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arXiv:2602.06963

Charge-$4e$ superconductor with parafermionic vortices: A path to universal topological quantum computation

Zhengyan Darius Shi, Zhaoyu Han, Srinivas Raghu, Ashvin Vishwanath

cond-mat.str-elhep-thquant-phSubmitted Feb 6, 2026• Announced Feb 9, 2026• 0 citations• Revised Mar 30, 2026 (v2)
v1 · Feb 6, 2026v2 · Mar 30, 2026
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Abstract

Topological superconductors (TSCs) provide a promising route to fault-tolerant quantum information processing. However, the canonical Majorana platform based on $2e$ TSCs remains computationally constrained. In this work, we find a $4e$ TSC that overcomes these constraints by combining a charge-$4e$ condensate with an Abelian chiral $\mathbb{Z}_3$ topological order in an intertwined fashion. Remarkably, this $4e$ TSC can be obtained by proliferating vortex-antivortex pairs in a stack of two $2e$ $p+ip$ TSCs, or by melting a $ν=2/3$ quantum Hall state. Specific to this TSC, the $hc/(4e)$ fluxes act as charge-conjugation defects in the topological order, whose braiding with anyons transmutes anyons into their antiparticles. This symmetry enrichment leads to $\mathbb{Z}_3$ parafermion zero modes trapped in the elementary vortex cores, which naturally encode qutrits. Braiding the parafermion defects alone generates the full many-qutrit Clifford group. We further show that a single-probe interferometric measurement enables topologically protected magic-state preparation, promoting Clifford operations to a universal gate set. Because the non-Abelian modes are bound to flux defects, they can, in principle, be externally controlled using superconducting circuit-based technology. More broadly, our results highlight hierarchical electron aggregation, the formation and condensation of higher-charge electron clusters, as a design principle for topological quantum matter with increased computational capability.

Comments
6 pages, 2 figures, 27 page appendices. v2: new appendix on effects of gapless phonons,

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•Aug 22, 2026•0 citations
Charge-4e4e4e superconductor with parafermionic vortices: A path to universal topological quantum computation
Topological superconductors (TSCs) provide a promising route to fault-tolerant quantum information processing. However, the canonical Majorana platform based on 2e2e2e TSCs remains computationally constrained. In this work, we find a 4e4e4e TSC that overcomes these constraints by combining a charge-4e4e4e condensate with an Abelian chiral Z3\mathbb{Z}_3Z3​ topological order in an intertwined fashion. Remarkably, this 4e4e4e TSC can be obtained by proliferating vortex-antivortex pairs in a stack of two 2e2e2e p+ipp+ipp+ip TSCs, or by melting a ν=2/3ν=2/3ν=2/3 quantum Hall state. Specific to this TSC, the hc/(4e)hc/(4e)hc/(4e) fluxes act as charge-conjugation defects in the topological order, whose braiding with anyons transmutes anyons into their antiparticles. This symmetry enrichment leads to Z3\mathbb{Z}_3Z3​ parafermion zero modes trapped in the elementary vortex cores, which naturally encode qutrits. Braiding the parafermion defects alone generates the full many-qutrit Clifford group. We further show that a single-probe interferometric measurement enables topologically protected magic-state preparation, promoting Clifford operations to a universal gate set. Because the non-Abelian modes are bound to flux defects, they can, in principle, be externally controlled using superconducting circuit-based technology. More broadly, our results highlight hierarchical electron aggregation, the formation and condensation of higher-charge electron clusters, as a design principle for topological quantum matter with increased computational capability.
We analyze the topological protection of a Clifford gate set generated by braiding four Majorana zero modes (MZMs) in a two-dimensional class-D topological superconductor. Encoding a single logical qubit in four MZMs bound to π\piπ-flux vortices, we show that elementary exchanges generate the single-qubit Clifford group up to a global phase. The residual gate error arises from finite-size Majorana hybridization and quasiparticle poisoning, scaling as ϵ∼exp⁡(−L/ξ)\epsilon \sim \exp(-L/\xi)ϵ∼exp(−L/ξ), where LLL is the inter-vortex separation and ξ\xiξ is the superconducting coherence length. We propose an interferometric measurement scheme to falsify this topological protection. By threading an Aharonov-Bohm loop around two MZMs, the conductance exhibits 4π4\pi4π-periodic oscillations with phase jumps of π/2\pi/2π/2 per braid. Any deviation from this quantized phase beyond the exp⁡(−L/ξ)\exp(-L/\xi)exp(−L/ξ) envelope at large L/ξL/\xiL/ξ falsifies the topological protection mechanism, indicating gap closing or quasiparticle poisoning. ## 1. Introduction Topological quantum computation seeks to process information non-locally to achieve intrinsic fault tolerance. In two-dimensional topological superconductors belonging to symmetry class D (such as a px+ipyp_x + i p_ypx​+ipy​ paired state or a proximitized semiconductor heterostructure), Majorana zero modes (MZMs) bind to π\piπ-flux vortices. Exchanging these vortices implements non-Abelian unitaries on the degenerate ground-state manifold. The central question addressed here is whether braiding four MZMs generates a topologically protected Clifford gate set with a residual error rate scaling as exp⁡(−L/ξ)\exp(-L/\xi)exp(−L/ξ), and whether this claim is falsifiable via interferometric measurement of the braiding phase. ## 2. Background The non-Abelian statistics of MZMs in class-D superconductors were established by Ivanov (2001), demonstrating that exchanging vortices realizes a braid group representation with π/2\pi/2π/2 rotations in Majorana Hilbert space [arXiv:cond-mat/0105040]. The topologi
Topological quantum computation encodes quantum information in the internal fusion space of non-Abelian anyonic quasiparticles, whose braiding implements logical gates. This goes beyond Abelian topological order (TO) such as the toric code, as its anyons lack internal structure. However, the simplest non-Abelian generalizations of the toric code do not support universality via braiding alone. Here we demonstrate that such minimally non-Abelian TOs can be made universal by treating anyon fusion as a computational primitive. We prepare a 54-qubit TO wavefunction associated with the smallest non-Abelian group, S3S_3S3​, on Quantinuum's H2 quantum processor. This phase of matter exhibits cyclic anyon fusion rules, known to underpin universality, which we evidence by trapping a single non-Abelian anyon on the torus. We encode logical qutrits in the nonlocal fusion space of non-Abelian fluxes and, by combining an entangling braiding operation with anyon charge measurements, realize a universal topological gate set and read-out, which we further demonstrate by topologically preparing a magic state. This work establishes S3S_3S3​ TO as simple enough to be prepared efficiently, yet rich enough to enable universal topological quantum computation.
Topological quantum computing seeks to store and manipulate information in a protected manner using topological phases of matter. Information encoded in the degenerate state space of pairs of non-Abelian anyons or defects is robust to local perturbations, reducing its susceptiblity to environmental errors and potentially providing a scalable approach to quantum computing. However, topological quantum computing faces significant challenges, not least of which is identifying an experimentally accessible platform supporting non-Abelian topological physics. In this thesis, we critically analyze topological quantum computing with Majorana zero modes, non-Abelian defects of a topological superconductor. We identify intrinsic error sources for Majorana-based systems and propose quantum computing architectures that minimize their effects. Additionally, we consider a new approach for realizing and detecting non-Abelian topological defects in fractional Chern insulators. Topological quantum computing is predicated on the idea that braiding non-Abelian anyons adiabatically can implement quantum gates fault tolerantly. However, any braiding experiment will necessarily depart from the strict adiabatic limit. We begin by analyzing the nature of diabatic errors for anyon braiding, paying particular attention to how such errors scale with braiding time. We find that diabatic errors are unfavorably large and worryingly sensitive to details of the time evolution. We present a measurement-based correction protocol for such errors, and illustrate its application in a particular Majorana-based qubit design. We next propose designs for Majorana-based qubits operated entirely by a measurement-based protocol, thereby avoiding the diabatic errors discussed above. Our designs can be scaled into large two dimensional arrays amenable to long-term quantum computing goals, whose core components are testable in near-term devices. These qubits are robust to quasiparticle poisoning, anticipated to be one of the dominant error sources coupling to Majorana zero modes. We demonstrate that our designs support topologically protected Clifford operations and can be augmented to a universal gate set without requiring additional control parameters.While topological protection greatly suppresses errors, residual coupling to noise limits the lifetimes of our proposed Majorana-based qubits. We analyze the dephasing times for our quasiparticle-poisoning-protected qubits by calculating their charge distribution using a particle number-conserving formalism. We find that fluctuations in the electromagnetic environment couple to an exponentially suppressed topological dipole moment. We estimate dephasing times due to 1/f1/f1/f noise, thermal quasiparticle excitations, and phonons for different qubit sizes.The residual errors discussed above will necessarily require error correction for a sufficiently long quantum computation. We develop physically motivated noise models for Majorana-based qubits that can be used to analyze the performance of a quantum error correcting code. We apply this noise model to estimate pseudo-thresholds for a small subsystem code, identifying the relative importance of difference error processes from a fault tolerance perspective. Our results emphasize the necessity of suppressing long-lived quasiparticle excitations that can spread across the code.Finally, we turn our attention to a different platform that could host non-Abelian topological defects: fractional Chern insulators in graphene. We study the edge states of fractional Chern insulators using the field theory of fractional quantum Hall edges supplemented with a symmetry action. We find that lattice symmetries impose a quantized momentum difference for edge electrons in a fractional state of a C=2C=2C=2 Chern band. This momentum difference can be used to selectively contact the different edge states, thereby allowing detection of topological defects in the bulk with a standard four terminal measurement. Our proposal could be implemented in graphene subject to an artificially patterned lattice.
Non-Abelian anyons promise to reveal spectacular features of quantum mechanics that could ultimately provide the foundation for a decoherence-free quantum computer. A key breakthrough in the pursuit of these exotic particles originated from Read and Green's observation that the Moore-Read quantum Hall state and a (relatively simple) two-dimensional p+ip superconductor both support so-called Ising non-Abelian anyons. Here we establish a similar correspondence between the Z_3 Read-Rezayi quantum Hall state and a novel two-dimensional superconductor in which charge-2e Cooper pairs are built from fractionalized quasiparticles. In particular, both phases harbor Fibonacci anyons that---unlike Ising anyons---allow for universal topological quantum computation solely through braiding. Using a variant of Teo and Kane's construction of non-Abelian phases from weakly coupled chains, we provide a blueprint for such a superconductor using Abelian quantum Hall states interlaced with an array of superconducting islands. Fibonacci anyons appear as neutral deconfined particles that lead to a two-fold ground-state degeneracy on a torus. In contrast to a p+ip superconductor, vortices do not yield additional particle types yet depending on non-universal energetics can serve as a trap for Fibonacci anyons. These results imply that one can, in principle, combine well-understood and widely available phases of matter to realize non-Abelian anyons with universal braid statistics. Numerous future directions are discussed, including speculations on alternative realizations with fewer experimental requirements.
New universal gates for topological quantum computation with Fibonacci-ε\boldsymbol{\varepsilon}ε composite Majorana edge modes on topological superconductor multilayers
We propose a new design of universal topological quantum computer device through a hybrid of the 1-, 2- and 7-layers of chiral topological superconductor (χ\chiχTSC) thin films. Based on the SO(7)1/(G2)1SO(7)_1/(G_2)_1SO(7)1​/(G2​)1​ coset construction, strongly correlated Majorana fermion edge modes on the 7-layers of χ\chiχTSC are factorized into the composite of the Fibonacci τ\tauτ-anyon and ε\varepsilonε-anyon modes in the tricritical Ising model. Furthermore, the deconfinement of τ\tauτ and ε\varepsilonε via the interacting potential gives the braiding of either τ\tauτ or ε\varepsilonε. Topological phase gates are assembled by the braidings. With these topological phase gates, we find a set of fully topological universal gates for the (τ,ε)(\tau,\varepsilon)(τ,ε) composite Majorana-Ising-type quantum computation. Because the Hilbert space still possesses a tensor product structure of quibts and is characterized by the fermion parities, encoding quantum information in this machine is more efficient and substantial than that with Fibonacci anyons. The computation results is easier to be read out by electric signals, so are the initial data inputted.
Background: Topological quantum computation is regarded as the most robust paradigm for quantum information processing, exploiting the global invariance of topological order to store information and resist local noise. However, the mainstream route—based on non-Abelian braiding statistics of Majorana zero modes—faces two inherent bottlenecks: (1) the fabrication of required topological superconductors depends critically on ultra-low temperatures (typically <100 mK) and precise external field tuning, with long-standing insufficient experimental reproducibility; (2) non-Abelian braiding operations have yet to yield reproducible zero-level experimental evidence. The common root of both bottlenecks is that the Majorana scheme grounds topological protection in a topological superconducting phase that has not been experimentally confirmed. Therefore, seeking an orthogonal pathway independent of Majorana particles is among the most urgent problems in topological quantum computing. Methods: This paper proposes a topological qubit scheme orthogonal to the Majorana route, based on chiral circulation topological defects predicted by Causal Set Gauge-Flow Theory (CSGFT). Within the CSGFT framework, material particles are identified as stable circulation topological defects, whose chirality marker σ = ±1 is a topological invariant protected by the discrete vector flux conservation law. We propose a qubit encoding using σ = +1 for |0⟩ and σ = −1 for |1⟩; phase circulation provides superposition state manipulation; flux conservation provides topological protection; and refractive index measurement provides readout. We present a complete encoding, manipulation, and readout scheme, prove the topological completeness of the chiral two-level system, and conduct quantitative comparisons with the Majorana route. Based on experimental data from the quantum Hall effect and quantum anomalous Hall effect, we extract the protection gap of chiral topological degrees of freedom; based on publicly available data from the FeSe/SrTiO₃ system, we fit the parameters of the CSGFT refractive index matching model and derive the quantitative relationship between refractive index matching and coherence time. Results: (1) The chirality marker σ = ±1 naturally forms a two-level system defined by topological invariants, whose ground-state discreteness is a direct consequence of topological properties and does not depend on any continuous parameters. (2) The flux conservation law guarantees that once a circulation defect forms, its chirality is invariant under continuous deformation, and local noise cannot alter the chirality marker—this is the most fundamental distinction from the Majorana scheme (the latter relies on the gap of a topological superconducting phase, while the former relies on a universal conservation principle). (3) The protection gap of chiral topological degrees of freedom is 1–10 meV, corresponding to an upper operating temperature limit of 10–100 K, which is 2–3 orders of magnitude higher than the Majorana scheme (<100 mK). (4) The FeSe/SrTiO₃ refractive index matching fit yields α ≈ 6.3 ± 1.5, T_c,0 ≈ 82 ± 12 K, with R² ≈ 0.89; from this we predict that each 10% improvement in refractive index matching increases the circulation defect coherence time by approximately 1.9-fold. (5) CSGFT predicts a decoherence time 2–4 orders of magnitude longer than standard theory (mesoscopic scale N = 10³–10⁵). (6) Three candidate material systems (InAs quantum rings, topological insulator rings, and FeSe/SrTiO₃ interfaces) are specified with concrete parameters and device designs. Conclusions: The core contribution of this paper is the proposal of a topological qubit scheme orthogonal to the Majorana route—using the chiral topological charge of CSGFT circulation defects as the encoding degree of freedom and flux conservation as the source of topological protection. This scheme does not require Majorana particles or a topological superconducting phase, and its material selection range is far broader than the Majorana route, with a theoretically predicted operating temperature 2–3 orders of magnitude higher. The chiral circulation topological defect qubit is a testable orthogonal pathway: it does not pursue Majorana fermions, but rather exploits the chiral topological charge protected by flux conservation. If experimentally verified, it would constitute an entirely new avenue in topological quantum computing.