The diagonal-coset scheme of Georgiev, Hadjiivanov and Matein (2024) encodes nn qubits in the fusion channels of 2n+22n+2 Fibonacci anyons — the ε\varepsilon primary fields of the Z3\mathbb{Z}_3 parafermion (Read–Rezayi k=3k=3) fractional quantum Hall state expected at filling factor ν=12/5\nu=12/5 — initialized on antidots threaded by one flux quantum and read out by Fabry–Pérot interferometry, with gates implemented by analytic-continuation (braid) matrices derived from the conformal field theory wave functions. The construction stops at the analytic level: the two-qubit braid generator B3(6)B_3^{(6)} couples the computational state ∣11⟩|11\rangle to a non-computational state (leakage), and the interferometric readout is analyzed only under the idealized assumption of zero electron tunneling into the antidot, where the chiral partition functions of the antidot edge reduce to their ideal single-character form. The open problem is to complete this scheme: construct leakage-free or explicitly leakage-bounded entangling gates for the 2n+22n+2-anyon pair encoding, and extend the initialization/readout analysis to the finite-electron-tunneling regime in which the ideal chiral partition functions must be replaced by their full simple-current-orbit sums.

Background

Topological quantum computation (TQC) encodes quantum information non-locally in the fusion channels of non-Abelian anyons so that gates implemented by braiding the anyons around one another are intrinsically protected against local noise. The Fibonacci anyon is the simplest non-Abelian anyon: it has two fusion channels governed by the rule ε×ε=I+ε\varepsilon\times\varepsilon=\mathbb{I}+\varepsilon, where I\mathbb{I} is the vacuum, and its braid group representations are dense in the unitary group, making braiding alone universal for quantum computation. The only established candidate physical realization is the Z3\mathbb{Z}_3 parafermion — the k=3k=3 Read–Rezayi — fractional quantum Hall (FQH) state, an incompressible electronic liquid in the second Landau level at filling factor ν=12/5\nu=12/5, for which large-scale density-matrix-renormalization-group (DMRG — a variational numerical method for strongly correlated systems on long cylinders) calculations support Read–Rezayi wave functions as the Coulomb ground state (Zhu, Gong, Haldane and Sheng 2015). A practical obstacle is that measured activation gaps — the energy cost of thermally exciting a quasiparticle pair, extracted from temperature-activated transport — in the second Landau level are roughly an order of magnitude smaller than at ν=5/2\nu=5/2 (Choi, Kang, Das Sarma, Pfeiffer and West 2008), so thermal quasiparticle creation and noise are correspondingly more severe. In two-dimensional conformal field theory (CFT), the language in which Read–Rezayi states are constructed, the quasiholes are represented by chiral primary fields, their multi-particle wave functions by conformal blocks, and exchanging quasiholes acts on these blocks as monodromy (analytic-continuation) matrices that furnish representations of the Artin braid group Bn\mathcal{B}_n. Georgiev, Hadjiivanov and Matein (2024), building on the explicit braid matrices of Hadjiivanov and Georgiev (2024), realize each qubit as the two fusion channels of a pair of ε\varepsilon fields — an nn-qubit register is 2n+22n+2 Fibonacci anyons — initialize anyons by threading antidots (potential hills inside the Hall liquid, forming an island whose edge can localize quasiparticles) with one quantum of Aharonov–Bohm flux, and propose non-destructive readout via Fabry–Pérot interferometry, in which quasiparticles tunnel between edges through quantum point contacts (QPCs) and the interference pattern reveals the monodromy — and hence the fusion channel — of the quasiparticle localized on the antidot. Single-qubit gates are approximated by explicit braid words; but for two qubits the generator B3(6)B_3^{(6)} mixes the computational basis state ∣11⟩|11\rangle with a non-computational fusion state ∣NC⟩|NC\rangle, causing leakage of information out of the computational subspace, and the authors only suggest excluding 'inert' anyons from braids without developing any systematic scheme. The operator-product-expansion (OPE — the short-distance expansion governing which fields are created together) of ε×ε\varepsilon\times\varepsilon contains, besides the vacuum, the higher-dimension primary ε′\varepsilon' (CFT dimension 7/57/5), so creating an ε\varepsilon anyon on an antidot necessarily co-creates ε′\varepsilon'; the source paper analyzes this field and concludes that ε′\varepsilon' is in the same topological sector as ε\varepsilon — described by the same parafermion character and having the same electric properties — so that this peculiarity does not change the encoding scheme, and that the co-creation of ε′\varepsilon' is not a problem for qubit initialization. The readout analysis assumes the antidot is so far from the outer edge that electron tunneling into it is negligible, so the chiral partition functions of the antidot edge reduce to their ideal single-character form rather than sums over the entire orbit of the simple current (a simple current is a chiral primary field whose fusion with any primary gives another primary, permuting them in a single orbit); finite electron tunneling would require replacing them by the full simple-current-orbit sums. Related but distinct questions in the wider literature: for the standard three-anyon-per-qubit Fibonacci encoding, no leakage-free entangling gate made by braiding alone was known as of 2018 (Bigelow and Levaillant 2018, who obtain an exact entangling gate only by supplementing braiding with measurements); numerically compiled, approximately leakage-free two-qubit gates appeared in 2024 (Burke, Aravanis, Aspman, Mareček and Vala 2024); fault-tolerant error correction for an abstract Fibonacci-anyon code on a torus at finite temperature has been simulated by Schotte, Burgelman and Zhu (2022); and Mach-Zehnder-interferometer probes of k=3k=3 parafermion statistics at ν=12/5\nu=12/5 were proposed by Law (2008). None of these addresses the pair encoding or the antidot initialization/readout scheme of the diagonal-coset proposal.

Problem Statement

For the diagonal-coset Fibonacci-anyon topological quantum computer — an nn-qubit register encoded in the fusion channels of 2n+22n+2 ε\varepsilon fields of the Z3\mathbb{Z}_3 parafermion fractional quantum Hall state at ν=12/5\nu=12/5, initialized by flux-threading of antidots and read out by Fabry–Pérot interferometry, with gates given by the analytic-continuation braid matrices of Hadjiivanov and Georgiev — complete the construction by answering, jointly: (1) Gate layer: construct, for the 2n+22n+2-anyon pair encoding, entangling two-qubit (and, if required for universality, multi-qubit) gate implementations that realize a specified target entangling gate of maximal entangling power up to a stated accuracy (so that their entangling strength is fixed and non-vanishing, not an arbitrarily weak near-identity perturbation) and whose leakage out of the computational subspace is exactly zero, or bounded by an explicit, quantified error that is non-trivial — strictly smaller than the leakage of the uncorrected two-qubit coupling of B3(6)B_3^{(6)}, so that the bound itself witnesses control of the non-computational coupling rather than merely restating the trivial bound of unity — whether by braid words that avoid or cancel the non-computational coupling of generators such as B3(6)B_3^{(6)}, by the scheme's non-destructive interferometric measurements, or by a proved impossibility showing that no route admitted here — braid words, the scheme's non-destructive interferometric measurements, or a combination of the two — achieves an entangling gate satisfying the same leakage standard; (2) Readout layer: extend the analysis of the antidot/interferometer initialization and readout beyond the idealized zero-tunneling limit, to the finite-electron-tunneling regime in which the ideal chiral partition functions K3 ch(Λˉ1+Λˉ2)K_3\,\mathrm{ch}(\bar\Lambda_1+\bar\Lambda_2) must be replaced by their full simple-current orbits, and determine how the finite tunneling modifies the interferometric readout of the localized quasiparticle's fusion channel. A complete answer supplies explicit protocols or constructions with proven or numerically certified leakage bounds meeting the non-degeneracy and non-triviality conditions of (1) (layer 1), and an analytic or numerically validated finite-tunneling partition-function and readout analysis (layer 2); a rigorous negative result on layer 1 — e.g. a proof that no entangling gate satisfying the layer-1 leakage standard can be achieved in this encoding by any of the admitted routes — equally resolves that part of the problem.

References

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