective Majorana zero modes or a genuinely fermionic stabilizer code. The distinction matters physically. The former can simulate Majorana braiding and fusion without requiring physical Majorana fermions; the latter incorporates fermion parity at the operator-algebra level. A further distinction is between codes that realize the bosonic $D(\mathbb Z_2)$ phase with twist defects and codes that realize intrinsically fermionic topological order. The first class includes matching codes and Majorana surface or color codes; the second includes the fermionic toric code in the hybrid Pauli–Majorana formalism (Wootton, 2015, Litinski et al., 2018, Sun et al., 23 Jun 2026).

Fermion parity is the common structural theme. In Majorana stabilizer codes, physical operations must commute with total parity, so stabilizers are even products of Majoranas. In qubit-based Majorana-Pauli encodings, parity reappears as a protected fusion or gauge degree of freedom. This parity constraint is not a cosmetic difference from ordinary Pauli stabilizer theory: it affects which Clifford operations are physical, which logical operators are observable, and how braiding protocols are implemented (Bravyi et al., 2010, Bettaque et al., 2024).

2. Matching codes and emergent Majoranas in qubit stabilizer models

A canonical Pauli realization is the family of matching codes introduced as qubit topological stabilizer codes on trivalent, edge-3-colored lattices. Qubits sit on vertices; each edge $l=(j,k)$ carries a label $\alpha\in\{x,y,z\}$, and one defines the two-qubit link operator

$K_l=\sigma^\alpha_j\sigma^\alpha_k.$

Plaquette stabilizers are loop products

$W_p \sim \prod_{l\in p} K_l,$

and a chosen matching $M$ of the vertices defines string stabilizers

$S_{j,k}\sim \prod_{l\in P(j,k)} K_l.$

The stabilizer group is generated by all $W_p$ and all $S_{j,k}$. Although there are $N$ such generators for $l=(j,k)$0 qubits, only $l=(j,k)$1 are independent, so on a torus the full construction is a two-qubit stabilizer code (Wootton, 2015).

The emergent anyon theory is $l=(j,k)$2. Violations of plaquette stabilizers realize the bosonic $l=(j,k)$3 and $l=(j,k)$4 charges, violations of string stabilizers realize the fermion $l=(j,k)$5, and the fusion rules are

$l=(j,k)$6

In this sense matching codes are a geometrically general Pauli stabilizer realization of the same anyon model as toric and surface codes. The nontrivial step is that the same construction also admits a Majorana interpretation: in the honeycomb mapping, each vertex carries a Majorana $l=(j,k)$7, plaquette operators live purely in the gauge sector, and in the no-plaquette-excitation subspace each string stabilizer is effectively a Majorana parity

$l=(j,k)$8

Removing a chosen $l=(j,k)$9 from the stabilizer group unpairs the corresponding Majoranas and produces Majorana zero modes as protected subsystem degrees of freedom (Wootton, 2015).

This realizes twist physics without geometric dislocations. In ordinary surface-code language, twist defects permute $\alpha\in\{x,y,z\}$0 and $\alpha\in\{x,y,z\}$1 and bind Majorana zero modes. In matching codes, the Majoranas are “always present” at vertices, and unpaired modes are created algebraically by omitting or modifying string stabilizers. Braiding can then be implemented by sequences of link-operator measurements. For neighboring computational Majoranas one obtains the standard braid unitary

$\alpha\in\{x,y,z\}$2

up to global phase (Wootton, 2015).

The same paper gives a minimal proof-of-principle demonstration on three qubits. After reduction from a six-qubit honeycomb fragment, one takes

$\alpha\in\{x,y,z\}$3

Preparing $\alpha\in\{x,y,z\}$4-basis states, performing two entangling two-qubit measurements, and measuring in the $\alpha\in\{x,y,z\}$5 basis reproduces the Majorana exchange rules on the fusion basis. This construction is often the clearest example of what “Majorana-Pauli” means in the Pauli-only setting: non-Abelian Majorana braiding emerges entirely from qubits and Pauli measurements (Wootton, 2015).

3. Algebraic and group-theoretic formulations

The fermionic formulation begins with $\alpha\in\{x,y,z\}$6 Majorana operators $\alpha\in\{x,y,z\}$7 or $\alpha\in\{x,y,z\}$8 satisfying the real Clifford algebra

$\alpha\in\{x,y,z\}$9

A Majorana stabilizer code is defined by an Abelian subgroup of even Majorana monomials that does not contain $K_l=\sigma^\alpha_j\sigma^\alpha_k.$0; the code space is their common $K_l=\sigma^\alpha_j\sigma^\alpha_k.$1 eigenspace. In binary form, a Majorana monomial is represented by a vector in $K_l=\sigma^\alpha_j\sigma^\alpha_k.$2, and commutation is encoded by a fermionic symplectic form or, in the Clifford formulation of Ising-anyon stabilizers, by the bilinear form

$K_l=\sigma^\alpha_j\sigma^\alpha_k.$3

A subspace is $K_l=\sigma^\alpha_j\sigma^\alpha_k.$4-isotropic iff $K_l=\sigma^\alpha_j\sigma^\alpha_k.$5 for all $K_l=\sigma^\alpha_j\sigma^\alpha_k.$6 in it, and this is exactly the condition that the associated even Clifford operators commute (Dutta, 11 Mar 2025).

This algebraic viewpoint yields a classification theorem: $K_l=\sigma^\alpha_j\sigma^\alpha_k.$7-isotropic subspaces in $K_l=\sigma^\alpha_j\sigma^\alpha_k.$8 are exactly punctured images of all-even, self-orthogonal binary codes in $K_l=\sigma^\alpha_j\sigma^\alpha_k.$9. Equivalently, commuting families of even Majorana or Clifford stabilizers are in one-to-one correspondence with punctured self-orthogonal classical codes of one higher length. The note of 2025 recasts and unifies earlier constructions of Bravyi and Vijay–Fu in this language, making the classical-coding input explicit (Dutta, 11 Mar 2025).

A complementary formulation uses a stabilizer matrix and parity-preserving fermionic Clifford gates. “Encoding Majorana codes” represents each stabilizer generator by a binary column and performs a Gaussian-elimination-like reduction using elementary fermionic Clifford operations, notably two-Majorana and four-Majorana gates. One universal algorithm uses two ancilla modes and works for all Majorana stabilizer codes; an ancilla-free algorithm exists when the total parity operator is not in the stabilizer group. Both have $W_p \sim \prod_{l\in p} K_l,$0 gate complexity for $W_p \sim \prod_{l\in p} K_l,$1 Majoranas and $W_p \sim \prod_{l\in p} K_l,$2 generators (Mudassar et al., 2024).

The parity-preserving Clifford group admits an especially sharp description. “The Structure of the Majorana Clifford Group” shows that parity-preserving fermionic Cliffords are represented by the orthogonal group $W_p \sim \prod_{l\in p} K_l,$3, rather than the full binary symplectic group. In this language, physical braid generators are

$W_p \sim \prod_{l\in p} K_l,$4

with $W_p \sim \prod_{l\in p} K_l,$5 an even Majorana string, and the resulting p-Clifford group can generate any even-parity Majorana stabilizer code. The same paper proves that the parity-restricted frame potential of the p-Clifford group equals the frame potential of the ordinary Clifford group acting on a fixed-parity sector, so on a fixed-parity sector it has the same 3-design behavior as the usual qubit Clifford group (Bettaque et al., 2024).

4. Classical-code correspondences and small-code families

The relation to classical coding theory is not merely formal. Bravyi’s original Majorana fermion codes and later work on small Majorana codes show that self-orthogonal binary codes can yield fermionic codes with parameters unavailable from direct qubit-to-Majorana lifts. In Bravyi’s construction, any $W_p \sim \prod_{l\in p} K_l,$6 qubit stabilizer code can be mapped to a Majorana fermion code on $W_p \sim \prod_{l\in p} K_l,$7 Majoranas with the same number of logical qubits and doubled distance $W_p \sim \prod_{l\in p} K_l,$8. The same framework also shows how to transform any qubit stabilizer code to a weakly self-dual CSS code (Bravyi et al., 2010).

Hastings studied small Majorana fermion codes under the assumption that fermion parity is included in the stabilizer group, so all logical operators have even weight and the distance must be even. For distance $W_p \sim \prod_{l\in p} K_l,$9, he derived the upper bound

$M$0

where $M$1 is the number of physical Majorana modes and $M$2 the number of logical qubits. He also constructed “Hamming Majorana codes” that saturate this bound when $M$3, with

$M$4

These are distance-4 Majorana stabilizer codes analogous to classical Hamming codes, and some of them encode more logical qubits than any Majorana code obtained from a qubit stabilizer code by the standard $M$5-Majorana-per-qubit mapping (Hastings, 2017).

The classical correspondence appears again in the 2025 classification note. A $M$6-isotropic subspace $M$7 defines a Clifford stabilizer code $M$8 with $M$9, and explicit families can be built from punctured all-even self-orthogonal classical codes. The paper’s Clifford Hamming construction yields

$S_{j,k}\sim \prod_{l\in P(j,k)} K_l.$0

codes with distance at least $S_{j,k}\sim \prod_{l\in P(j,k)} K_l.$1, detecting all weight-1 and weight-2 even Clifford errors (Dutta, 11 Mar 2025).

A plausible implication is that “Majorana-Pauli stabilizer code” should not be restricted to geometrically defined lattice models. The classical-code route shows that purely algebraic constructions, including high-rate families and punctured self-orthogonal codes, belong to the same subject whenever commuting even Clifford operators or parity-preserving Pauli surrogates define the code space (Dutta, 11 Mar 2025).

5. Topological, subsystem, and locality-preserving realizations

The topological branch of the subject includes Majorana surface codes, Majorana color codes, Majorana loop stabilizer codes, and subsystem constructions derived from interacting Majorana Hamiltonians. Litinski and von Oppen formulated Majorana surface codes on 3-colorable tilings with Majoranas on vertices and face stabilizers given by products of all Majoranas around a face. Tetrons realize $S_{j,k}\sim \prod_{l\in P(j,k)} K_l.$2 codes, hexons realize $S_{j,k}\sim \prod_{l\in P(j,k)} K_l.$3 codes, and larger 2D patches yield logical tetrons and hexons whose logical Pauli operators are even products of Majoranas along colored strings. Their framework implements all logical Clifford gates with zero time overhead by Pauli product measurements or by twist-based lattice surgery using only local few-Majorana parity measurements (Litinski et al., 2018).

The same work introduced Majorana color codes by concatenating Majorana surface codes with small Majorana fermion codes. Representative examples include the 4.8.8 $S_{j,k}\sim \prod_{l\in P(j,k)} K_l.$4 Majorana color code, the 4.8.8 $S_{j,k}\sim \prod_{l\in P(j,k)} K_l.$5 code, and the 4.8.8 $S_{j,k}\sim \prod_{l\in P(j,k)} K_l.$6 code. Compared with bosonic color codes at matched stabilizer weight, these constructions reduce Majorana overhead while preserving a surface-code-like measurement structure (Litinski et al., 2018).

For fermionic simulation, the Majorana loop stabilizer code defines stabilizers as products of Majorana edge operators around closed paths of the hopping graph. On a 2D square lattice it uses one qubit per edge, keeps encoded number operators at weight $S_{j,k}\sim \prod_{l\in P(j,k)} K_l.$7 and nearest-neighbor hopping at weight $S_{j,k}\sim \prod_{l\in P(j,k)} K_l.$8 or $S_{j,k}\sim \prod_{l\in P(j,k)} K_l.$9, and achieves code distance $W_p$0. The paper proves that on the 2D square lattice the MLSC can correct all single-qubit errors, whereas previous geometric locality-preserving codes such as BKSF can only detect all single-qubit errors on the same lattice (Jiang et al., 2018).

The subsystem branch is exemplified by the Majorana-XYZ code, derived from the Li–Franz Majorana Hamiltonian on the honeycomb lattice. After dimerization and a Jordan–Wigner-type mapping, quartic Majorana interactions become local 3-qubit Pauli triangle operators $W_p$1 and $W_p$2 on a triangular lattice. As a subsystem code it has parameters

$W_p$3

with 3-local nearest-neighbour physical check operators. The stabilizer group is generated by nonlocal double loops, the gauge group by local triangle operators, and logical operators are winding double strings around the torus. The code therefore combines topological logical operators with a macroscopically large gauge subsystem (Busse et al., 27 Mar 2026).

These realizations also clarify a common misconception. Local physical checks do not imply local stabilizer generators. The Majorana-XYZ code has local 3-body gauge checks but nonlocal stabilizers of weight $W_p$4, while Majorana surface codes have local face stabilizers but their logical operators are noncontractible strings. The Majorana-Pauli label therefore covers both stabilizer and subsystem architectures, provided the protected degrees of freedom are Majorana-like and the algebra is controlled by Pauli or hybrid Pauli–Majorana measurements (Litinski et al., 2018, Busse et al., 27 Mar 2026).

6. Intrinsically fermionic stabilizer phases and duality webs

The 2026 paper “Majorana-Pauli stabilizer codes and duality webs of fermionic topological phases” gives the term its most literal meaning: stabilizers are built simultaneously from generalized Pauli operators on qudits and Majorana operators on physical fermions. In this framework, the Hilbert space is

$W_p$5

and commuting, local, fermion-parity-even stabilizer generators define an exactly solvable commuting-projector Hamiltonian. The main example is an exactly solvable stabilizer realization of the fermionic toric code using $W_p$6 qudits on edges and one complex fermion on each plaquette (Sun et al., 23 Jun 2026).

In that construction, the vertex terms $W_p$7, plaquette terms $W_p$8, and edge condensate terms $W_p$9 generate the stabilizer group. The resulting theory realizes $S_{j,k}$0, with anyons $S_{j,k}$1, $S_{j,k}$2, $S_{j,k}$3, and the transparent physical fermion $S_{j,k}$4, satisfying

$S_{j,k}$5

and topological spin

$S_{j,k}$6

String operators, fusion rules, and braiding phases follow directly from the stabilizer algebra: for example, $S_{j,k}$7 and $S_{j,k}$8 have mutual phase $S_{j,k}$9, while $N$0 braids trivially (Sun et al., 23 Jun 2026).

The same paper embeds the fermionic toric code in a duality web generated by anyon condensation and by gauging bosonic or fermion-parity symmetries. The web connects bosonic topological orders, symmetry-enriched topological phases, fermionic and bosonic symmetry-protected topological phases, and twisted gauge theories, all within a common stabilizer description. A central claim is that the construction extends to all Abelian fermionic topological orders with gapped boundaries and to all supercohomology fermionic SPT phases in $N$1 dimensions (Sun et al., 23 Jun 2026).

Going beyond order-2 Majoranas, the paper introduces fermionic clock and shift operators for $N$2 symmetries with even $N$3. For $N$4, one defines local operators $N$5 satisfying

$N$6

together with clock-shift commutation relations. This yields an exact bosonization map for $N$7 symmetries and supports, among other examples, a stabilizer model for a nontrivial $N$8 fermionic SPT phase with no free-fermion analog (Sun et al., 23 Jun 2026).

This hybrid formalism separates two issues that were often conflated. A Pauli code with emergent Majorana defects need not realize intrinsically fermionic topological order, and an intrinsically fermionic stabilizer model need not reduce to a defect picture in a bosonic code. The 2026 framework places both within a broader stabilizer-code paradigm while keeping the distinction explicit (Sun et al., 23 Jun 2026).

7. Encoding, logical operations, and limitations

Preparation and manipulation of Majorana-Pauli codes require parity-preserving encoders and fault-tolerant gate primitives. For direct Majorana stabilizer codes, unitary encoding circuits can be computed from the stabilizer matrix by fermionic Gaussian elimination. One algorithm uses an additional ancilla mode pair and works for all Majorana stabilizer codes; a second, ancilla-free algorithm works only when the total parity operator is not contained in the stabilizer group. Both use elementary fermionic Clifford operations in place of ordinary row operations, and both produce encoding circuits with $N$9 gate complexity (Mudassar et al., 2024).

Logical Clifford operations can be implemented either through measurement-based protocols or through explicit motion of Majorana degrees of freedom. In planar Pauli stabilizer codes re-expressed microscopically in terms of Majorana particles, logical information is stored in pairwise fermion parities of undimerized Majoranas. “Practical gates by Majorana fermion motion” develops fault-tolerant Majorana motion, dense memory layouts, and braiding-based logical gates in this language. For 2-qubit Clifford gates it reports that the braiding-based protocol uses $l=(j,k)$00 physical qubits, compared with $l=(j,k)$01 for lattice surgery, and finds numerically that the protocol outperforms lattice surgery for near-term error rates and realistic device constraints (Lensky et al., 2 Jun 2026).

The role of parity gives Majorana codes a distinctive protection profile but does not remove standard locality constraints. Bravyi’s 2010 analysis emphasizes that two-dimensional local Majorana codes still have string-like even logical operators and therefore do not evade the usual no-go logic for thermally self-correcting memories in two dimensions. What parity conservation adds is a second protection scale: odd logical operators may be forbidden physically, so even when the smallest logical operator is odd and low-weight, the smallest even logical operator can be geometrically large. Kitaev’s one-dimensional chain is the basic example, with $l=(j,k)$02 but $l=(j,k)$03 (Bravyi et al., 2010).

A further limitation is conceptual rather than physical. Stabilizer methods are powerful, but they do not exhaust all code possibilities even in Pauli language. The negative Ramsey-type result for Pauli channels shows that some quantum graphs admit nontrivial cliques or anticliques but no nontrivial ones that are stabilizer codes. A plausible implication is that the same caution applies to Majorana-Pauli settings under fermion-to-qubit mappings: stabilizer descriptions are broad and systematic, but not universal (Bousba et al., 2020).

Taken together, these developments define a mature research area rather than a single code family. Matching codes show how Majorana defects emerge inside bosonic Pauli topological order; Majorana stabilizer and Clifford formalisms provide the fermionic algebraic backbone; surface, color, loop, and subsystem constructions supply concrete architectures; and hybrid Pauli–Majorana stabilizers extend the paradigm to intrinsically fermionic topological phases. The unifying idea is that Majorana parity, fusion, and braiding can be encoded, classified, and manipulated within stabilizer theory, while fermion parity remains the organizing symmetry across all variants (Wootton, 2015, Bettaque et al., 2024, Sun et al., 23 Jun 2026).