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Chiral Circulation Topological Defect Qubits: A Fault-Tolerant Encoding Pathway Bypassing the Majorana Bottleneck — Topological Encoding, Dynamical Operations, and Candidate Material Systems within the CSGFT Framework

Yunquan Liu

Published Aug 22, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.5281/zenodo.22059379

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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.
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.
The chiral Majorana fermion is a massless self-conjugate fermion which can arise as the edge state of certain 2D topological matters. It has been theoretically predicted and experimentally observed in a hybrid device of a quantum anomalous Hall insulator and a conventional superconductor. Its closely related cousin, the Majorana zero mode in the bulk of the corresponding topological matter, is known to be applicable in topological quantum computations. Here we show that the propagation of chiral Majorana fermions leads to the same unitary transformation as that in the braiding of Majorana zero modes and propose a platform to perform quantum computation with chiral Majorana fermions. A Corbino ring junction of the hybrid device can use quantum coherent chiral Majorana fermions to implement the Hadamard gate and the phase gate, and the junction conductance yields a natural readout for the qubit state.
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.
Quantum computers are devices that can solve certain problems faster than ordinary, classical computers. The fundamental units of quantum information are qubits, superpositions of two states, a "zero" state and a "one" state. There are various approaches to construct such two-level systems, among others, using superconducting circuits, trapped ions or photons. A common feature of these physical systems is that their coherence times are relatively short compared to the length of useful computations. Superconducting qubits, for instance, are currently the most advanced solid-state qubits, but they decohere after around 100 microseconds, and any information stored in these qubits is lost. On the other hand, useful quantum computations may require quantum information to survive on time scales that are many orders of magnitude longer, as their runtimes can reach several hours or even days. Topological quantum computing is an approach to construct qubits that survive for the entire duration of such a long computation. Topological quantum computing comes in two flavors. The condensed matter approach is to build error-resilient qubits using exotic quasiparticles in topological materials, most prominently Majorana zero modes in topological superconductors. Even though no such qubit has been built to date, the hope is that their coherence times may be significantly longer than the coherence times of currently available solid-state qubits, but are still expected to be too short for large-scale quantum computing. The quantum information approach is to combine many error-prone qubits to build more robust logical qubits using topological error-correcting codes, e.g., surface codes. Even though the first approach is hardware-based and the second approach is software-based, they are deeply related. With Majorana-based qubits, the main logical operations are Majorana fermion parity measurements. By replacing Majorana-based qubits with surface-code patches and parity measurements with lattice-surgery operations, schemes for quantum computation with Majorana-based qubits or with surface codes can be identical. In this thesis, we explore how to construct a large-scale topological fault-tolerant quantum computer that can perform useful quantum computations. Here, topological refers to the nature of the quantum error-correcting code, while the underlying hardware may be based on non-topological qubits, but could also be composed of Majorana-based qubits. We provide a complete picture of such a large-scale device, breaking down large quantum computations into logical qubits and logical operations, describing how these logical operations are performed on the level of physical qubits and physical gates, and finally discussing how these physical qubits can be pieced together in a Majorana-based system using topological superconducting nanowires.
Majorana zero modes (MZMs) are promising candidates for topologically-protected quantum computing hardware, however their large-scale use will likely require quantum error correction. Majorana surface codes (MSCs) have been proposed to achieve this. However, many MSC properties remain unexplored. We present a unified framework for MSC "twist defects" – anyon-like objects encoding quantum information. We show that twist defects in MSCs can encode twice the amount of topologically protected information as in qubit-based codes or other MSC encoding schemes. This is due to twists encoding both logical qubits and "logical MZMs," with the latter enhancing the protection microscopic MZMs can offer. We explain how to perform universal computation with logical qubits and logical MZMs while potentially using far fewer resources than in other MSC schemes. All Clifford gates can be implemented on logical qubits by braiding twist defects. We introduce lattice-surgery-based techniques for computing with logical MZMs and logical qubits, achieving the effect of Clifford gates with zero time overhead. We also show that logical MZMs may result in improved spatial overheads for sufficiently low rates of quasi-particle poisoning. Finally, we introduce a novel MSC analogue of transversal gates that achieves encoded Clifford gates in small codes by braiding microscopic MZMs. MSC twist defects thus open new paths towards fault-tolerant quantum computation.
A quantum computer can perform exponentially faster than its classical counterpart. It works on the principle of superposition. But due to the decoherence effect, the superposition of a quantum state gets destroyed by the interaction with the environment. It is a real challenge to completely isolate a quantum system to make it free of decoherence. This problem can be circumvented by the use of topological quantum phases of matter. These phases have quasiparticles excitations called anyons. The anyons are charge-flux composites and show exotic fractional statistics. When the order of exchange matters, then the anyons are called non-Abelian anyons. Majorana fermions in topological superconductors and quasiparticles in some quantum Hall states are non-Abelian anyons. Such topological phases of matter have a ground state degeneracy. The fusion of two or more non-Abelian anyons can result in a superposition of several anyons. The topological quantum gates are implemented by braiding and fusion of the non-Abelian anyons. The fault-tolerance is achieved through the topological degrees of freedom of anyons. Such degrees of freedom are non-local, hence inaccessible to the local perturbations. In this dissertation, we provide a comprehensive review of the fundamentals of logic design in topological quantum computing. The braid group and knot invariants in the skein theory are discussed. The physical insight behind the braiding is explained by the geometric phases and the gauge transformation. The mathematical models for the fusion and braiding are presented in terms of the category theory and the quantum deformation of the recoupling theory. The topological phases of matter are described by the topology of band structure. The wave function of quasiparticles in the quantum Hall effect and the theory of Majorana fermions in topological superconductors are also discussed. The dynamics of the charge-flux composites and their Hilbert space are expressed through the Chern-Simons theory and the two-dimensional topological quantum field theory. The Ising and Fibonacci anyonic models for binary gates are briefly given. Ternary logic gates are more compact than their binary counterparts and naturally arise in a type of anyonic model called the metaplectic anyons. We reduced the quantum cost of the existing ternary quantum arithmetic gates and proposed that these gates can be implemented with the metaplectic anyons.