Majorana Stabilizer Codes Overview

• Majorana stabilizer codes are fermionic analogues defined by commuting, parity-preserving Majorana operators that enforce fermion-parity superselection.
• They leverage algebraic constructions from classical self-orthogonal codes and embeddings from qubit stabilizer codes to achieve robust error correction.
• Extensions include Majorana-native architectures, Floquet schemes, and hybrid constructions that integrate Majorana defects within topological codes.

Majorana stabilizer codes are fermionic analogues of stabilizer codes in which the elementary physical operators are Majorana modes and the code space is defined by commuting, parity-preserving Majorana operators. In a broader but closely related usage, the term also covers stabilizer-style descriptions of Majorana-based qubits, Majorana defect encodings in bosonic topological codes, and hybrid Pauli–Majorana constructions for intrinsically fermionic topological order. Across these variants, the distinctive structural feature is fermion-parity superselection: even operators define measurable stabilizers, whereas odd operators model parity-changing processes such as quasiparticle poisoning and can also appear as logical operators [(Bravyi et al., 2010); (Sun et al., 23 Jun 2026)].

1. Algebraic foundations

A system of $2n$ Majorana modes is generated by Hermitian operators $c_1,\dots,c_{2n}$ obeying

$c_u c_v + c_v c_u = 2\delta_{u,v} I,$

or, in equivalent notation,

$$\gamma_j=\gamma_j^\dag,\qquad \gamma_j^2=\mathbb I,\qquad \{\gamma_j,\gamma_k\}=2\delta_{j,k}.$$

Any Majorana monomial is specified by a support set $A$ or a binary vector, and its weight is the size of that support. For supports $A,B$, the commutation rule is

$c_A c_B = (-1)^{|A|\cdot |B| + |A\cap B|} c_B c_A,$

so commuting even operators are characterized by even overlap, while odd operators introduce the specifically fermionic sign structure absent from ordinary Pauli stabilizer theory (Bravyi et al., 2010).

A Majorana stabilizer code is defined by an Abelian subgroup ${\cal S}_{\mathrm{maj}}$ of even Majorana operators with $-I\notin{\cal S}_{\mathrm{maj}}$. The code space is the joint $+1$ eigenspace of the stabilizers, equivalently the ground space of a commuting Hamiltonian $c_1,\dots,c_{2n}$0. Logical operators are elements of the centralizer $c_1,\dots,c_{2n}$1, and the distance is the minimum Majorana weight of a nontrivial logical operator (Bravyi et al., 2010).

Total fermion parity plays a special role. For $c_1,\dots,c_{2n}$2 Majoranas,

$c_1,\dots,c_{2n}$3

A code has an odd logical operator iff $c_1,\dots,c_{2n}$4. This criterion is central because parity-preserving environments can suppress odd logical processes even when the algebraic distance is small; Bravyi, Terhal, and Leemhuis therefore introduced the additional scale $c_1,\dots,c_{2n}$5, the minimum diameter of an even logical operator, to quantify protection that is invisible to the usual distance parameter (Bravyi et al., 2010).

A later algebraic reformulation expresses commuting even Clifford or Majorana stabilizers through $c_1,\dots,c_{2n}$6-isotropic subspaces of $c_1,\dots,c_{2n}$7, with

$c_1,\dots,c_{2n}$8

Such subspaces are in one-to-one correspondence with punctured all-even self-orthogonal classical codes in one higher dimension. This gives a precise classical-code dictionary for parity-preserving fermionic stabilizers and recasts error detectability as membership in $c_1,\dots,c_{2n}$9 or outside $c_u c_v + c_v c_u = 2\delta_{u,v} I,$0 (Dutta, 11 Mar 2025).

2. Relation to qubit stabilizer codes and to Majorana-like defects

Majorana stabilizer codes are closely related to ordinary stabilizer codes, but the relationship is not exhaustive. A standard embedding maps every $c_u c_v + c_v c_u = 2\delta_{u,v} I,$1 qubit stabilizer code to a Majorana fermion code on $c_u c_v + c_v c_u = 2\delta_{u,v} I,$2 modes encoding $c_u c_v + c_v c_u = 2\delta_{u,v} I,$3 logical qubits with distance $c_u c_v + c_v c_u = 2\delta_{u,v} I,$4; one realization assigns four Majoranas to each qubit and adds a local four-Majorana parity constraint to reproduce the Pauli algebra (Bravyi et al., 2010). This embedding is structurally useful, but it produces only parity-even logicals and therefore does not exploit specifically fermionic logical structure.

Conversely, not every Majorana code arises from a qubit code. Hastings showed that small genuinely fermionic codes can encode more logical qubits than any Majorana code obtained from a qubit stabilizer construction at the same number of modes and distance; examples occur already for $c_u c_v + c_v c_u = 2\delta_{u,v} I,$5 at distance $c_u c_v + c_v c_u = 2\delta_{u,v} I,$6, and at $c_u c_v + c_v c_u = 2\delta_{u,v} I,$7 for distance $c_u c_v + c_v c_u = 2\delta_{u,v} I,$8 (Hastings, 2017). This distinguishes Majorana stabilizer coding as a genuine coding-theoretic extension rather than a mere reparametrization.

A different but related usage of “Majorana stabilizer code” appears in qubit topological codes with twist defects. Matching codes are qubit stabilizer codes on trivalent lattices whose underlying anyon model is $c_u c_v + c_v c_u = 2\delta_{u,v} I,$9, the same as the surface code, but whose defect sector contains unpaired modes with Majorana fusion and braiding behavior. In that setting the “Majoranas” are extrinsic defects or unpaired endpoint modes rather than elementary physical fermions (Wootton, 2015).

A further reinterpretation applies to planar Pauli stabilizer codes themselves. Recent work reformulates planar Pauli codes, including the surface code, in terms of Majorana graphs whose undimerized defects are point-like Majoranas $$\gamma_j=\gamma_j^\dag,\qquad \gamma_j^2=\mathbb I,\qquad \{\gamma_j,\gamma_k\}=2\delta_{j,k}.$$0; logical information is stored in nonlocal fermion parities of separated defects, and conventional corner twists of a planar patch become four undimerized Majoranas (Lensky et al., 2 Jun 2026). This suggests that the modern literature uses the phrase across at least three layers: native fermionic stabilizer codes, defect-based Majorana encodings inside bosonic codes, and Majorana-native implementations of otherwise Pauli-level codes.

3. Construction methods and representative code families

A large part of the subject concerns explicit families derived from classical coding theory. Vijay and Fu developed a generic construction from weakly self-dual classical binary codes. If $$\gamma_j=\gamma_j^\dag,\qquad \gamma_j^2=\mathbb I,\qquad \{\gamma_j,\gamma_k\}=2\delta_{j,k}.$$1 is a generator matrix with $$\gamma_j=\gamma_j^\dag,\qquad \gamma_j^2=\mathbb I,\qquad \{\gamma_j,\gamma_k\}=2\delta_{j,k}.$$2, its rows define commuting parity-even Majorana stabilizers, yielding a fermion code

$$\gamma_j=\gamma_j^\dag,\qquad \gamma_j^2=\mathbb I,\qquad \{\gamma_j,\gamma_k\}=2\delta_{j,k}.$$3

This framework treats both parity-violating single-Majorana poisoning errors and parity-conserving errors on equal footing, and it produces the shortest one-qubit code correcting elementary poisoning, $$\gamma_j=\gamma_j^\dag,\qquad \gamma_j^2=\mathbb I,\qquad \{\gamma_j,\gamma_k\}=2\delta_{j,k}.$$4, together with translationally invariant families such as $$\gamma_j=\gamma_j^\dag,\qquad \gamma_j^2=\mathbb I,\qquad \{\gamma_j,\gamma_k\}=2\delta_{j,k}.$$5 and $$\gamma_j=\gamma_j^\dag,\qquad \gamma_j^2=\mathbb I,\qquad \{\gamma_j,\gamma_k\}=2\delta_{j,k}.$$6 (Vijay et al., 2017).

Hastings analyzed finite-mode optimality for small Majorana codes. For distance $$\gamma_j=\gamma_j^\dag,\qquad \gamma_j^2=\mathbb I,\qquad \{\gamma_j,\gamma_k\}=2\delta_{j,k}.$$7, he proved the bound

$$\gamma_j=\gamma_j^\dag,\qquad \gamma_j^2=\mathbb I,\qquad \{\gamma_j,\gamma_k\}=2\delta_{j,k}.$$8

and constructed Hamming-like Majorana codes saturating it whenever $$\gamma_j=\gamma_j^\dag,\qquad \gamma_j^2=\mathbb I,\qquad \{\gamma_j,\gamma_k\}=2\delta_{j,k}.$$9. Thus $A$0 yields $A$1, and $A$2 yields $A$3, with both outperforming qubit-derived alternatives at the same mode count (Hastings, 2017).

A related Clifford-theoretic family arises from $A$4-isotropic subspaces. The “Clifford Hamming” construction gives

$A$5

codes derived from the dual of the classical Hamming code, with all weight-1 and weight-2 even Clifford errors detectable when the associated dual distance is at least $A$6 (Dutta, 11 Mar 2025).

Family	Representative parameters	Distinctive feature
Weakly self-dual fermion codes	$A$7, $A$8, $A$9	Correct poisoning and parity-conserving errors (Vijay et al., 2017)
Hamming Majorana codes	$A,B$0, $A,B$1 at $A,B$2	Saturate the distance-4 bound (Hastings, 2017)
Clifford Hamming codes	$A,B$3	$A,B$4-isotropic Clifford construction (Dutta, 11 Mar 2025)

These constructions support a common inference: classical self-orthogonality remains the basic combinatorial organizing principle, but parity constraints alter both the admissible stabilizer sets and the physically relevant logical sector.

4. Majorana-native architectures and odd-error correction

One major line of work builds stabilizer codes directly from Majorana hardware. The Majorana fermion surface code uses one Majorana zero mode per site of a honeycomb lattice and six-Majorana plaquette stabilizers

$A,B$5

Its topological order is fermion-parity graded, with global relations tying products of plaquettes on the three honeycomb sublattices to total fermion parity. A central hardware claim is that each plaquette stabilizer can be measured in a single step without ancilla qubits by using charging-energy-induced quantum phase slips on mesoscopic superconducting islands (Vijay et al., 2015).

A more recent hardware-oriented development concerns the tetron architecture. Each tetron is a superconducting island hosting four Majorana zero modes $A,B$6, with fixed even total parity so that the island encodes one qubit. The experimentally natural measurements act on zero or two Majoranas per tetron, and direct four-Majorana tetron parity measurement is treated as difficult. The key result is that these restricted measurements are nevertheless sufficient to correct both bosonic even-parity errors and fermionic odd-parity errors. The construction compares two Majorana representatives of the tetron Pauli algebra,

$A,B$7

and from any bosonic $A,B$8 stabilizer code builds a tetron Majorana code

$A,B$9

whose stabilizer group contains the tetron parity operators as derived, not directly measured, stabilizers (Kundu et al., 2023). In this way odd single-Majorana errors become syndrome-visible without ever measuring the four-Majorana parity of a tetron.

The paper gives explicit examples. From the Steane code it derives a $c_A c_B = (-1)^{|A|\cdot |B| + |A\cap B|} c_B c_A,$0 code on seven tetrons, and from larger color and surface codes it obtains $c_A c_B = (-1)^{|A|\cdot |B| + |A\cap B|} c_B c_A,$1, $c_A c_B = (-1)^{|A|\cdot |B| + |A\cap B|} c_B c_A,$2, and $c_A c_B = (-1)^{|A|\cdot |B| + |A\cap B|} c_B c_A,$3 families, together with concrete syndrome-extraction schedules that obey the zero-or-two-Majoranas-per-tetron rule (Kundu et al., 2023).

A distinct locality-preserving route is the Majorana loop stabilizer code. There qubits live on edges of a fermionic hopping graph, logical fermionic operators are images of quadratic Majorana generators $c_A c_B = (-1)^{|A|\cdot |B| + |A\cap B|} c_B c_A,$4 and $c_A c_B = (-1)^{|A|\cdot |B| + |A\cap B|} c_B c_A,$5, and stabilizers are products of $c_A c_B = (-1)^{|A|\cdot |B| + |A\cap B|} c_B c_A,$6-operators around closed loops. On a two-dimensional square lattice the construction achieves distance $c_A c_B = (-1)^{|A|\cdot |B| + |A\cap B|} c_B c_A,$7 and can correct all single-qubit errors, whereas previous locality-preserving codes of the same type could only detect them (Jiang et al., 2018).

5. Clifford structure, encoding circuits, and braiding operations

The parity-preserving fermionic Clifford group is more constrained than the ordinary qubit Clifford group. For $c_A c_B = (-1)^{|A|\cdot |B| + |A\cap B|} c_B c_A,$8 Majoranas, the full Majorana Clifford action is symplectic, but the physically relevant parity-preserving subgroup is represented by the binary orthogonal group $c_A c_B = (-1)^{|A|\cdot |B| + |A\cap B|} c_B c_A,$9. In this setting even-parity braiding operators

${\cal S}_{\mathrm{maj}}$0

implement Householder reflections on binary labels, and weight-2 and weight-4 braids suffice to generate the parity-preserving Clifford group. Every even-parity Majorana stabilizer code can be reached from a canonical stabilizer by such p-Clifford operations, with two ancilla Majorana modes used to remove the obstruction associated with the fixed parity vector ${\cal S}_{\mathrm{maj}}$1 (Bettaque et al., 2024).

Encoding can therefore be posed as a fermionic Clifford synthesis problem. One algorithm takes a binary stabilizer matrix for a Majorana code and performs a Gaussian-elimination-like reduction using quadratic and quartic fermionic Clifford gates. An ancilla-assisted version works for all Majorana stabilizer codes, while an ancilla-free version works only when total parity is not itself in the stabilizer group. The stated gate complexity is ${\cal S}_{\mathrm{maj}}$2 for ${\cal S}_{\mathrm{maj}}$3 stabilizer generators on ${\cal S}_{\mathrm{maj}}$4 Majorana modes (Mudassar et al., 2024).

Logical operations can be expressed either in native Majorana language or in defect language. In matching codes, computational Majoranas are created by removing pairing stabilizers, their parity operators satisfy the expected Majorana algebra, and exchanges implement the unitary

${\cal S}_{\mathrm{maj}}$5

which is the standard Majorana braid transformation up to phase (Wootton, 2015). In the more general planar-Majorana picture, local swaps

${\cal S}_{\mathrm{maj}}$6

move neighboring Majoranas, and braid paths among point defects implement logical Clifford operations. A benchmarked two-qubit Clifford protocol based on such motion uses ${\cal S}_{\mathrm{maj}}$7 qubits rather than the ${\cal S}_{\mathrm{maj}}$8 of lattice surgery and numerically outperforms lattice surgery in the reported near-term regime (Lensky et al., 2 Jun 2026).

6. Subsystem, Floquet, and locality-preserving extensions

Not all Majorana-based error-correcting schemes are static stabilizer codes. Planar Floquet codes implemented on tetron qubits use only two-qubit Pauli measurements, but the protected subspace is generated dynamically by a time-ordered cycle of measurements rather than a fixed commuting check set. The relevant instantaneous stabilizer group changes during the measurement period, yet the syndrome data remain stabilizer-like and decode by minimum-weight perfect matching. On the Majorana measurement-only hardware model studied, the planar honeycomb and ${\cal S}_{\mathrm{maj}}$9 Floquet codes require no auxiliary qubits, have shallow measurement schedules, and exhibit thresholds in the $-I\notin{\cal S}_{\mathrm{maj}}$0–$-I\notin{\cal S}_{\mathrm{maj}}$1 range, compared with roughly $-I\notin{\cal S}_{\mathrm{maj}}$2 for the compared Majorana-surface-code implementations (Paetznick et al., 2022).

The Majorana-XYZ subsystem code occupies a different point in the landscape. It is derived from a honeycomb Majorana model with local four-Majorana interactions, mapped to a triangular-lattice qubit model with local three-body gauge checks $-I\notin{\cal S}_{\mathrm{maj}}$3 and $-I\notin{\cal S}_{\mathrm{maj}}$4. As a subsystem code it has

$-I\notin{\cal S}_{\mathrm{maj}}$5

with no local stabilizer generators but with logical operators tied to noncontractible loop sectors. As a non-gauge stabilizer code on the same qubits it encodes asymptotically $-I\notin{\cal S}_{\mathrm{maj}}$6 logical qubits, but then the check operators have weight $-I\notin{\cal S}_{\mathrm{maj}}$7 (Busse et al., 27 Mar 2026).

The Majorana loop stabilizer code also belongs in this extension class because it combines locality-preserving fermion-to-qubit mapping with nontrivial code distance. Its stabilizers are directly inherited from closed-path relations among quadratic Majorana operators on the hopping graph, so the code subspace is defined by fermionic loop consistency rather than by an arbitrary Pauli presentation (Jiang et al., 2018).

7. Hybrid fermionic phases and broader generalizations

The stabilizer paradigm now extends beyond free-fermion-style Majorana codes. Majorana-Pauli stabilizer codes are exactly solvable fermionic lattice models whose stabilizers mix generalized Pauli operators and Majorana operators. Their main example is a stabilizer realization of the fermionic toric code using $-I\notin{\cal S}_{\mathrm{maj}}$8 edge qudits and plaquette Majoranas; within that framework the anyons, string operators, fusion rules, and braiding statistics all follow from the stabilizer algebra. The construction is then extended to all Abelian fermionic topological orders with gapped boundaries and to all supercohomology fermionic SPT phases in $-I\notin{\cal S}_{\mathrm{maj}}$9 dimensions (Sun et al., 23 Jun 2026).

Projection methods provide another bridge between Majorana stabilizer states and bosonic topological order. For free-fermion, nondegenerate, parity-even Majorana stabilizer groups with four Majoranas per site, Gutzwiller projection onto the physical spin Hilbert space yields an exact qubit stabilizer description. In particular, a square-lattice Majorana dimer code projects to the Wen plaquette model and hence to $+1$0 topological order, while a trivial on-site pairing code projects to a polarized product state. The same work shows that adiabatically connected free-fermion Majorana codes can project to distinct spin phases, so the projected topological order cannot be read off solely from the unprojected free-fermion phase (Macedo et al., 5 May 2025).

The parafermion generalization places Majorana codes in a wider hierarchy. Parafermion stabilizer codes specialize to Majorana stabilizer codes at $+1$1, preserve the same emphasis on parity-conserving stabilizers, and introduce the parity-aware logical length scale

$+1$2

for the minimum diameter of a parity-preserving logical operator. That formulation makes explicit that Majorana codes are the $+1$3 endpoint of a broader stabilizer theory in which superselection can supplement ordinary code-distance protection (Güngördü et al., 2014).

Taken together, these developments show that “Majorana stabilizer code” no longer names a single narrowly defined object. It denotes a family of fermionic coding frameworks unified by commuting parity constraints, Majorana algebra, and topological nonlocality, but diversified across native fermionic codes, Majorana-native qubit architectures, subsystem and Floquet schemes, defect-based interpretations of bosonic codes, and hybrid stabilizer realizations of intrinsically fermionic topological phases.