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arxiv: 2606.03916 · v1 · pith:RSBPLLNJ · submitted 2026-06-02 · quant-ph · cond-mat.other

Practical gates by Majorana fermion motion

Yuri D. Lensky , Bryce Kobrin , Kostyantyn Kechedzhi , Igor Aleiner This is my paper

Reviewed by Pith T0 review T1 audit T2 compute T3 formal T4 kernel 2026-06-28 09:22 UTCgrok-4.3pith:RSBPLLNJrecord.jsonopen to challenge →

classification quant-ph cond-mat.other
keywords Majorana fermionsquantum error correctionstabilizer codeslogical gatesbraidinglattice surgeryfault tolerance
0
0 comments X

The pith

Majorana fermion motion in stabilizer codes enables compact braiding-based logical gates with lower space overhead than lattice surgery.

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

The paper shows how planar Pauli stabilizer codes can be described using point-like Majorana fermions whose pairwise parities hold the logical information. This description holds at every length scale down to the lattice spacing. The authors introduce fault-tolerant motion of these fermions as a basic operation and use it to construct braiding gates. Numerical comparisons indicate that the resulting 2-qubit Clifford gates achieve better logical error rates than lattice surgery when the number of physical qubits is fixed and error rates are in the near-term regime.

Core claim

A complete description of planar Pauli stabilizer codes in terms of Majorana fermions captures logical information at all scales and allows fault-tolerant motion of the fermions; this motion serves as a primitive for braiding-based logical gates that pack logical operations more densely in spacetime and thereby reduce space overhead.

What carries the argument

Majorana fermions that encode logical information via pairwise parities, with their motion used to implement braiding gates.

If this is right

  • Logical operations occupy less physical space because information can be packed densely using the motion primitive.
  • Braiding gates built from fermion motion outperform lattice surgery at realistic near-term error rates for the same number of physical qubits.
  • The same motion primitive can be applied to design other Clifford gates beyond the 2-qubit examples shown.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The motion primitive may extend to non-Clifford gates or to codes on higher-dimensional lattices.
  • Hardware implementations could test whether the dense packing actually yields the predicted reduction in logical error rate.
  • The approach suggests a route to redesigning entire error-correction protocols around compact spacetime motion rather than static patches.

Load-bearing premise

The Majorana-fermion picture of the code remains accurate when the fermions are moved at the scale of the lattice spacing.

What would settle it

A simulation or device measurement that shows additional uncorrectable errors appear when Majorana fermions are moved at lattice-constant distances would falsify the claimed performance gain.

Figures

Figures reproduced from arXiv: 2606.03916 by Bryce Kobrin, Igor Aleiner, Kostyantyn Kechedzhi, Yuri D. Lensky.

Figure 1
Figure 1. Figure 1: FIG. 1. Several memories written in terms of Majorana [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Path of a distance-preserving braid in the space of 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Outline of the steps of the Majorana embedding, described in more detail in [2]. First, a set of arrowed tetrahedra [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. An example code where code distance is determined [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Representative Wilson line generating the motion of [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Virtual particle motion for Majorana fermions. We move the Majorana fermion at the top middle to the location [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Fermion error analysis. The support of an error [PITH_FULL_IMAGE:figures/full_fig_p008_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Left: qubit layout for lattice surgery. Right: qubit layout and Majorana fermion trajectories for braiding. [PITH_FULL_IMAGE:figures/full_fig_p009_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Stabilizer configurations of an unoptimized virtual particle-based braid, following steps 1-5 of Fig. 2. Time order is [PITH_FULL_IMAGE:figures/full_fig_p009_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: VIII. CONCLUSION In conclusion, we demonstrated that macroscopic Ma￾jorana degrees of freedom are a powerful and versatile tool for the analysis of the local and global building blocks of fault-tolerant quantum computation in 2D. It allows for optimization and code distance analysis of fault-tolerant memories and gates. We studied and microscopically justified macroscopic properties of Majorana graphs. Us… view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. The gate fidelity (or [PITH_FULL_IMAGE:figures/full_fig_p010_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Construction of the Majorana graph of a distance 3 surface code. From left to right: (i) Mapping a qubit into [PITH_FULL_IMAGE:figures/full_fig_p013_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Left: A region of the Majorana graph of a surface code. Right: The same as left with a single Majorana dimer [PITH_FULL_IMAGE:figures/full_fig_p013_14.png] view at source ↗
Figure 7
Figure 7. Figure 7: It consists of two repeated cycles, which between them extract all the stabilizers and fusion outcomes. [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Measurement circuits for the 5th stabilizer configuration of the braid shown in Fig. 11. The single-qubit gates [PITH_FULL_IMAGE:figures/full_fig_p015_15.png] view at source ↗
read the original abstract

Quantum error correction protocols protect against local errors by storing logical information non-locally. This poses a challenge: how to design efficient logical gates on the non-local ``hidden'' logical information, and how to implement these gates using the local physical operations. We develop a general description of planar Pauli stabilizer codes and protocols for logical operations in terms of point-like particles called Majorana fermions. Information is stored in the pairwise fermion parities of spatially separated Majorana fermions. The description in terms of Majorana fermions captures not only large distance asymptotics, but also all scales down to the lattice constants. We exploit this locality to densely pack logical information in spacetime. The simplest application is to a static case: dense memory. More importantly, we implement fault-tolerant Majorana motion and leverage this primitive to design braiding-based logical gates. This approach reduces space overhead of logical operations resulting in an improved logical error rate given fixed number of physical qubits. We illustrate a practical use of our approach by designing and benchmarking of 2-qubit Clifford gates. We find numerically that our protocol outperforms lattice surgery in this setting for near-term error rates and realistic device constraints. More generally, introduction of compact motion of Majorana fermions as an efficient computational primitive opens a promising new route for the design of low overhead error correction protocols.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript develops a general Majorana-fermion description of planar Pauli stabilizer codes in which logical information is encoded in pairwise parities of spatially separated point-like fermions. This description is claimed to be accurate at all scales down to the lattice constant. The authors exploit the resulting locality to introduce a fault-tolerant Majorana-motion primitive, use it to construct braiding-based logical gates, and report that the resulting protocols reduce space overhead relative to lattice surgery, yielding improved logical error rates for fixed physical-qubit count. They illustrate the approach with explicit 2-qubit Clifford gates and state that numerical benchmarking shows outperformance over lattice surgery for near-term error rates and realistic device constraints.

Significance. If the numerical comparison is reproducible and the motion primitive preserves the effective Majorana description without unmodeled errors at lattice scale, the work supplies a new computational primitive that could systematically lower the space-time overhead of logical operations in stabilizer codes.

major comments (2)
  1. [Abstract / Majorana-motion section] Abstract and the section describing the motion primitive: the headline claim that the protocol outperforms lattice surgery rests on the assertion that the Majorana-fermion representation remains accurate and complete when motion is executed at distances comparable to the lattice spacing. The text states that the description “captures all scales down to the lattice constants,” yet supplies no explicit error-channel analysis, leakage bounds, or Hamiltonian simulation demonstrating that the concrete motion circuit introduces no additional terms that would invalidate the dense-packing advantage.
  2. [Numerical results] Numerical benchmarking paragraph: the statement that the protocol outperforms lattice surgery is presented without the accompanying error model, noise parameters, or raw data tables. Because the central performance claim cannot be verified from the supplied information, the quantitative advantage remains unconfirmed.
minor comments (1)
  1. Notation for the pairwise parity operators should be introduced with an explicit equation before being used in the gate constructions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. We address the two major comments point by point below, indicating where revisions will be made to improve clarity and verifiability.

read point-by-point responses

  1. Referee: [Abstract / Majorana-motion section] Abstract and the section describing the motion primitive: the headline claim that the protocol outperforms lattice surgery rests on the assertion that the Majorana-fermion representation remains accurate and complete when motion is executed at distances comparable to the lattice spacing. The text states that the description “captures all scales down to the lattice constants,” yet supplies no explicit error-channel analysis, leakage bounds, or Hamiltonian simulation demonstrating that the concrete motion circuit introduces no additional terms that would invalidate the dense-packing advantage.

    Authors: We agree that the manuscript would benefit from an explicit error-channel analysis of the motion primitive at lattice scales. While the Majorana-fermion description follows directly from the underlying stabilizer code (which encodes all local degrees of freedom by construction), we did not supply leakage bounds or a Hamiltonian simulation of the concrete motion circuit. In the revised manuscript we will add a dedicated subsection deriving bounds on additional error terms and leakage, together with a small-scale numerical check confirming that the effective description remains valid under the motion protocol. revision: yes

  2. Referee: [Numerical results] Numerical benchmarking paragraph: the statement that the protocol outperforms lattice surgery is presented without the accompanying error model, noise parameters, or raw data tables. Because the central performance claim cannot be verified from the supplied information, the quantitative advantage remains unconfirmed.

    Authors: The benchmarking employs a standard depolarizing noise model whose parameters are stated in the methods; the comparison is performed at near-term physical error rates. To make the quantitative claim verifiable, the revised version will include an explicit table of noise parameters, the precise simulation settings, and raw logical-error-rate data (or a pointer to supplementary material containing them). revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is modeling-based

full rationale

The paper develops a Majorana-fermion representation of planar stabilizer codes as a general description that is asserted to hold at all scales, then uses this representation to define motion primitives and benchmark gates. No equations, fitted parameters, or self-citations are quoted that reduce a claimed prediction or uniqueness result to the input by construction. The numerical outperformance claim rests on simulation within the adopted model rather than on any self-referential loop. This is the common case of a domain-specific modeling paper whose central content is independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; ledger is necessarily incomplete. Main modeling choice is the equivalence between stabilizer codes and Majorana parity storage.

axioms (1)
  • domain assumption Planar Pauli stabilizer codes admit a complete description in terms of point-like Majorana fermions whose pairwise parities store logical information at all length scales.
    Invoked to justify dense packing and motion primitives.

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discussion (0)

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Reference graph

Works this paper leans on

34 extracted references · 10 canonical work pages · 7 internal anchors

  1. [1]

    circuit-level

    In fact, this is a maximally dense packing at distancedaccording to the metrics above, the fact apparently overlooked in the previous literature. For example, both Wilson lines shown are lengthd. It is denser than the first example by a factor of3/2. The density can be further improved for encod- ings of more qubits by combining the dense packing of Major...

  2. [2]

    To generate the full Clifford group, one additionally needs either the swap of a horizontal pair, or a cyclic permutation of 4 Majo- ranas

    The Manhattan metrics for the two types of paths induced by the square grid suggests that in fact the braid is distance-preserving, and therefore op- timal (as far as code distance). To generate the full Clifford group, one additionally needs either the swap of a horizontal pair, or a cyclic permutation of 4 Majo- ranas. There is a simpler bulk braid (whi...

  3. [3]

    isentropic

    for more details on the formal structure). First, we assign 4 Majorana fermions to each qubit, and the first gauge constraint is to fix the parity sector to be even for each such group. Majorana bilinears on a single qubit (shown with colors in Fig. 3 and elsewhere) are gauge- invariant and satisfy a Pauli algebra on each qubit. The Majorana graph is a co...

  4. [4]

    A. Y. Kitaev, Fault-tolerant quantum computation by anyons, Annals of Physics303, 2 (2003), arXiv:quant- ph/9707021

    work page arXiv 2003

  5. [5]

    Y. D. Lensky, K. Kechedzhi, I. Aleiner, and E.-A. Kim, Graph gauge theory of mobile non-Abelian anyons in a qubit stabilizer code, Annals of Physics452, 169286 (2023)

    2023

  6. [6]

    Google Quantum AI and Collaborators, Quantum error correction below the surface code threshold,638, 920

  7. [7]

    Bluvstein, S

    D. Bluvstein, S. J. Evered, A. A. Geim, S. H. Li, H. Zhou, T. Manovitz, S. Ebadi, M. Cain, M. Kalinowski, D. Hangleiter, J. P. Bonilla Ataides, N. Maskara, I. Cong, X. Gao, P. Sales Ro- driguez, T. Karolyshyn, G. Semeghini, M. J. Gullans, M. Greiner, V. Vuletić, and M. D. Lukin, Logical quan- tum processor based on reconfigurable atom arrays, Na- ture626,...

    2023

  8. [8]

    Dennis, A

    E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological quantum memory, Journal of Mathemati- cal Physics43, 4452 (2002)

    2002

  9. [9]

    Thee, m, εanyons of the surface code [5]

  10. [10]

    The construction of compact braids contradicts the con- clusions of Ref. [28]. However, this reference discussed braiding of defects rather than point-like Majorana fermions. The former apparently requires significantly larger overhead

  11. [11]

    Anyons in an exactly solved model and beyond,

    A. Kitaev, Anyons in an exactly solved model and be- yond, Annals of Physics321, 2 (2006), arXiv:cond- mat/0506438

    work page arXiv 2006

  12. [12]

    H. Bombin, Topological Order with a Twist: Ising Anyons from an Abelian Model, Physical Review Let- ters105, 030403 (2010), arXiv:1004.1838 [cond-mat, physics:hep-th, physics:quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  13. [13]

    Models for gapped boundaries and domain walls

    A. Kitaev and L. Kong, Models for gapped boundaries and domain walls, Communications in Mathematical Physics313, 351 (2012), arXiv:1104.5047 [cond-mat]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  14. [14]

    Barkeshli, C.-M

    M. Barkeshli, C.-M. Jian, and X.-L. Qi, Theory of de- fects in Abelian topological states, Physical Review B 88, 235103 (2013)

    2013

  15. [15]

    Projective non-Abelian Statistics of Dislocation Defects in a Z_N Rotor Model

    Y.-Z. You and X.-G. Wen, Projective non-Abelian Statistics of Dislocation Defects in a Z_N Rotor Model, Physical Review B86, 161107 (2012), arXiv:1204.0113 [cond-mat, physics:quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  16. [16]

    M. B. Hastings and A. Geller, Reduced space-time and time costs using dislocation codes and arbitrary ancillas (2015), arXiv:1408.3379 [quant-ph]. 11

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  17. [17]

    Benhemou, J

    A. Benhemou, J. K. Pachos, and D. E. Browne, Non- Abelian statistics with mixed-boundary punctures on the toric code, arXiv:2103.08381 [quant-ph] (2021), arXiv:2103.08381 [quant-ph]

    work page arXiv 2021

  18. [18]

    Demonstrating non-Abelian statistics of Majorana fermions using twist defects

    H. Zheng, A. Dua, and L. Jiang, Demonstrating non-Abelian statistics of Majorana fermions using twist defects, Physical Review B92, 245139 (2015), arXiv:1508.04166

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  19. [19]

    B. J. Brown, K. Laubscher, M. S. Kesselring, and J. R. Wootton, Poking holes and cutting corners to achieve Clifford gates with the surface code, Physical Review X7, 021029 (2017), arXiv:1609.04673 [cond- mat, physics:quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  20. [20]

    In other topologies, one has to account for additional paths of both types that wrap handles, holes, or go be- tween boundaries

  21. [21]

    As we will see in Section III, counted as the number of vertices along the path

  22. [22]

    augmented

    In [2], it was important to distinguish between 2 types of Wilson lines, while we will only be concerned with 1 type. When we refer to Wilson lines in this work, we are referring to what are called “augmented” Wil- son lines in [2]. The “unaugmented” Wilson lines are the conventional parallel transport operators, whereas “aug- mented” Wilson lines are gau...

  23. [23]

    G. H. Low, W. J. Huggins, D. W. Berry, T. Khattar, A. F. White, N. C. Rubin, and R. Babbush, A denser planar surface code, (2026), arXiv:2605.30455 [quant- ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  24. [24]

    If we embed in a general manifoldMand only take topologically trivial loops as stabilizers, the number of logical qubits scales as∼nσ/2−χ(M)withO(1)correc- tions (related to threaded Majorana “fluxes”) discussed in Ref. [2]

  25. [25]

    In the absence of coherent errors it is not important that we make the choice that does not create flux, only that we know whether or not we did

  26. [26]

    T. I. Andersen, Y. D. Lensky, K. Kechedzhi, I. K. Drozdov, A. Bengtsson, S. Hong, A. Morvan, X. Mi, A. Opremcak, R. Acharya, R. Allen, M. Ansmann, F. Arute, K. Arya, A. Asfaw, J. Atalaya, R. Babbush, D. Bacon, J. C. Bardin, G. Bortoli, A. Bourassa, J. Bo- vaird, L. Brill, M. Broughton, B. B. Buckley, D. A. Buell, T. Burger, B. Burkett, N. Bushnell, Z. Che...

    2023

  27. [27]

    As discussed below, we actually add additional restric- tions, such as readout only on certain “ancilla” qubits and square grid connectivity, which are already satisfied by lattice surgery

  28. [28]

    Gidney, M

    C. Gidney, M. Newman, A. Fowler, and M. Broughton, A Fault-Tolerant Honeycomb Memory, Quantum5, 605 (2021)

    2021

  29. [29]

    Gidney, Stim: a fast stabilizer circuit simulator, Quantum5, 497 (2021)

    C. Gidney, Stim: a fast stabilizer circuit simulator, Quantum5, 497 (2021)

    2021

  30. [30]

    Higgott and C

    O. Higgott and C. Gidney, Sparse Blossom: correcting a million errors per core second with minimum-weight matching, Quantum9, 1600 (2025)

    2025

  31. [31]

    Horsman, A

    D. Horsman, A. G. Fowler, S. Devitt, and R. V. Me- ter, Surface code quantum computing by lattice surgery, New Journal of Physics14, 123011 (2012)

    2012

  32. [32]

    Jordan and E

    P. Jordan and E. Wigner, Über das paulische Äquiv- alenzverbot, Zeitschrift für Physik (in German)47, 631 (1928). 12 Appendix A: Basics of Majorana code theory The purpose of this section is to summarize notions from Majorana graph theory that are not standard knowledge in surface code error correction theory. For detailed discussion of theory [2]

    1928

  33. [33]

    Majorana fermions, fermions and Pauli matrices Consider a qubit, which is nothing but a two level sys- tem. All qubit operations can be described in the lan- guage of fermion creation/annihilation operatorsc †, c, that satisfy, c†c+cc † = 1, c 2 = 0.(A1) The same description is also realized in terms of a pair of Majorana fermion operators,α1, α2, whereα ...

  34. [34]

    Majorana vs Pauli stabilizer code Consider a stabilizer of a standard surface code, Fig. 13, that has a form in a symmetric basis, B=Z 1X2Z3X4.(A7) SubstitutingMajoranarepresentationfromeq.(A5)and regrouping Majorana operators we obtain, B=τ 12τ23τ34τ41,(A8) where we introduced link parity operators, τjk =iα jαk,(A9) connecting qubitsjandk. Here the Major...