Fermionic Clifford Transformations

• Fermionic Clifford transformations are algebra-preserving maps that maintain the structure of fermionic operator strings via Majorana braid conjugations, parity-preserving groups, and native second-quantized schemes.
• They are applied in topological quantum computation and tensor-network algorithms to efficiently handle fermionic statistics and preserve physical symmetries like fermion parity.
• These transformations highlight the key differences between native fermionic operations and qubit Clifford transformations, emphasizing many-body rank preservation and the role of parity superselection.

Searching arXiv for recent and foundational papers on fermionic Clifford transformations and Majorana/fermionic Clifford structures. Searching for Majorana Clifford groups, fermionic Clifford transformations, and Clifford-based fermionic formalisms on arXiv. Fermionic Clifford transformations are algebra-preserving transformations in fermionic theories that send distinguished fermionic operator strings to operator strings of the same type under conjugation, or act as internal Clifford-algebra automorphisms on fermionic generators. In the cited literature, the phrase does not designate a single universally fixed construction. It includes Majorana braid conjugations generated inside a real Clifford algebra, parity-preserving Majorana Clifford operators represented by binary orthogonal transformations, setwise stabilizers of fermionic creation–annihilation string monoids, and Grassmann-even Clifford circuits used as native fermionic disentanglers in tensor-network algorithms (Kauffman et al., 2016, Bettaque et al., 2024, Magoulas et al., 27 Oct 2025, Yosprakob et al., 5 Oct 2025).

1. Terminological scope and basic algebra

The common algebraic starting point is the fermionic decomposition into Majorana or creation–annihilation variables. For a fermionic annihilation operator $\psi$ and its adjoint $\psi^\dagger$, one has

$$\psi^2=0,\qquad (\psi^\dagger)^2=0,\qquad \psi\psi^\dagger+\psi^\dagger\psi=1.$$

For Majorana operators $c_i$, the defining relations are

$c_i^2=1,\qquad c_ic_j=-c_jc_i\quad (i\neq j),$

so a family of Majorana modes is already a real Clifford algebra. This is the point of departure for the braid-theoretic construction of Majorana exchange operators (Kauffman et al., 2016).

The literature then splits into several non-equivalent but related notions.

Setting	Elementary objects	Preservation law
Majorana braiding	$c_i$, $c_{i+1}c_i$	$T_k(x)=\tau_k x\tau_k^{-1}$
Majorana Clifford group	$\mu(v)$	$\gamma\mu(v)\gamma^\dagger=c(v)\mu(Sv)$
Fermionic Clifford group	$\psi^\dagger$0 strings	$\psi^\dagger$1
Grassmann Clifford circuits	Grassmann Pauli operators	$\psi^\dagger$2

This suggests that “fermionic Clifford transformations” is best understood as a family of preservation principles rather than a single definition. In the Majorana case, the preserved objects are Hermitian Majorana strings; in the second-quantized fermionic case they are ordered monomials in $\psi^\dagger$3; in the Grassmann tensor-network case they are Grassmann-Pauli operators acting on fermionic modes without Jordan–Wigner strings (Bettaque et al., 2024, Magoulas et al., 27 Oct 2025, Yosprakob et al., 5 Oct 2025).

A further distinction is physical admissibility. In Majorana systems, fermion parity is an unbreakable symmetry, so the physically distinguished subgroup is parity-preserving. In Grassmann tensor networks, the analogous admissibility condition is Grassmann-evenness, which enforces fermion-parity superselection. In encoded qubit approaches, by contrast, Clifford transformations are applied after Jordan–Wigner, parity, or Bravyi–Kitaev mapping and need not be native fermionic transformations in second quantization (Bettaque et al., 2024, Yosprakob et al., 5 Oct 2025, Chang et al., 10 Jun 2026).

2. Majorana exchange as a Clifford-algebra transformation

A foundational construction begins with the Majorana realization of a fermionic mode. From two Majorana operators $\psi^\dagger$4, one builds

$\psi^\dagger$5

Conversely, a fermionic mode splits into two Hermitian Majorana modes. Because distinct Majoranas anticommute and square to $\psi^\dagger$6, the operator algebra of Majorana modes is a real Clifford algebra (Kauffman et al., 2016).

The basic fermionic Clifford transformation in this setting is generated by

$\psi^\dagger$7

acting by conjugation

$\psi^\dagger$8

Its action on generators is

$\psi^\dagger$9

This is the characteristic Majorana exchange law: neighboring Majoranas are exchanged, but with a minus sign on one of them. It is therefore not a simple permutation. The cited work interprets this as the algebraic shadow of fermionic exchange statistics in the plane (Kauffman et al., 2016).

The resulting braid representation is formalized by the Clifford Braiding Theorem. For generators $$\psi^2=0,\qquad (\psi^\dagger)^2=0,\qquad \psi\psi^\dagger+\psi^\dagger\psi=1.$$0 satisfying Clifford relations, the elements

$$\psi^2=0,\qquad (\psi^\dagger)^2=0,\qquad \psi\psi^\dagger+\psi^\dagger\psi=1.$$1

form a representation of the circular Artin braid group. The proof is an explicit Clifford-algebra calculation using $$\psi^2=0,\qquad (\psi^\dagger)^2=0,\qquad \psi\psi^\dagger+\psi^\dagger\psi=1.$$2, leading to

$$\psi^2=0,\qquad (\psi^\dagger)^2=0,\qquad \psi\psi^\dagger+\psi^\dagger\psi=1.$$3

The significance is that braid statistics arise directly from the intrinsic algebra of Majorana generators rather than being imposed externally (Kauffman et al., 2016).

The three-Majorana case exhibits the quaternionic structure already present in the Clifford algebra. For

$$\psi^2=0,\qquad (\psi^\dagger)^2=0,\qquad \psi\psi^\dagger+\psi^\dagger\psi=1.$$4

one has

$$\psi^2=0,\qquad (\psi^\dagger)^2=0,\qquad \psi\psi^\dagger+\psi^\dagger\psi=1.$$5

The braid generators

$$\psi^2=0,\qquad (\psi^\dagger)^2=0,\qquad \psi\psi^\dagger+\psi^\dagger\psi=1.$$6

satisfy braid relations such as $$\psi^2=0,\qquad (\psi^\dagger)^2=0,\qquad \psi\psi^\dagger+\psi^\dagger\psi=1.$$7. In the same language, the one-parameter element

$$\psi^2=0,\qquad (\psi^\dagger)^2=0,\qquad \psi\psi^\dagger+\psi^\dagger\psi=1.$$8

acts by

$$\psi^2=0,\qquad (\psi^\dagger)^2=0,\qquad \psi\psi^\dagger+\psi^\dagger\psi=1.$$9

At $c_i$0, this reduces to the Majorana exchange

$c_i$1

while for $c_i$2, $c_i$3, it is a genuine rotation in the $c_i$4-$c_i$5 plane. In this sense, Majorana exchange is a distinguished quarter-turn generated by a Clifford bivector (Kauffman et al., 2016).

When represented on a Hilbert space, these braid elements become unitary exchange operators. For four Majoranas, the paper states that the resulting operators act on $c_i$6, satisfy the Yang–Baxter equation, and are entangling; with arbitrary single-qubit unitaries, they are universal two-qubit gates by the Brylinski criterion. The article’s topological significance is therefore double: it is both a braid-group construction and a gate-theoretic construction relevant to topological quantum computation (Kauffman et al., 2016).

3. The parity-preserving Majorana Clifford group

For $c_i$7 Majorana operators $c_i$8, the modern stabilizer-theoretic formulation packages products into Hermitian Majorana strings

$c_i$9

Their commutation law is

$c_i^2=1,\qquad c_ic_j=-c_jc_i\quad (i\neq j),$0

where $c_i^2=1,\qquad c_ic_j=-c_jc_i\quad (i\neq j),$1 is the binary symplectic form governing commutation. The full Majorana Clifford group consists of unitaries $c_i^2=1,\qquad c_ic_j=-c_jc_i\quad (i\neq j),$2 such that

$c_i^2=1,\qquad c_ic_j=-c_jc_i\quad (i\neq j),$3

with $c_i^2=1,\qquad c_ic_j=-c_jc_i\quad (i\neq j),$4 linear and symplectic; thus the full group is represented projectively by $c_i^2=1,\qquad c_ic_j=-c_jc_i\quad (i\neq j),$5 (Bettaque et al., 2024).

Fermion parity changes this structure decisively. Writing

$c_i^2=1,\qquad c_ic_j=-c_jc_i\quad (i\neq j),$6

the parity operator is

$c_i^2=1,\qquad c_ic_j=-c_jc_i\quad (i\neq j),$7

and it acts by

$c_i^2=1,\qquad c_ic_j=-c_jc_i\quad (i\neq j),$8

Even-parity Majorana strings commute with fermion parity and are the physical observables; odd strings anticommute with parity. Requiring a Clifford transformation to preserve parity of strings forces

$c_i^2=1,\qquad c_ic_j=-c_jc_i\quad (i\neq j),$9

hence

$c_i$0

The physically relevant subgroup is therefore represented not by the full binary symplectic group but by the binary orthogonal group $c_i$1 (Bettaque et al., 2024).

This yields the parity-preserving Majorana Clifford group, or p-Clifford group. Its elements are of the form

$c_i$2

with $c_i$3 and $c_i$4 even-parity. The p-Clifford group commutes with $c_i$5, preserves the even observable algebra, and respects the decomposition of Hilbert space into fixed-parity sectors. This is the paper’s central group-theoretic correction to the naive import of qubit Clifford theory into fermionic systems (Bettaque et al., 2024).

A constructive generating set is provided by even Majorana braiding operators

$c_i$6

whose induced binary action is the Householder reflection

$c_i$7

Because the binary orthogonal group is generated by such reflections, the p-Clifford group is generated, modulo even Majorana strings and excluding $c_i$8, by products of $c_i$9 and $c_{i+1}c_i$0. The same framework classifies even-parity Majorana stabilizer codes and proves that, on a fixed-parity sector, the p-Clifford frame potential equals that of the ordinary Clifford group on one fewer qubit; in that restricted sense it is a 3-design but not a 4-design (Bettaque et al., 2024).

A common misconception is that the fermionic Majorana Clifford group is simply the qubit Clifford group in another basis. Algebraically, Majorana and Pauli strings are related by a binary change of basis, but parity superselection changes the physically relevant subgroup from $c_{i+1}c_i$1 to $c_{i+1}c_i$2. The difference is therefore not merely representational (Bettaque et al., 2024).

4. Clifford transformations on fermionic creation–annihilation strings

A distinct notion arises directly in second quantization. For one mode, the fermionic monoid is

$c_{i+1}c_i$3

For $c_{i+1}c_i$4 modes,

$c_{i+1}c_i$5

Because $c_{i+1}c_i$6 and $c_{i+1}c_i$7 are nilpotent, $c_{i+1}c_i$8 is not a group. The correct fermionic analogue of a Clifford transformation is therefore the setwise stabilizer of $c_{i+1}c_i$9 inside the unitary group: a unitary $T_k(x)=\tau_k x\tau_k^{-1}$0 such that

$T_k(x)=\tau_k x\tau_k^{-1}$1

for every fermionic string $T_k(x)=\tau_k x\tau_k^{-1}$2 (Magoulas et al., 27 Oct 2025).

The main characterization is highly restrictive. Fermionic Clifford transformations are generated by Hermitian and anti-Hermitian linear combinations of half-body and pair operators,

$T_k(x)=\tau_k x\tau_k^{-1}$3

with discrete angles

$T_k(x)=\tau_k x\tau_k^{-1}$4

except that

$T_k(x)=\tau_k x\tau_k^{-1}$5

is Clifford for arbitrary $T_k(x)=\tau_k x\tau_k^{-1}$6. The resulting generating set is

$T_k(x)=\tau_k x\tau_k^{-1}$7

This is the central structural result of the cited work (Magoulas et al., 27 Oct 2025).

The completeness argument rests on many-body rank preservation. A one-body string and a two-body string have different Frobenius norms, such as

$T_k(x)=\tau_k x\tau_k^{-1}$8

so unitary conjugation cannot map one to the other. Fermionic Clifford transformations therefore preserve many-body rank, and hence preserve fermionic parity as well. This sharply distinguishes them from general Majorana Clifford transformations, which may change Majorana-string length (Magoulas et al., 27 Oct 2025).

The allowed generators realize only a few elementary moves. Half-body generators implement particle–hole conjugation on a shared index,

$T_k(x)=\tau_k x\tau_k^{-1}$9

Single-excitation generators implement index swaps while preserving operator type,

$\mu(v)$0

Pair creation–annihilation generators combine index swap with particle–hole conjugation,

$\mu(v)$1

This makes the fermionic Clifford group much smaller than the Majorana Clifford group, even though every fermionic mode can be decomposed into Majoranas (Magoulas et al., 27 Oct 2025).

The relation to Majoranas is explicit: $\mu(v)$2 so half-body and pair generators become linear combinations of one- and two-Majorana terms. Because the relevant fermionic Clifford angles are odd multiples of $\mu(v)$3, the corresponding Majorana angles are multiples of $\mu(v)$4, exactly as in ordinary Majorana Clifford theory. The converse implication does not hold: not every Majorana or qubit Clifford is Clifford in the fermionic-string sense (Magoulas et al., 27 Oct 2025).

This yields several counterintuitive consequences. The fermionic phase gate

$\mu(v)$5

is Clifford for arbitrary $\mu(v)$6, so under Jordan–Wigner the fermionic $\mu(v)$7 gate is Clifford, whereas the qubit $\mu(v)$8 gate is not Clifford on Pauli strings. Conversely, the fermionic realizations of qubit $\mu(v)$9 and CNOT are not Clifford in the fermionic sense. In qubit tapering, a transformation that is Clifford in Pauli space may cease to be Clifford in second quantization; for the $\gamma\mu(v)\gamma^\dagger=c(v)\mu(Sv)$0 example discussed in the paper, the number of fermionic strings increases from $\gamma\mu(v)\gamma^\dagger=c(v)\mu(Sv)$1 to $\gamma\mu(v)\gamma^\dagger=c(v)\mu(Sv)$2 after the tapering transformation (Magoulas et al., 27 Oct 2025).

5. Tensor-network and encoded realizations

A computationally distinct line of work embeds Clifford transformations directly into fermionic tensor-network formalisms. In a Grassmann tensor-network approach, fermionic degrees of freedom are encoded by Grassmann generators $\gamma\mu(v)\gamma^\dagger=c(v)\mu(Sv)$3 satisfying canonical anticommutation rules, and local operator algebras are represented by Grassmann analogues of Pauli matrices,

$\gamma\mu(v)\gamma^\dagger=c(v)\mu(Sv)$4

A Grassmann Clifford circuit is defined by the normalizer condition

$\gamma\mu(v)\gamma^\dagger=c(v)\mu(Sv)$5

but only Grassmann-even local unitaries are physically admissible, because they commute with total fermion parity

$\gamma\mu(v)\gamma^\dagger=c(v)\mu(Sv)$6

Imposing this evenness reduces the candidate two-site disentangler set from $\gamma\mu(v)\gamma^\dagger=c(v)\mu(Sv)$7 qubit-style Clifford representatives to $\gamma\mu(v)\gamma^\dagger=c(v)\mu(Sv)$8 allowed fermionic ones (Yosprakob et al., 5 Oct 2025).

These Grassmann-even Clifford circuits are embedded into a two-site GMPS optimization. The effective Hamiltonian is written as a sum of Grassmann-Pauli strings, candidate local Clifford circuits are tested bond by bond, and the selected circuit minimizes the bipartite entanglement across the updated bond. Because Clifford circuits map Pauli strings to Pauli strings, the Hamiltonian update

$\gamma\mu(v)\gamma^\dagger=c(v)\mu(Sv)$9

remains efficient. In the benchmarks reported, CAGMPS systematically outperforms ordinary GMPS in ground-state energy error for the $\psi^\dagger$00-$\psi^\dagger$01 model, lowers bipartite entanglement entropy across every cut, and preserves the expected central charge $\psi^\dagger$02 in the tight-binding chain. The central conceptual point is native fermionic locality: the state and the Clifford disentanglers are both encoded in Grassmann calculus, so no Jordan–Wigner strings appear (Yosprakob et al., 5 Oct 2025).

A separate line applies qubit Clifford disentanglers to fermionic molecular Hamiltonians only after fermion-to-qubit encoding. There the starting Hamiltonian is mapped by Jordan–Wigner, parity, or Bravyi–Kitaev encoding to a Pauli sum

$\psi^\dagger$03

and Clifford unitaries are used because they preserve the Pauli-string form and the number of Hamiltonian terms under conjugation,

$\psi^\dagger$04

The search over local Cliffords is reduced by classifying them by their action on the Schmidt spectrum across a bipartition, yielding $\psi^\dagger$05 equivalence classes for two-qubit Cliffords and $\psi^\dagger$06 for four-qubit Cliffords. Embedded in CAMPS, these disentanglers reduce energy errors at fixed bond dimension, mitigate sensitivity to orbital orderings and fermion-to-qubit mappings, and improve shallow-circuit VQE performance on the test cases studied (Chang et al., 10 Jun 2026).

The distinction between these two computational notions is essential. The Grassmann construction defines fermionic Clifford circuits natively, with parity and locality built into the fermionic formalism. The molecular CAMPS construction uses encoded-state qubit Clifford similarity transformations; those are explicitly described as not being native fermionic canonical transformations in second quantization and not generally preserving particle-number symmetry (Yosprakob et al., 5 Oct 2025, Chang et al., 10 Jun 2026).

6. Categorical, representation-theoretic, and algebraic generalizations

Beyond operator conjugation, Clifford transformations also appear in categorical and representation-theoretic form. A categorical boson–fermion correspondence constructs a triangulated monoidal Clifford category $\psi^\dagger$07, generated by a diagrammatic DG algebra $\psi^\dagger$08 and bimodules $\psi^\dagger$09, acting on a categorified fermionic Fock space. The categorical generators satisfy analogues of fermionic relations: $\psi^\dagger$10 and adjacent-index relations are expressed by cones. Under the charge-zero equivalence $\psi^\dagger$11, Heisenberg generators become explicit complexes of Clifford generators, categorifying the classical fact that bosonic operators are quadratic in fermions (Tian, 2017).

A different algebraic generalization starts from a finite-dimensional vector space $\psi^\dagger$12 with scalar product. The Clifford algebra

$\psi^\dagger$13

has neutral version

$\psi^\dagger$14

and the cited work identifies $\psi^\dagger$15 with fermionic statistics, or fermionic parastatistics of order $\psi^\dagger$16. It also distinguishes transformation groups: $\psi^\dagger$17 This provides a precise algebraic answer to which linear transformations preserve full Clifford relations and which preserve only their homogeneous, neutral version. The same work develops higher-order generalizations $\psi^\dagger$18 and $\psi^\dagger$19, together with Green-ansatz embeddings into tensor products of exterior or Clifford algebras (Dubois-Violette et al., 10 Feb 2026).

In the classification of gapped quadratic fermionic Hamiltonians, Clifford transformations arise on real Nambu-space multiplicity sectors. Decomposing the real Nambu space into isotypic components for the unitary symmetry subgroup produces multiplicity spaces $\psi^\dagger$20 with endomorphism algebra $\psi^\dagger$21. Antiunitary symmetries are transferred to unitary chiral symmetries, and the flattened Hamiltonian becomes an additional Clifford generator. The ten Altland–Zirnbauer symmetry classes are thereby put into one-to-one correspondence with the ten Morita equivalence classes of real and complex Clifford algebras (Abramovici et al., 2011).

These developments broaden the subject from “which unitaries map strings to strings?” to “which categorical, representation-theoretic, or symmetry-module structures make fermionic transformations Clifford?” The answer depends on context: cone relations in the categorical case, $\psi^\dagger$22 or $\psi^\dagger$23 in the algebraic-homogeneous case, and Clifford-module extensions in the Nambu-space classification of quadratic Hamiltonians (Tian, 2017, Dubois-Violette et al., 10 Feb 2026, Abramovici et al., 2011).

7. Foundational constructions and interpretive boundaries

Several foundational constructions treat fermionic creation and annihilation directly inside Clifford modules. In complex modules over real even-dimensional Clifford algebras, one introduces a Witt-type basis

$\psi^\dagger$24

satisfying

$\psi^\dagger$25

The canonical primitive idempotent

$\psi^\dagger$26

is a Hermitian primitive vacuum, and the minimal left ideal $\psi^\dagger$27 is the spinor space. In this framework, the variables $\psi^\dagger$28 and $\psi^\dagger$29 are vacuum-relative creation and annihilation operators, and different primitive idempotents define equivalent Clifford vacua (Monakhov, 2016).

Related work on second quantization in Clifford space uses two mutually anticommuting Clifford sets $\psi^\dagger$30 and $\psi^\dagger$31, with Lorentz generators

$\psi^\dagger$32

Odd Clifford products are identified with one-fermion creation operators, $\psi^\dagger$33 acts within a family, and $\psi^\dagger$34 acts between families. The cited work’s claim is that one-particle states written as products of nilpotents and projectors of odd Clifford character already satisfy the anticommutation relations required by second quantization, with the second Clifford copy supplying family structure (Borstnik et al., 2018).

These foundational programs delimit what the term does and does not mean. In Clifford-bundle bosonization and fermionization, a fermionic field can be represented by an equivalence class of bosonic Clifford-bundle fields, and a bosonic Maxwell field by an equivalence class of spinor representatives. The cited work explicitly states that this is a formal and geometric equivalence of representations, not a change of particle statistics (Jr, 2016). Likewise, in the higher-order Clifford and meson-algebra work, the phrase “fermionic Clifford transformations” is not a standard term, and the analysis concerns automorphisms, homogeneous algebras, and Green-ansatz embeddings rather than Bogoliubov transformations or a general Hamiltonian canonical-transformation theory (Dubois-Violette et al., 10 Feb 2026).

A concise summary is therefore possible. Fermionic Clifford transformations comprise several mathematically coherent but non-identical notions: Majorana braid conjugations in real Clifford algebras; parity-preserving Majorana Cliffords represented by $\psi^\dagger$35; second-quantized fermionic string stabilizers generated by half-body and pair operators; native Grassmann-even Clifford circuits in fermionic tensor networks; and categorical or module-theoretic Clifford actions in boson–fermion correspondence and quadratic-fermion symmetry classification. What unifies them is not a single presentation, but the recurring principle that fermionic structure is preserved by transformations internal to a Clifford, Majorana, or Grassmann operator calculus (Kauffman et al., 2016, Bettaque et al., 2024, Magoulas et al., 27 Oct 2025, Yosprakob et al., 5 Oct 2025, Tian, 2017, Abramovici et al., 2011).