Non-Abelian quantum Hall states at strong Landau level mixing: finding anyons with first-principles AI

Pierre-Antoine Graham; Ahmed Abouelkomsan; Liang Fu

cond-mat.mes-hallOct 13, 2026

Non-Abelian quantum Hall states at strong Landau level mixing: finding anyons with first-principles AI

Authors: Pierre-Antoine Graham, Ahmed Abouelkomsan, Liang Fu

Organizations: Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

Abstract

Non-Abelian anyons, strange particles forming the basis of fault-tolerant topological quantum computing, are predicted to emerge in fractional quantum Hall states at even-denominator filling factors. Identifying which topological order is realized has proven to be a challenging problem that requires an accurate, non-perturbative treatment of realistic Landau level mixing, which has so far been missing from theoretical studies. Here we present the first complete solution of non-Abelian quantum Hall problem that includes all electrons and all Landau levels, for spin-polarized electrons at filling factor $ν= 3/2$ . We develop a momentum-resolved neural-network variational wave function, optimized solely on the microscopic Hamiltonian, which gives direct access to the topologically degenerate ground states. Without any physics input about the expected outcome, it discovers a degenerate ground-state manifold whose entanglement reveals anyons with quantum dimension $\sqrt{2}$ , the hallmark of Ising-type non-Abelian order. By directly computing the overlap with trial wave functions, we further show that Landau level mixing favors the anti-Pfaffian over other candidate states. Our results demonstrate the power of neural-network wave functions in finding and characterizing non-Abelian anyons from first principles, in regimes of strong Landau level mixing, relevant to many recent experiments.

Figures & tables

Figure 1: Schematic illustration of the momentum-resolved MagNet ansatz. Electron coordinates are mapped to high-dimensional features

\{\boldsymbol{h}_{i}\}

that are periodic on the torus and then passed through

L

layers combining self-attention and multilayer perceptrons. The network outputs the periodic amplitudes

F^{n}_{j}

and the winding maps

\eta^{(\alpha,n)}_{j}

, which parametrize the zeros (crosses) of the quasiperiodic factor

\chi^{n}_{j}

, shown on the supercell with its phase encoded in color. Together,

F^{n}_{j}

and

\chi^{n}_{j}

form the many-body orbitals of the variational wave function

\Psi_{\theta}

in Eq. ( 2 ). The momentum eigenstate

\Phi_{\mathbf{k}}

is then obtained by summing the translated copies of

\Psi_{\theta}

over the

N_{\phi}

center-of-mass magnetic translations

T(\mathbf{a})

, as in Eq. ( 5 ).

Figure 2: (a) Lowest energy in each momentum sector

\mathbf{k}=(k_{1},k_{2})

, labeled by

k_{1}+N_{1}k_{2}

, for

N=21

electrons and

\kappa=1.0

, obtained from NN-VMC and from ED restricted to the LLL and 1LL. We use a hexagonal supercell spanned by

\mathbf{L}_{1}=N_{1}\mathbf{a}_{1}

and

\mathbf{L}_{2}=N_{2}\mathbf{a}_{2}

with

N_{1}=7

and

N_{2}=2

, so that

k_{i}=0,\ldots,N_{i}-1

. The NN-VMC energies lie well below the ED energies in all sectors, and both show a twofold degenerate ground state at

\mathbf{k}=(0,0)

and

(0,1)

. (b) Pair correlation function

g(\mathbf{r})

of the two NN-VMC ground states.

Figure 3: (a) Lowest energies in each momentum sector

\mathbf{k}=(k_{1},k_{2})

, labeled by

k_{1}+N_{1}k_{2}

, for

N=18

electrons and

\kappa=1.0

, obtained from NN-VMC and from ED restricted to the LLL and 1LL. We use a hexagonal supercell spanned by

\mathbf{L}_{1}=N_{1}\mathbf{a}_{1}

and

\mathbf{L}_{2}=N_{2}\mathbf{a}_{2}

with

N_{1}=4

and

N_{2}=3

. Numbers indicate twofold degenerate ED levels within a sector. The blue labeled states are orthogonal ground states obtained with NN-VMC. (b) Differences

\Delta S_{n}

between the second Rényi entropies of the six MESs and the smallest one, for three inequivalent noncontractible cuts of the torus (bottom). For each cut, four MESs have

\Delta S_{n}\approx 0

, consistent with Abelian sectors, while the remaining two lie close to

\log 2

(dashed line), as expected for

d_{\sigma}=\sqrt{2}

Figure 4: (a) Occupation $\nu_{n}$ of Landau level $n$ , on a logarithmic scale, for $N=18$ and $N=21$ electrons at $\kappa=1,2$ and $5$ . Inset: deviation $\Delta\nu_{n}$ from the unmixed occupations $\nu_{0}=1$ , $\nu_{1}=1/2$ , and $\nu_{n\geq 2}=0$ . Both the LLL and the 1LL are depleted, with the weight transferred to higher levels. (b) Pair correlation function computed at $\kappa=5$ and $20$ (see [ 56 ] for more details).

Figure 5: Overlap

\mathcal{O}

between the NN-VMC ground-state manifold and the Moore–Read Pfaffian, the anti-Pfaffian, the PH-Pfaffian, and the exact Coulomb ground state of the 1LL-projected Hamiltonian, as a function of Landau level mixing

\kappa

, for

N=18

electrons. Open symbols with dashed lines show overlaps with the bare trial states, defined within the LLL and 1LL, and filled symbols with solid lines show overlaps with dressed trial states, which include a cloud of virtual inter-Landau-level excitations [ 56 ] . In both cases, the overlap with the anti-Pfaffian exceeds that with the other trial states at all

\kappa

Architecture		Optimization		Monte Carlo
Layers $L$	4	Optimizer	MinSR	Walkers	1024
Heads	4	Learning rate	0.02	Steps/sample	10
Attention dim.	32	Iterations	$10^{3}$ to $10^{4}$	Move width	$\ell_{B}$
Perceptron dim.	128			Burn-in	300
Perceptrons/layer	2
Determinants	2

Table 1: Hyperparameters used in this work.

Figure S1: Supercell tilings and fine-projected spectra, one row per system. (a) The tiling of

N=18

(

4\times 3

cells) and (d) of

N=21

(

7\times 2

cells). The one-flux cell vectors

\mathbf{a}_{1}

\mathbf{a}_{2}

are in orange, the fine lattice

\Lambda_{\phi}

in grey, and the maximal commuting set of translations in blue. In one cell the generator

T(\mathbf{a}_{2}/n_{2})

of the fine projection is drawn with its

n_{2}

points in dark blue. The unused

\mathbf{a}_{1}/3

N=18

is dashed. (b, c) NN-VMC spectra of

N=18

\kappa=2

and

\kappa=5

. Each bar is the energy of the fine-projected state

\Phi_{\mathbf{k},l}

of Eq. ( S25 ) in one momentum sector, with its statistical error. Energies are relative to the mean

E_{0}

of the sixfold ground multiplet,

E_{0}=-4.9736\pm 0.0001\,\hbar\omega_{c}

(b) and

-37.1050\pm 0.0002\,\hbar\omega_{c}

(c). The sectors

(1,0)

and

(3,0)

hold two states: the second is drawn in blue, shorter, over the first. (e, f) The same for

N=21

\kappa=2

and

\kappa=5

. Here

E_{0}

is the mean of the ground doublet (

k_{1}+N_{1}k_{2}=0,7

E_{0}=-6.5028\pm 0.0003\,\hbar\omega_{c}

(e) and

-45.0552\pm 0.0003\,\hbar\omega_{c}

(f).

Figure S2: Wigner crystals at $N=21$ , $\kappa=20$ . (a, b) The two $21$ -site triangular lattices $A$ and $B$ of the $7\times 2$ torus, rotated by $\pm 10.9^{\circ}$ from $\mathbf{L}_{1}$ , and the fine translation that preserves each of them: $T_{A}=T(\mathbf{a}_{1})\,T(\mathbf{a}_{2}/7)^{4}$ , by $\mathbf{a}_{1}+4\mathbf{a}_{2}/7$ , and $T_{B}=T(\mathbf{a}_{1})\,T(\mathbf{a}_{2}/7)^{8}$ , by $\mathbf{a}_{1}+8\mathbf{a}_{2}/7$ (arrows). (c) The fine sectors $(k_{1},l)$ of each sector $k_{2}=0,1$ , both labels modulo $7$ . A rigid crystal of orientation $A$ ( $B$ ) can exist only on the red (blue) line, Eq. ( S31 ); the lines cross only at the origin $k_{1}=0$ , $l=3k_{2}$ (black). (d) Spectrum relative to $E_{0}=-252.819\,\hbar\omega_{c}$ , the mean energy of the two ground states $G$ : in each sector the lowest fine-projected state on the $A$ line (red) and on the $B$ line (blue); at the origin, columns $0$ and $7$ , the cat pair, the ground state $G$ (A $+$ B) and its penalty partner $P$ (A $-$ B), in black, and the energies of the De-Cat states $\Psi_{\pm}=(G\pm e^{-i\phi_{A}}P)/\sqrt{2}$ (open diamonds). (g) The sign flip in the sector $k_{2}=0$ : the phase of $P(x)/G(x)$ , measured from its value on the $A$ side, against the orientation $O(x)=[S_{A}(x)-S_{B}(x)]/[S_{A}(x)+S_{B}(x)]$ of the sampled configurations $x$ , with $S_{X}$ of Eq. ( S32 ). (e) Pair correlation $g(\mathbf{r})$ and Bragg weights $S_{A}$ , $S_{B}$ of $G$ and $P$ : both superpose the two orientations. (f) The same as (e) for $\Psi_{+}$ and $\Psi_{-}$ : single crystals $A$ and $B$ .

Quantum Many-Body Physics

Quantum many-body physics studies how interacting quantum particles produce collective behavior, phases of matter, and complex dynamics.

See all 55 papers

Sep 7, 2026

Topology Obstructs Pure Foundation Neural Quantum States

Foundation models for ground states in spin-1/2 systems are a promising method for problems ranging from quantum chemistry to identifying new phase diagrams. Nearly all such models are currently pure-states that condition on the Hamiltonian's parameters, whose Monte Carlo samples give energy estimates according to the variational principle. In this contribution, we show that this representation is topologically obstructed. For any gapped Hamiltonian family whose ground-state bundle is non-trivial, every continuous normalized state-vector model has zero fidelity with the ground state at some parameter value in the Hamiltonian family. For that value, the energy is at least one spectral gap,

Δ

, with an

O(Δ)

gap in an open-neighbourhood of that point. We show that this is a sufficient no-go also in the case of degenerate ground-state manifolds, time dynamics, and periodic systems with mixed space-time topology, demonstrating these obstructions on one- and two-qubit systems. We discuss how this causes a spike in the fidelity susceptibility, giving a numerical signature of a phase-transition where there is none. We then show that operator-valued models canonically avoid these obstructions and preserve topological information, implying a structural necessity in representation for foundation neural quantum states.

Timothy Heightman, Elena Orlova, Philip Mantrov +1

1ICFO–Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Barcelona, Spain · 2Simulacra Research Inc., London, UK and Chicago, USA

Neural Quantum States Quantum Many-Body Physics

Ask PDF

May 21, 2026

Comment on "Spin-1/2 Kagome Heisenberg Antiferromagnet: Machine Learning Discovery of the Spinon Pair-Density-Wave Ground State"

A recent article [Phys. Rev. X 15, 011047 (2025)] utilizes group-equivariant convolutional neural networks to study the ground state of the kagome Heisenberg antiferromagnet. On the largest finite-size cluster studied to date (

N=108

), the authors report variational energies significantly lower than other numerical methods, including state-of-the-art density matrix renormalization group (DMRG) calculations. In contrast to previous results suggesting a possible spin-liquid ground state, the authors observe a spinon pair-density-wave ground state. We find that: (i) the reported low energies are artifacts of broken ergodicity in the Metropolis--Hastings sampling, since the single-spin-flip update rule utilized by the authors effectively freezes the Markov chains; and (ii) when ergodic sampling is enforced via spin-exchange updates, the neural network converges to energies significantly higher than existing DMRG results, calling the paper's claims into question.

Helia Kamal, Dominik Kufel, DinhDuy Vu +2

Department of Physics, Harvard University, Cambridge, MA 02138, USA · Harvard Quantum Initiative, Cambridge, MA 02138, USA · Department of Physics, Boston University, Boston, MA 02215, USA

Neural Quantum States Markov Chain Monte Carlo Variational Monte Carlo Quantum Many-Body Physics

Ask PDF

Jun 14, 2026

Learning ground state observables from quantum computing experiments

Recent theoretical progress has established conditions under which machine learning models can efficiently predict ground-state properties of gapped local Hamiltonians when trained on quantum-generated data. Previous experimental demonstrations in this paradigm, however, have largely been limited to small systems or highly structured states, due to the difficulty of preparing many-body ground states on quantum processors. In this work, we demonstrate learning from experimental quantum data generated from approximate ground states of the two-dimensional Heisenberg XXZ model with system sizes up to 115 qubits. We construct a dataset of single-site expectation values, two-point correlations, and 12-body loop correlations across the antiferromagnetic phase. We then train neural networks on this data and show that they can accurately predict spatially resolved observables for previously unseen Hamiltonian parameters, both within the training distribution and in an out-of-distribution regime approaching the phase boundary. Our results demonstrate the practical realization of learning from quantum data for an interacting two-dimensional many-body system at scale, motivating a path toward regimes where quantum processors could provide training data beyond the reach of classical approximation methods.

Ben Jaderberg, Freya Shah, Minjun Jeon +3

IBM Quantum, IBM Research Europe, Hursley, Winchester, SO21 2JN, United Kingdom · Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, United Kingdom · IBM Quantum, T. J. Watson Research Center, Yorktown Heights, NY 10598, USA +4

Quantum Machine Learning Quantum Many-Body Physics

Ask PDF

Abstract

Quantum Many-Body Physics

Explore similar work

Topology Obstructs Pure Foundation Neural Quantum States

Comment on "Spin-1/2 Kagome Heisenberg Antiferromagnet: Machine Learning Discovery of the Spinon Pair-Density-Wave Ground State"

Learning ground state observables from quantum computing experiments