Generating non-Clifford gate operations through exact mapping between Majorana fermions and $\mathbb{Z}_4$ parafermions

Ali Hamed Safwan; Raditya Weda Bomantara

REVIEW 2 major objections 5 minor 79 references

Generating non-Clifford gate operations through exact mapping between Majorana fermions and $\mathbb{Z}_4$ parafermions

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Braiding Majorana fermions and Z4 parafermions together can generate non-Clifford quantum gates in either representation, pointing toward topologically protected universal quantum computation.

desk verdict Solid algebraic observation about cross-representation braids, but the universality claim is asserted rather than proven. read the letter →

arxiv 2411.18736 v3 pith:UTUZMG3A submitted 2024-11-27 quant-ph cond-mat.mes-hall

Ali Hamed Safwan , Raditya Weda Bomantara This is my paper · ORCID

classification quant-phcond-mat.mes-hall MSC 81P6881P45

keywords MajoranafermionsZ4parafermionstopologicalquantumcomputationnon-Cliffordgatesanyonbraidingquditcomputingparitymeasurementexactmapping

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Topological quantum computing protects qubits by storing information nonlocally, but the braiding operations available are known to generate only Clifford gates, which are classically simulable. This paper develops an exact algebraic mapping between two anyon species—Majorana fermions, which pair into qubits, and Z4 parafermions, which pair into four-dimensional qudits—and shows that a braid of one species becomes a non-Clifford gate when read in the other species' representation. Majorana braids yield non-Clifford qudit gates, and Z4 parafermion braids yield non-Clifford qubit gates such as the T gate, a controlled-S gate, and controlled rotations. Since each species' own braids remain Clifford in its native representation, the known limitation is not violated; what the mapping makes possible is a universal gate set obtained by combining both braid families in one device. The paper also supplies parity-measurement protocols that enact these braids, with timing estimates based on current experiments.

What carries the argument

The load-bearing object is the algebraic dictionary between the Jordan-Wigner representation of Majoranas, $\gamma_{2j-1}=\sigma_y^{(j)}\prod_{i<j}\sigma_z^{(i)}$ and $\gamma_{2j}=\sigma_x^{(j)}\prod_{i<j}\sigma_z^{(i)}$, and the Fradkin-Kadanoff representation of Z4 parafermions, $\psi_{2j-1}=T_y^{(j)}\prod_{i<j}T_z^{(i)}$ and $\psi_{2j}=T_x^{(j)}\prod_{i<j}T_z^{(i)}$. Combining these transformations gives explicit identities such as $T_z=\sigma_z^{(2)}[(1+\sigma_z^{(1)})/2+i(1-\sigma_z^{(1)})/2]$ and $T_x=(\sigma_x^{(1)}+i\sigma_y^{(1)})/2+\sigma_x^{(2)}(\sigma_x^{(1)}-i\sigma_y^{(1)})/2$, together with the inverse formulas expressing each Majorana operator as a polynomial in parafermions. These identities are what allow a braiding operator, which is defined by how it permutes one species, to be re-expressed in the other species' operators, where its non-Clifford action on that representation becomes visible.

What would settle it

In a device that realizes the periodically driven Majorana-chain model of Ref. [52], implement the four-step parity-measurement protocol for braiding a pair of Z4 parafermions and tomographically reconstruct the logical unitary on the two-qubit Majorana subspace; the central claim predicts a non-Clifford gate containing a T gate with the specific outcome probabilities of Sec. V, so observing a Clifford gate (or gate fidelity that collapses when the interaction strength is varied) would falsify it.

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Extended reading notes

Core claim

The paper's central discovery is that an exact mapping exists between the operator algebras of four Majorana fermions and two Z4 parafermions, and between six Majorana fermions and four Z4 parafermions under conserved total parity, and that this mapping transfers non-Clifford character across the two representations. In the minimal parity-free setting, the generalized Pauli matrices of the parafermion qudit are written in terms of the two-qubit Pauli matrices, and the Majorana operators are expressed as polynomials in the parafermions. The Majorana braid $U^{(MF)}_{i,j}=(I+\gamma_i\gamma_j)/\sqrt2$, a Clifford gate on the two-qubit space, becomes, in the qudit picture, an operator that conjugates $T_x$ into $\frac{1}{2}(T_x+T_x^3+iT_z^2(T_x-T_x^3))$, hence lies outside the four-dimensional Clifford group; explicitly, $U^{(MF)}_{3,4}=S_4^\dagger H_4^2 T_z^3 S_4^6 R_4(-3\pi/4) H_4^2$. Conversely, the single Z4 parafermion braid $U^{(P)}=\frac12(I+\psi_2^3\psi_1+\psi_2^2\psi_1^2+\psi_2\psi_1^3)$, a Clifford gate on the qudit, maps to a two-qubit gate containing the T gate, and in the parity-conserving setting the parafermion braids contain controlled-S and controlled-rotation gates. The paper concludes that braiding Majorana fermions and braiding Z4 parafermions are complementary rather than redundant: combined, they generate a universal gate set, while each family alone remains confined to its own Clifford group.

Load-bearing premise

The physical route to universality assumes that Z4 parafermions can be formed from Majorana fermions in a tunable way and that both Majorana and parafermion braids can be performed in the same device; the paper's implementation rests on a periodically driven chain proposal and a geometric (not strictly topological) adiabatic protocol, with no explicit Hamiltonian or noise analysis for the coupling step.

Editorial extensions

If this is right

A Majorana-based topological quantum computer can be made universal without magic-state injection: form Z4 parafermions from the same Majoranas and braid them, and the resulting gates include the missing T gate.
A Z4-parafermion qudit computer gains universality by supplementing parafermion braids with Majorana braids; in the parity-conserving setting the parafermion braids supply entangling non-Clifford operations such as controlled-S and controlled rotations.
The exact mapping holds both without parity conservation (four Majoranas to two parafermions) and with it (six Majoranas to four parafermions), so the universality route survives in closed systems with fixed total parity.
Both braid families can be enacted by sequences of four parity measurements with ancillas: the Majorana protocol gives the desired braid directly with probability 1/4 and reaches 99.9% within 92 measurements, while the parafermion protocol gives it with probability 1/16 and reaches 98% within 248 measurements.
The classic result that braiding alone yields only Clifford gates is preserved for each species in its own representation; the non-Clifford gates appear only when the two representations are combined.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If a hybrid Floquet platform hosting Majorana and Z4 parafermion edge modes simultaneously is realized, the non-Clifford gates obtained this way would inherit braiding's topological protection, since the only non-topological step in the paper's implementation proposal is the geometric adiabatic coupling between the two sectors.
The same mechanism is likely to generalize beyond Z4: the mapping's nonlinearity (powers of operators and parity projectors) is what turns a Clifford braid into a non-Clifford one, so exploring Z3 or Z8 parafermions might reveal non-Clifford cross-representation gates with different qudit dimensions, including a possible link to Fibonacci anyons.
A direct numerical test of the periodically driven chain of Ref. [52] could quantify how gate fidelity degrades as interaction strength is tuned, since the paper's minimal-system analysis does not include error rates.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

First thing to know: the core algebraic result here is real. The authors write down exact mappings between four Majoranas and two Z4 parafermions (and six Majoranas with four parafermions under parity conservation), and they show explicitly that braiding operators in one representation can decompose into non-Clifford gates in the other. That cross-representation observation is new as far as I can tell, and it is not a trivial corollary of the mapping; the decompositions in Eqs. (29) and (31) are concrete and checkable. The paper is worth reading for that.

The appendices help, and the measurement-based braiding protocol in Sec. V is a genuine plus. The authors give explicit operator sequences and estimate success probabilities and timescales, which makes the proposal more tangible than most mapping papers.

Where I part company: the advertised route to universal topological quantum computing is not established. The stress-test note is right. Exhibiting a braid whose decomposition contains a T gate or a rotation by -3π/4 does not prove that the full available gate set (Majorana braids plus parafermion braids) generates a dense subgroup. Non-Cliffordness is not enough; the generated group could still be finite. For qubits, Clifford+T universality requires that the Clifford part actually be available and that the T is the right T; the paper does not verify that the Clifford gates appearing in its decompositions are all realizable by braiding in their setting. For the 4D qudit, no analogue of the Clifford+T theorem is cited or proved. So 'suggests universality' is honest; 'may be possible' is honest; but the abstract's phrasing could lead readers to think more is proven.

The second soft spot is physical. The paper itself concedes in Sec. VI that coupling the Majorana and parafermion sectors requires a geometric, non-topological adiabatic protocol whose construction is beyond its scope. That is a real caveat, and I appreciate them stating it. It means the 'topologically protected' part of the story is not there yet.

Minor issues: the notation s2 is used for both a measurement outcome and s^2 in places, and the appendix proof in App. B has a typo ('two orthogonal eigenstates' for a 4x4 matrix). Nothing load-bearing.

Verdict: this deserves a serious referee. The algebraic mapping and the non-Clifford decompositions are solid and should be published, but the authors should be asked to either prove or explicitly leave open the universality question, and to temper the abstract. I'd send it to review.

Referee Report

2 major / 5 minor

Summary. The paper establishes explicit algebraic mappings between systems of Majorana fermions and Z4 parafermions, in both an unconstrained four-Majorana/two-parafermion setting and a parity-conserving six-Majorana/four-parafermion setting. It expresses Majorana parity, projector, and braiding operators in terms of parafermions and vice versa, then computes the corresponding quantum gates in both the two-qubit Pauli representation and the 4D generalized-Pauli (qudit) representation. The central observation is that while braiding Majorana fermions (respectively Z4 parafermions) generates only Clifford gates in their native representation, the same braid operators become non-Clifford gates when viewed in the other representation. Examples include a 4D rotation R4(-3π/4) in Eq. (29) and a T gate in Eq. (31). The paper also presents measurement-based protocols for braiding both types of quasiparticles and discusses possible physical implementations via periodically driven systems.

Significance. If the algebraic claims are correct, the paper provides a concrete counterexample to the common expectation that braiding of Majorana fermions or Z4 parafermions can only produce Clifford operations, and it opens a potential route to topological quantum universality by hybrid braiding. The derivation is explicit and checkable: the mappings in Eqs. (20)-(27) and Eqs. (36)-(41) are written out in full, and the non-Clifford decompositions in Eqs. (29), (31), (43), and (45) are concrete. The paper is self-contained and does not rely on fitted parameters or circular reasoning. The main weakness is that the passage from 'these gates are non-Clifford' to 'the generated gate set is universal' is asserted rather than proved, especially for the 4D qudit case. The physical implementation section also honestly discloses that coupling the two sectors requires a non-topological geometric protocol, so the advertised 'topologically protected universal computing' remains an algebraic suggestion rather than a demonstrated physical scheme.

major comments (2)

[Sec. III.C and Sec. IV.C] The paper claims that the exhibited non-Clifford gates lead to universal quantum computation, but it never proves that the braid-generated Clifford subgroup together with these specific non-Clifford gates generates a dense subgroup of the relevant unitary group. For the qubit case, the claim is repairable: Eq. (28) already shows that H, S, and CNOT are generated by Majorana braids (hence the full Clifford group is generated), and Eq. (31) expresses a Clifford conjugate of the T gate, so the standard 'Clifford + T' universality theorem applies. This argument should be stated explicitly, with a citation. For the 4D qudit case, Eq. (29) only shows that U_{3,4}^{(MF)} contains a factor R4(-3π/4); since a non-Clifford gate can still lie in a finite subgroup, the authors must either prove (or cite a proof) that the qudit Clifford group together with R4(-3π/4) generates a dense subgroup of SU(4), or explicitly label the qudit universality as a conjecture. As written, the stronger statements 'may also lead to universal gate operations' and 'universal quantum computing ... can be achieved' in Secs. III.C and IV.C are not supported by the given derivations.
[Sec. VI] The implementation discussion concedes that coupling the Majorana and parafermion sectors requires a geometric, non-topological adiabatic protocol whose construction is 'beyond the scope of this present work.' Consequently, the claim of 'topologically protected universal quantum computing' is not established at the physical level: even if the algebraic universality were proven, the coupling step would introduce a non-topological error channel. The paper does hedge in the abstract ('may be possible'), but the body should more sharply distinguish the proven algebraic statement from the prospective physical implementation. The authors should either provide a concrete Hamiltonian or protocol for the tunable coupling, or explicitly state that the physical realization of such a coupling is an open problem.

minor comments (5)

[Eq. (10) and Eq. (24)] The notation 's2' and 's3' is ambiguous; these should be typeset as powers s^2 and s^3, because s itself can be ±1 or ±i, and the reader cannot always tell whether 's2' denotes s^2 or a subscript label. The same ambiguity appears in Eq. (24) and in the measurement-outcome formulas of Sec. V.B.
[Sec. IV.A] The first sentence says 'Section III A mathematically details the exact mapping between the two systems,' but the mapping under total parity conservation is presented in Sec. IV A, not Sec. III A; the introduction contains a similar mislabeled cross-reference.
[Eq. (16)] The definition S_N = diag(d0, d1, ..., d_N) should have indices running from 0 to N-1 for an N-dimensional gate; as written it suggests N+1 entries.
[Sec. II.B] The sentence 'For the case of n = 4, Tx and Ty take the explicit form' is followed by matrices for Tx and Tz, not Ty; the text should say Tx and Tz, or Ty should be displayed.
[Sec. VI] The word 'brading' appears instead of 'braiding' in the opening sentence of the concluding section.

Circularity Check

0 steps flagged · score 0.0 of 10

The derivation is self-contained: the Majorana–Z4 parafermion mapping is explicitly constructed from the defining algebras, and the non-Clifford gate decompositions are computed directly from the braiding operators; no fitted parameter or target result is used as an input.

full rationale

The paper's central claim is an algebraic mapping between Majorana fermions and Z4 parafermions, with the non-Clifford character of certain braid operators demonstrated by explicit unitary decompositions. The mapping is constructed from first principles: effective Pauli matrices are defined in Eqs. (18) and (34), generalized Pauli matrices in Eqs. (19) and (35), and invertible relations between the two sets are written out explicitly in Eqs. (20)-(25) and Eqs. (36)-(41). The braiding operators are then computed directly from the defining exchange relations, e.g., U_{3,4}^{(MF)} in Eq. (29) is decomposed as S4† H4^2 Tz^3 S4^6 R4(-3π/4) H4^2, and U^{(P)} in Eq. (31) is decomposed in terms of Clifford gates and a T gate. These are direct algebraic computations, not predictions extracted from fitted parameters, and the target result is not assumed in the derivation. The only self-citation, Ref. [52], concerns a periodically driven system that could host Z4 parafermions; this supports physical feasibility rather than the mathematical mapping, and the paper itself concedes that the adiabatic coupling protocol is geometric rather than strictly topological and its detailed construction is 'beyond the scope of this present work.' The inference from individual non-Clifford gates to universality is not fully proved, but that is a completeness or correctness gap, not circularity. No load-bearing step reduces to its own input by construction, so the paper merits a score of 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central mathematical claim rests on standard braiding formulas and basis transformations from the literature, plus the domain assumption that a hybrid Majorana-parafermion system can be physically realized. There are no fitted free parameters and no newly invented physical entities.

assumptions (4)

standard math Existence and correctness of the Majorana braiding operator U_{ij}^{(MF)} = (I + γ_i γ_j)/√2 (Eq. 6).
This expression is cited from Refs [55-58] and is standard in the literature.
standard math Existence and correctness of the Z4 parafermion braiding operator U_{jk}^{(P)} = (1/2)(I + ψ_k^3 ψ_j + ψ_k^2 ψ_j^2 + ψ_k ψ_j^3) (Eq. 13).
This is the standard braiding operator for Z4 parafermions, cited from Ref [9].
standard math Jordan-Wigner and Fradkin-Kadanoff transformations mapping Majoranas to Paulis and parafermions to generalized Paulis (Eqs. 7 and 15).
These are standard transformations cited from Ref [59].
domain assumption A physical system hosting both Majorana and Z4 parafermion edge modes exists.
The paper relies on Ref [52] for a periodically driven system with Z4 parafermion edge modes, but no concrete Hamiltonian for the hybrid system is provided.

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Pith. "Pith review of Generating non-Clifford gate operations through exact mapping between Majorana fermions and $\mathbb{Z}_4$ parafermions." pith.science (2026). https://pith.science/paper/UTUZMG3A

@misc{pith2026241118736,
  author       = {Pith},
  title        = {Pith review of: Generating non-Clifford gate operations through exact mapping between Majorana fermions and $\mathbbZ_4$ parafermions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UTUZMG3A}},
  note         = {Machine review of arXiv:2411.18736}
}

abstract

Majorana fermions and their generalizations to $\mathbb{Z}_n$ parafermions are considered promising building blocks of fault-tolerant quantum computers for their ability to encode quantum information nonlocally. In such topological quantum computers, highly robust quantum gates are obtained by braiding pairs of these quasi-particles. However, it is well-known that braiding Majorana fermions or parafermions only leads to a Clifford gate, hindering quantum universality. This paper establishes an exact mapping between Majorana fermions to $\mathbb{Z}_4$ parafermions in systems under total parity non-conserving and total parity conserving setting. It is revealed that braiding of Majorana fermions may lead to non-Clifford quantum gates in the 4-dimensional qudit representation spanned by $\mathbb{Z}_4$ parafermions, whilst braiding of $\mathbb{Z}_4$ parafermions may similarly yield non-Clifford quantum gates in the qubit representation spanned by Majorana fermions. This finding suggests that topologically protected universal quantum computing may be possible with Majorana fermions ($\mathbb{Z}_4$ parafermions) by supplementing the usual braiding operations with the braiding of $\mathbb{Z}_4$ parafermions (Majorana fermions) that could be formed out of Majorana fermions ($\mathbb{Z}_4$ parafermions) via the mapping prescribed here. Finally, the paper discusses how braiding of Majorana fermions or $\mathbb{Z}_4$ parafermions could be obtained via a series of parity measurements.

Figures

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Works this paper leans on

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(30) Unsurprisingly, such a braiding leads only to a diagonal quantum gate operation in the qudit space

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Assuming U = A and Substituting in Eq

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Pith tools

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(30) Unsurprisingly, such a braiding leads only to a diagonal quantum gate operation in the qudit space

= − 1 + i√ 2 S4. (30) Unsurprisingly, such a braiding leads only to a diagonal quantum gate operation in the qudit space. Interestingly, 6 when viewed from the two-qubit perspective spanned by four Majorana fermions, the resulting quantum gate op- eration does not belong to the Clifford gate set for qubits, as it contains the sought-after T gate, i.e., U ...

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(25) 5 Consequently, the Majorana fermion parity operators can be written in terms of parafermions as P (M F) 12 = −ψ2 2ψ2 1, P (M F) 13 = 1 + i 2 √ 2 (−ψ3 2 + iψ2 + i(−ψ3 2 + iψ2)ψ2 1), P (M F) 14 = − 1√ 2 (ψ3 1 + ψ1 + (ψ3 1 − ψ1)ψ2 2), P (M F) 23 = 1 + i 2 √ 2 (−iψ3 2 + ψ2 − (ψ3 2 + iψ2)ψ2 1), P (M F) 24 = i√ 2 (ψ1 − ψ3 1 + (ψ1 + ψ3 1)ψ2 2), P (M F) 34 ...

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(40) 7 The corresponding projectors are p(M F),s 12 = 1 2 (I − sψ2 1ψ2 2) p(M F),s 13 = 1 2 [I − s 1 + i 2 √ 2 (ψ1ψ3 3 + ψ3 1ψ3 + ψ1ψ2 2ψ3 − ψ3 1ψ2 2ψ3 3)], p(M F),s 45 = 1 2 (I − s 1√ 2 (ψ3 1ψ2 + iψ1ψ3 2)), p(M F),s 46 = 1 2 (I − sψ2 1ψ2 3). (41) B. Braiding Majorana fermions There are totally 15 possible braidings with six Majo- rana fermions. The six b...

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Assuming U = A and Substituting in Eq

and c4 = 1. Assuming U = A and Substituting in Eq. (C3) gives s(I + cψ3 1ψ2 + c2ψ2 1ψ2 2 + c3ψ1ψ3

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This implies that : ic′ = c3 & −icc′ = 1 (C4) since c4 = 1 , this implies ic′c = 1 & −icc′ = 1 (C5) which is a contradiction

= c′s(iψ1ψ3 2 − icI +ic2ψ3 1ψ2 − ic3ψ2 1ψ2 2). This implies that : ic′ = c3 & −icc′ = 1 (C4) since c4 = 1 , this implies ic′c = 1 & −icc′ = 1 (C5) which is a contradiction. To complete the proof, one should consider the three remaining cases ofU containing ψ1, ψ2 1, and ψ3

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12 Appendix D: Braiding of Majorana fermions under a conserved total parity In a system of six Majorana fermions where the total parity is conserved to 1, i.e

Repeating the same steps as above, all of them leads to a similar inconsistent set of equations. 12 Appendix D: Braiding of Majorana fermions under a conserved total parity In a system of six Majorana fermions where the total parity is conserved to 1, i.e. the states are restricted to the +1 eigenstates of the total parity, the action of the total parity ...

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