Making the universal $D(S_3)$ topological gate set fault-tolerant: decoders, thresholds, and robust anyon-manipulation protocols under realistic noise including measurement errors

The quantum double of the smallest non-Abelian group, $D(S_3)$, was recently identified as a sweet spot for topological quantum computation: simple enough to be prepared by finite-depth adaptive circuits, yet powerful enough to support a universal gate set from anyon braiding and fusion/charge measurements alone. A pedagogical exposition (Lo, Lyons, Verresen, Vishwanath, Tantivasadakarn, arXiv:2502.14974) consolidates this gate set and its lattice implementation, but explicitly leaves open how to make it robust: the source asks, verbatim, "given near-term, noisy devices, how can we make such a gate set robust?" To date, decoders tailored to $D(S_3)$ have been proposed but not demonstrated to preserve the encoded computation, no error-correction threshold (numerical or analytic) exists for any non-Abelian quantum double with noisy syndrome measurements, and noise-robust, constant-depth coherent transport and active stabilization protocols remain undeveloped, even as adjacent pieces (an incremental adaptive anyon-movement protocol with local-code concatenation, heralded and Bayes-optimal decoders for $D_4$ under perfect measurement, and an experimental realization of the idealized $S_3$ gate set on a 54-qubit quantum processor) have been developed. The surviving problem is stated at the source's generality: a quantitative error-correction and fault-tolerance theory — decoders, thresholds including measurement errors, compatible movement/stabilization protocols, and the induced resource overhead — for universal computation with the $S_3$ quantum double.

Background

Topological order (TO) is a quantum phase of matter whose degenerate ground state and quasiparticle excitations are protected against local noise. Its pointlike excitations can have exchange statistics beyond bosons and fermions ("anyons"); when the exchange and fusion of an anyon type act non-commutatively on a multi-anyon state, the anyon is non-Abelian, and its internal fusion space can encode quantum information. Kitaev's quantum double construction associates to any finite group $G$ an exactly solvable lattice model whose deconfined phase has anyonic gauge charges, fluxes, and dyons — particles carrying both a charge and a flux — (Kitaev, Ann. Phys. 303, 2 (2003)). For $G=\mathbb{Z}_2$ this is the toric code, whose anyons are Abelian and not universal for quantum computation by braiding; Mochon (Phys. Rev. A 69, 032306 (2004)) showed that already the smallest non-Abelian group, the symmetric group $S_3$, supports universal anyonic computation. In the $D(S_3)$ model, anyons carry a flux labeled by a conjugacy class of $S_3$ (trivial $A$, Abelian $B$, non-Abelian $C$ classes) and a charge labeled by an irreducible representation; fusion of two anyons can have several admissible outcomes with different probabilities — the defining feature that makes non-Abelian anyons computationally powerful and, simultaneously, hard to error-correct.

Lo, Lyons, Verresen, Vishwanath, and Tantivasadakarn (arXiv:2502.14974) review how to compute with $D(S_3)$: logical qutrits (three-level quantum units) are encoded in pairs of non-Abelian fluxes; a universal gate set — Hadamard and phase gates plus the non-Clifford controlled-controlled-$Z$ (CCZ) — is assembled from three topological primitives: a braiding-derived "pull-through" entangling gate, a qutrit $Z$-basis measurement (winding a charge singlet around a flux and re-fusing), and a qutrit $X$-basis measurement (fusion/charge measurement). Because several primitives succeed only probabilistically (fusion outcomes are non-deterministic), gates and state preparations use repeat-until-success repetition, e.g. about 35 repetitions of the sign-flip construction for a 99% success rate. At the lattice level these primitives are realized by ribbon operators, generalized string-like operators that create and transport anyons; the notes introduce an extended ribbon operator for state initialization. Crucially, $S_3$ is a solvable group, so the $D(S_3)$ ground state and ribbon operations can be implemented by adaptive (measurement-and-feedforward) circuits of constant depth (Ren, Tantivasadakarn, Williamson, arXiv:2411.04985; Lyons et al., arXiv:2411.04181), making the order accessible to near-term digital devices — an advantage no nonsolvable universal order (e.g. Fibonacci anyons) shares. This positioning has been validated experimentally: a 54-qubit $S_3$ topological order with a universal braiding-plus-measurement gate set and topological preparation of magic states (non-stabilizer resource states that boost a Clifford gate set to universality) was demonstrated on the Quantinuum H2 trapped-ion processor (Lo et al., arXiv:2601.20956).

What remains missing is the layer that would make this a genuine fault-tolerant architecture. For Abelian codes such as the toric code, active error correction is theoretically understood: syndromes (local violation flags) are extracted by repeated measurement, a decoder (a classical algorithm, e.g. minimum-weight perfect matching, MWPM) infers the most likely error, and a threshold error rate $p_c$ — below which increasing system size suppresses logical failures — can be computed from a mapping to a classical statistical-mechanics model. For non-Abelian orders none of this is in place for $S_3$: fusion outcomes are probabilistic so anyon pairs cannot simply be paired and annihilated; error information can hide in the internal fusion degrees of freedom of non-Abelian anyons and become invisible to syndrome measurements; and the repeated-measurement, feedforward structure of the gate primitives means syndrome extraction is itself noisy. Earlier work established pieces for other anyon theories: Monte-Carlo error-correction studies of non-Abelian codes (Wootton, Burri, Iblisdir, Loss, Phys. Rev. X 4, 011051 (2014)), a teleportation-based fault-tolerant scheme for general non-Abelian anyons (Dauphinais, Poulin, Commun. Math. Phys. 355, 519 (2017)), and decoding thresholds for the universal Fibonacci Turaev–Viro code (a lattice code realizing Fibonacci-anyon topological order) (Schotte, Zhu, Burgelman, Verstraete, Phys. Rev. X 12, 021012 (2022)). Recent analyses of the related $D_4$ order showed that non-Abelian orders can be unexpectedly robust against decoherence (Sala, Alicea, Verresen, arXiv:2409.12948; Sala, Verresen, arXiv:2409.12230), and intrinsic heralding — using non-deterministic fusion outcomes themselves as noise indicators — yields decoders and Bayes-optimal thresholds for $D_4$ under perfect measurement (Jing, Sala, Jiang, Verresen, arXiv:2507.23765). But the source notes' concluding question — how to make the $S_3$ universal gate set robust on noisy, measurement-imperfect devices — remains without an answer, as the notes themselves acknowledge when deferring stabilization protocols, non-Abelian error-correction theory, and the fault tolerance of measurement-based protocols to future work, and when deferring coherent anyon transport (moving an anyon without leaving remnant excitations at its original site) to an upcoming work.

Problem Statement

The source question, stated verbatim in its conclusion, is: given near-term, noisy devices, how can the universal $D(S_3)$ topological gate set be made robust? Concretely, construct and analyze an active error-correction and fault-tolerance framework for universal quantum computation with the $S_3$ quantum double — at the source's generality, without restricting to perfect syndrome measurement, to a special noise model, or to a subset of the gate set — comprising jointly supporting components that serve the single common objective of a robust, in-practice fault-tolerant realization of the universal $S_3$ gate set: (i) a decoding and correction scheme for the non-Abelian anyon error model of $D(S_3)$ (errors create probabilistic-fusing anyon pairs and can hide in anyon internal fusion degrees of freedom), demonstrated or proven to preserve the encoded computation; (ii) an error-correction threshold for realistic noise models that include syndrome-measurement errors, i.e. the repeated-measurement, feedforward setting in which the repeat-until-success primitives (flux/charge measurements, sign-flip constructions) actually operate — either a numerical threshold determination (e.g. via statistical-mechanics mapping and finite-size scaling, including the three-dimensional space-time decoding problem) or a proof of threshold existence; (iii) protocols compatible with (i)-(ii) for coherently moving non-Abelian anyons (preserving internal state, leaving no remnant excitations) and for actively stabilizing the prepared $S_3$ topological order, including their noise robustness; and (iv) the resource accounting this framework induces — physical qubit/qutrit counts, circuit depth, and measurement/feedback repetitions per logical gate, e.g. for magic-state preparation and a non-Clifford gate — quantifying the overhead of executing a representative nontrivial algorithm fault-tolerantly in a $D(S_3)$ device. A complete resolution must address all four components (i)–(iv); since (iii) and (iv) presuppose (i) and (ii), a resolution that omits the decoding/correction scheme (i) or the error-correction threshold (ii) is incomplete. Answers may cover the more general class of solvable-group quantum doubles provided the $S_3$ universal-gate-set case is treated.

References

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