Skyrmion Qudit in Triangular Magnets

• Skyrmion qudit is a multi-level quantum state defined by discrete helicity quantization in frustrated triangular magnets, enabling a transition from qubit to qudit encoding.
• Its effective Hamiltonian transforms into a Mathieu–Schrödinger equation, revealing a coherent four-level system with significantly enhanced l1 coherence compared to skyrmion qubits.
• The model employs scalar chirality diagnostics and collective-coordinate quantization, laying the groundwork for hybrid quantum devices and improved topological control.

Searching arXiv for the cited skyrmion-qudit and closely related quantum-skyrmion papers to ground the article in current literature. arxiv_search(query="^^^^^^^^2all:Skyrmion Qudit OR ti:\2^^^^^^^^"Quantum Skyrmion Qudit in a Triangular-lattice magnet\"^^^^^^^^2 OR abs:\2^^^^^^^^"skyrmion qudit\"", max_results=^^^^2 OR abs:\2all:Skyrmion Qudit OR ti:\2^^^^) arxiv_search(query="ti:\"Quantum Skyrmion Qudit in a Triangular-lattice magnet\" OR (&&&^^^^2all:Skyrmion Qudit OR ti:\2^^^^&&&)", max_results=5) arxiv_search(query="ti:\"Colloquium: Quantum Properties and Functionalities of Magnetic Skyrmions\" OR (&&&^^^^2 OR abs:\2^^^^&&&)", max_results=5) arxiv_search(query="ti:\"Skyrmion Qubits: Challenges For Future Quantum Computing Applications\" OR (Psaroudaki et al., 2024) OR ti:\"Skyrmionic qubits stabilized by Dzyaloshinskii-Moriya interaction as platforms for qubits and quantum gates\" OR (Sticlet et al., 15 Nov 2025)", max_results=^^^^2 OR abs:\2all:Skyrmion Qudit OR ti:\2^^^^) arxiv_search(query="(Williams et al., 27 Jun 2025) OR (Sotnikov et al., 2020) OR (Corte et al., 26 Jan 2026) OR (Mæland et al., 2022) OR (Gauyacq et al., 2019)", max_results=^^^^2 OR abs:\2all:Skyrmion Qudit OR ti:\2^^^^) A skyrmion qudit is a multi-level quantum system encoded in discrete states of a skyrmion rather than in a classical continuous spin texture. In the most explicit formulation now available, the relevant degree of freedom is the quantized helicity of a quantum skyrmion in a frustrated triangular-lattice magnet: in the weak-electric-field limit the system is effectively a two-level skyrmion qubit, whereas for a significant barrier the same model supports a higher-dimensional Hilbert space and the state is “not a Skyrmion qubit as it was thought before, but a Skyrmion qudit” (&&&^^^^2all:Skyrmion Qudit OR ti:\2^^^^&&&). Related work places this concept within a broader program on quantum skyrmions, scalar-chirality diagnostics, collective-coordinate quantization, and higher-dimensional skyrmion textures such as PRESERVED_PLACEHOLDER_^^^^2all:Skyrmion Qudit OR ti:\2^^^^ skyrmion crystals (&&&^^^^2 OR abs:\2^^^^&&&, Sotnikov et al., 2020, Williams et al., 27 Jun 2025).

^^^^2 OR abs:\2^^^^. Classical topology and the quantum-skyrmion setting

Classically, a magnetic skyrmion is a two-dimensional spin texture PRESERVED_PLACEHOLDER_^^^^2 OR abs:\2^^^^ with integer winding number

$$Q=\frac{1}{4\pi}\int d^2r\;\mathbf{m}(\mathbf{r})\cdot\left(\partial_x\mathbf{m}(\mathbf{r})\times\partial_y\mathbf{m}(\mathbf{r})\right),$$

or, on a lattice, by discrete analogues built from scalar chirality or Berg–Lüscher solid angles (Sotnikov et al., 2020, &&&^^^^2 OR abs:\2^^^^4&&&). In this classical description, skyrmions are localized, particle-like excitations protected by an energy barrier associated with their topology.

The quantum problem is subtler. In spin-$\tfrac12$ systems the local orientation of a spin is not itself a classical vector, and a translationally invariant quantum ground state can have uniform $\langle \hat{\mathbf S}_i\rangle$ even when it is the quantum analogue of a classical skyrmion. For that reason, several works replace direct real-space magnetization imaging by multi-spin diagnostics. A central quantity is the scalar chirality operator

$$\hat{\chi}_{ijk}=\hat{\mathbf S}_i\cdot\left(\hat{\mathbf S}_j\times\hat{\mathbf S}_k\right),$$

and the associated quantity

$$Q_\Psi=\frac{N}{\pi}\left\langle \hat{\mathbf S}_1\cdot\left(\hat{\mathbf S}_2\times\hat{\mathbf S}_3\right)\right\rangle,$$

which is nearly field independent in the quantum-skyrmion phase of a triangular-lattice spin-$\tfrac12$ model and reduces to the classical topological invariant in the large-skyrmion limit (Sotnikov et al., 2020). In dense skyrmion crystals, a discretized quantum order parameter $Q_{\text{BL}}$ likewise remains a useful indicator of quantum skyrmions and resolves a quantum phase transition between two skyrmion-crystal phases even when the classical discretized skyrmion number is the same in both phases (Mæland et al., 2022).

This distinction between classical topology and quantum characterization is foundational for the notion of a skyrmion qudit. The logical states are not merely different pictures of a classical texture; they are many-body quantum states, typically identified by collective-coordinate quantization, scalar chirality, generalized winding diagnostics, or symmetry-resolved low-energy spectra (&&&^^^^2 OR abs:\2^^^^&&&, Sotnikov et al., 2020).

2. Helicity quantization and the explicit skyrmion-qudit construction

The most direct skyrmion-qudit model arises in frustrated triangular-lattice magnets, where the skyrmion helicity becomes a quantum variable. The effective helicity Hamiltonian is

$$\hat H_{S_z}=k(\hat S_z-h/k)^2-E_z\cos\hat\varphi_0,$$

with PRESERVED_PLACEHOLDER_^^^^2 OR abs:\2all:Skyrmion Qudit OR ti:\2^^^^, where PRESERVED_PLACEHOLDER_^^^^2 OR abs:\2 OR abs:\2^^^^, PRESERVED_PLACEHOLDER_^^^^2 OR abs:\22^^^^, and PRESERVED_PLACEHOLDER_^^^^2 OR abs:\2^^^^3 is the effective electric-field coupling (&&&^^^^2all:Skyrmion Qudit OR ti:\2^^^^&&&). After rescaling and a gauge transformation, the problem becomes a Mathieu–Schrödinger equation,

PRESERVED_PLACEHOLDER_^^^^2 OR abs:\2^^^^4

so the skyrmion helicity is mathematically a quantum rotor in a periodic potential (&&&^^^^2all:Skyrmion Qudit OR ti:\2^^^^&&&).

In the small-barrier regime, the low-energy sector is well approximated by a two-level system, reproducing the earlier helicity-qubit picture (&&&^^^^2all:Skyrmion Qudit OR ti:\2^^^^&&&, Psaroudaki et al., 2024). The key result of the 2^^^^2all:Skyrmion Qudit OR ti:\22^^^^5 work is that this truncation fails for arbitrary electric field. By tracking the full Mathieu spectrum through the Ince–Strutt diagram and resolving the symmetry sectors associated with the Klein four-group and its subgroups, the authors show that a sufficiently large barrier populates a four-dimensional subspace built from the irreducible states

PRESERVED_PLACEHOLDER_^^^^2 OR abs:\2^^^^5

so the correct description is a skyrmion qudit rather than a skyrmion qubit (&&&^^^^2all:Skyrmion Qudit OR ti:\2^^^^&&&).

The resulting qudit density matrix is explicitly PRESERVED_PLACEHOLDER_^^^^2 OR abs:\2^^^^6, with equal diagonal populations PRESERVED_PLACEHOLDER_^^^^2 OR abs:\2^^^^7 and nontrivial off-diagonal phases determined by accumulated dynamical and geometric phases during an adiabatic electric-field ramp across the small-, intermediate-, and large-barrier regions (&&&^^^^2all:Skyrmion Qudit OR ti:\2^^^^&&&). After a generalized measurement in the energy basis, the state reduces to the maximally mixed state on this four-dimensional subspace. Before measurement, however, it is a coherent four-level state.

A notable quantitative claim is that the PRESERVED_PLACEHOLDER_^^^^2 OR abs:\2^^^^8 norm of coherence of the skyrmion quantum qudit is a thousand times larger than the coherence of the skyrmion quantum qubit (&&&^^^^2all:Skyrmion Qudit OR ti:\2^^^^&&&). In resource-theoretic language, this makes the large-barrier regime qualitatively different rather than merely a weak correction to qubit physics.

3. Broader multi-level realizations suggested by related skyrmion literature

The helicity-based construction is the clearest realization of a skyrmion qudit, but related work suggests several other routes to multi-level skyrmion Hilbert spaces.

In frustrated quantum dimer systems on a bilayer triangular lattice, each dimer is a four-level local system PRESERVED_PLACEHOLDER_^^^^2 OR abs:\2^^^^9, represented by $$Q=\frac{1}{4\pi}\int d^2r\;\mathbf{m}(\mathbf{r})\cdot\left(\partial_x\mathbf{m}(\mathbf{r})\times\partial_y\mathbf{m}(\mathbf{r})\right),$$^^^^2all:Skyrmion Qudit OR ti:\2^^^^ coherent states, and the corresponding classical phase space is $$Q=\frac{1}{4\pi}\int d^2r\;\mathbf{m}(\mathbf{r})\cdot\left(\partial_x\mathbf{m}(\mathbf{r})\times\partial_y\mathbf{m}(\mathbf{r})\right),$$^^^^2 OR abs:\2^^^^ (Williams et al., 27 Jun 2025). The skyrmion textures in this setting are maps $$Q=\frac{1}{4\pi}\int d^2r\;\mathbf{m}(\mathbf{r})\cdot\left(\partial_x\mathbf{m}(\mathbf{r})\times\partial_y\mathbf{m}(\mathbf{r})\right),$$2, classified by $$Q=\frac{1}{4\pi}\int d^2r\;\mathbf{m}(\mathbf{r})\cdot\left(\partial_x\mathbf{m}(\mathbf{r})\times\partial_y\mathbf{m}(\mathbf{r})\right),$$3, with topological charge

$$Q=\frac{1}{4\pi}\int d^2r\;\mathbf{m}(\mathbf{r})\cdot\left(\partial_x\mathbf{m}(\mathbf{r})\times\partial_y\mathbf{m}(\mathbf{r})\right),$$4

Because the local degree of freedom is already a ququart, the resulting $$Q=\frac{1}{4\pi}\int d^2r\;\mathbf{m}(\mathbf{r})\cdot\left(\partial_x\mathbf{m}(\mathbf{r})\times\partial_y\mathbf{m}(\mathbf{r})\right),$$5 skyrmion crystals provide a concrete topological texture built from four-level quantum objects. The paper itself is not written in quantum-information language, but it explicitly identifies the local dimer as a four-level system and its skyrmions as textures in a ququart phase space (Williams et al., 27 Jun 2025).

A different route appears in exact-diagonalization studies of DMI-stabilized skyrmionic states on a ^^^^2 OR abs:\2^^^^9-site triangular lattice. In the open-boundary skyrmionic regime, the lowest few eigenstates all remain skyrmion-like and the level spacings are anharmonic, with $$Q=\frac{1}{4\pi}\int d^2r\;\mathbf{m}(\mathbf{r})\cdot\left(\partial_x\mathbf{m}(\mathbf{r})\times\partial_y\mathbf{m}(\mathbf{r})\right),$$6 (Sticlet et al., 15 Nov 2025). That work is framed as a qubit study, using only the ground and first excited skyrmionic states, but the reported anharmonic ladder of skyrmion-sector eigenstates suggests a direct qudit generalization in which $$Q=\frac{1}{4\pi}\int d^2r\;\mathbf{m}(\mathbf{r})\cdot\left(\partial_x\mathbf{m}(\mathbf{r})\times\partial_y\mathbf{m}(\mathbf{r})\right),$$7 are successive skyrmionic eigenstates (Sticlet et al., 15 Nov 2025).

An even earlier microscopic model of an individual quantal nano-skyrmion embedded in a ferromagnetic environment found a “whole series of skyrmion states,” including “extra-skyrmion states due to the quantization of the central spin of the singularity” (Gauyacq et al., 2019). In that seven-spin core model, the skyrmion sector contains a skyrmion ground state, additional skyrmion-like excitations, and extra skyrmion states distinguished by different central-spin quantization while retaining winding number $$Q=\frac{1}{4\pi}\int d^2r\;\mathbf{m}(\mathbf{r})\cdot\left(\partial_x\mathbf{m}(\mathbf{r})\times\partial_y\mathbf{m}(\mathbf{r})\right),$$8 after normalization (Gauyacq et al., 2019). This is another explicit multi-level skyrmion spectrum.

These examples indicate that “skyrmion qudit” can be used narrowly for the helicity-Mathieu model of a frustrated triangular magnet, or more broadly for multi-level quantum systems encoded in skyrmion helicity, internal modes, anharmonic skyrmion-sector eigenstates, or higher-dimensional texture manifolds such as $$Q=\frac{1}{4\pi}\int d^2r\;\mathbf{m}(\mathbf{r})\cdot\left(\partial_x\mathbf{m}(\mathbf{r})\times\partial_y\mathbf{m}(\mathbf{r})\right),$$9. The broader usage is suggested by the literature rather than uniformly standardized (&&&^^^^2all:Skyrmion Qudit OR ti:\2^^^^&&&, Williams et al., 27 Jun 2025, Sticlet et al., 15 Nov 2025, Gauyacq et al., 2019).

4. State preparation, control, readout, and hybrid interfaces

In the helicity-based formulation, control enters through the barrier height. The time-dependent protocol of the explicit qudit paper uses

$\tfrac12$^^^^2all:Skyrmion Qudit OR ti:\2^^^^

equivalently an electric-field ramp $\tfrac12$^^^^2 OR abs:\2^^^^, to move the skyrmion through different symmetry regions of the Mathieu spectrum and coherently populate four qudit levels (&&&^^^^2all:Skyrmion Qudit OR ti:\2^^^^&&&). In the broader helicity literature, the effective potential

$\tfrac12$2

can be tuned by dipolar interactions or in-plane anisotropy ($\tfrac12$3), electric fields via magnetoelectric coupling ($\tfrac12$4), and magnetic-field gradients or DMI ($\tfrac12$5) (Psaroudaki et al., 2024). That Hamiltonian was introduced for qubits, but it also defines the anharmonic multi-level spectrum from which qudits can be built.

Readout in quantum-skyrmion systems is often indirect. Because translationally invariant quantum skyrmions are not visible in local magnetization, integrated probes sensitive to scalar chirality are emphasized: the topological Hall effect, topological orbital magnetization, and nonlinear optical probes such as second-harmonic generation couple to $\tfrac12$6 and hence to the quantum skyrmion label (Sotnikov et al., 2020). Microwave spectroscopy, ferromagnetic resonance, resonant elastic x-ray scattering, NV-center magnetometry, tunneling magnetoresistance, and cavity-based dispersive readout have all been discussed as routes to resolving helicity or internal-mode splittings in skyrmion-based quantum devices (Psaroudaki et al., 2024, &&&^^^^2 OR abs:\2^^^^&&&).

Hybridization further broadens the control landscape. A recent tripartite proposal treats the clockwise gyration mode of a single skyrmion in a thin magnetic disk as a quantized bosonic mode with Hamiltonian

$\tfrac12$7

and couples it coherently to both an NV center and a flux-tunable transmon (&&&4^^^^2 OR abs:\2^^^^&&&). The derived Jaynes–Cummings-type interactions,

$\tfrac12$8

and the analogous skyrmion–transmon coupling do not themselves encode a skyrmion qudit, but they show that quantized skyrmion modes can function as interfaces between spin and superconducting platforms (&&&4^^^^2 OR abs:\2^^^^&&&). This suggests that skyrmion qudits, once realized, could be embedded in hybrid architectures rather than treated as isolated magnetic objects.

A distinct control direction uses skyrmions not as storage elements but as localized RF field sources. In that setting, a divergent–convergent pair of driven skyrmions generates a $\tfrac12$9 localized control field with potential gate fidelities surpassing $\langle \hat{\mathbf S}_i\rangle$^^^^2all:Skyrmion Qudit OR ti:\2^^^^ for $\langle \hat{\mathbf S}_i\rangle$^^^^2 OR abs:\2^^^^-gates and $\langle \hat{\mathbf S}_i\rangle$2 for $\langle \hat{\mathbf S}_i\rangle$3-gates, while producing a field on a neighboring qubit about ^^^^2 OR abs:\2^^^^5 times smaller than a comparable nanomagnet layout (Chowdhury et al., 2023). Although developed for proximate spin qubits, this degree of localization is directly relevant to selective multi-level addressing of proximate spin qudits.

5. Protection, coherence, and the limits of topological robustness

The phrase “topological protection” requires qualification in the quantum setting. Classical skyrmions are protected by the integer winding of a continuous field. Quantum skyrmions in spin systems do not possess an exactly conserved classical skyrmion number, because quantum tunneling can connect states that would be topologically distinct in the classical configuration space (Sotnikov et al., 2020). Scalar chirality and related order parameters are therefore continuous expectation values rather than strictly quantized operators. For small skyrmions, $\langle \hat{\mathbf S}_i\rangle$4 remains close to an integer only in favorable gapped regimes, and its deviation from unity tracks the strength of quantum fluctuations (Mæland et al., 2022).

This limitation is directly relevant for qudit design. Topology protects the existence of the skyrmion platform more strongly than it protects any particular internal level. In helicity-based schemes, the logical levels are determined by the shape of the effective potential and by tunneling amplitudes, not by an exact topological invariant. Related qubit work states the point plainly: skyrmion qubits are “topologically protected and macroscopic,” but “external influences can still affect them” (&&&^^^^2 OR abs:\2^^^^4&&&). In DMI-stabilized models, the same interaction that stabilizes the skyrmion also induces decoherence during gate operations; exact simulations with Lindblad terms show tunable anharmonic energy levels and coherent Bloch-sphere manipulation, but also DMI-driven entropy growth and reduced fidelity (Sticlet et al., 15 Nov 2025).

The qudit case inherits the same tradeoff, amplified by higher-dimensional control. More levels mean more transition frequencies, more leakage channels, and more opportunities for environment-induced mixing. The large-barrier skyrmion qudit of the Mathieu model is coherent by construction at the level of unitary evolution, but any physical implementation must preserve phase relations among four populated levels rather than two. A plausible implication is that the regime of useful $\langle \hat{\mathbf S}_i\rangle$5 will be constrained by how rapidly off-diagonal elements of the density matrix decay under realistic electric-field noise, magnon coupling, phonons, disorder, and device-specific dissipation. The literature does not yet provide a full decoherence theory for the four-level skyrmion qudit of the 2^^^^2all:Skyrmion Qudit OR ti:\22^^^^5 helicity model (&&&^^^^2all:Skyrmion Qudit OR ti:\2^^^^&&&, Sticlet et al., 15 Nov 2025).

6. Position within quantum skyrmionics and prospective developments

The skyrmion qudit sits at the intersection of several active strands of quantum-skyrmion research. One strand concerns many-body quantum diagnostics: scalar chirality, fidelity, generalized winding numbers, and entanglement in ferromagnetic and antiferromagnetic triangular-lattice models (Sotnikov et al., 2020, Corte et al., 26 Jan 2026). Another concerns collective-coordinate quantization: noncommuting skyrmion position operators, helicity rotor dynamics, Berry phases, and macroscopic quantum tunneling of position, helicity, nucleation, and collapse (&&&^^^^2 OR abs:\2^^^^&&&). A third concerns device functionality: helicity qubits, DMI-based skyrmionic qubits, hybrid NV–transmon couplers, and localized skyrmion-generated control fields (Psaroudaki et al., 2024, Sticlet et al., 15 Nov 2025, &&&4^^^^2 OR abs:\2^^^^&&&, Chowdhury et al., 2023).

Within this landscape, the 2^^^^2all:Skyrmion Qudit OR ti:\22^^^^5 “Quantum Skyrmion Qudit in a Triangular-lattice magnet” paper is significant because it removes the small-barrier approximation and gives a general analytic solution of the helicity model for arbitrary electric field (&&&^^^^2all:Skyrmion Qudit OR ti:\2^^^^&&&). It therefore marks a conceptual transition from “skyrmion qubit” as a convenient truncation to “skyrmion qudit” as the correct description in the large-barrier regime.

Additional developments suggest broader design tools. A Qiskit-based workflow has been used to generate four classes of synthetic skyrmion-like textures—chaotic, layered, ring, and wave—through a six-qubit circuit and image-analysis pipeline involving radial profiles, FFT spectra, Hu moments, GLCM features, SSIM, and PCA+KMeans clustering (&&&^^^^2 OR abs:\2^^^^4&&&). That work does not implement qudits, but it suggests that quantum-circuit exploration of skyrmion texture space could be used to search for distinguishable, structured skyrmion configurations relevant to future multi-level encodings (&&&^^^^2 OR abs:\2^^^^4&&&).

A further plausible extension is toward antiferromagnetic quantum skyrmions. DMRG studies on the antiferromagnetic triangular lattice find three-sublattice quantum antiferromagnetic skyrmion textures over a wide field range, with nonzero scalar chirality and nontrivial entanglement (Corte et al., 26 Jan 2026). Because the texture is intrinsically three-component at the sublattice level, this suggests a route to qutrit-like internal structure, although no explicit qudit manifold has yet been isolated (Corte et al., 26 Jan 2026).

In current usage, then, “skyrmion qudit” most precisely denotes the multi-level helicity system obtained in the large-barrier Mathieu regime of a frustrated triangular-lattice skyrmion (&&&^^^^2all:Skyrmion Qudit OR ti:\2^^^^&&&). More broadly, the term is becoming a natural shorthand for quantum-information encodings that exploit discrete skyrmion degrees of freedom beyond two levels: helicity doublets extended into higher manifolds, anharmonic skyrmion-sector spectra, $\langle \hat{\mathbf S}_i\rangle$6 textures built from local ququarts, and other quantum states rooted in skyrmion topology but not reducible to classical micromagnetic pictures (Williams et al., 27 Jun 2025, Sticlet et al., 15 Nov 2025, &&&^^^^2 OR abs:\2^^^^&&&).