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arXiv:2508.11230 · Chinese Physics B

Fault-tolerant mixed boundary punctures on the toric code

Yao Shen, Fulin Zhang

Published Apr 8, 2026• 0 citations
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Abstract

Abstract Defects on the toric code, a well-known exactly solvable Abelian anyon model, can exhibit non-Abelian statistical properties, which can be classified into punctures and twists. Benhemou et al. [ Phys. Rev. A 105 042417 (2022)] introduced a mixed boundary puncture model that integrates the advantages of both punctures and twists. They proposed that non-Abelian properties could be realized in the symmetric subspace {|++〉, |−−〉}. This work demonstrates that the nontrivial antisymmetric subspace {|+−〉, |−+〉} also supports non-Abelian statistics. The mixed boundary puncture model is shown to be fault-tolerant in both subspaces, offering resistance to collective dephasing noise and collective rotation noise. In addition, we propose and validate a quantum information masking scheme within the three-partite mixed boundary puncture model.

DOI
10.1088/1674-1056/ae5c78

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Abstract Defects on the toric code, a well-known exactly solvable Abelian anyon model, can exhibit non-Abelian statistical properties, which can be classified into punctures and twists. Benhemou et al. [ Phys. Rev. A 105 042417 (2022)] introduced a mixed boundary puncture model that integrates the advantages of both punctures and twists. They proposed that non-Abelian properties could be realized in the symmetric subspace {|++〉, |−−〉}. This work demonstrates that the nontrivial antisymmetric subspace {|+−〉, |−+〉} also supports non-Abelian statistics. The mixed boundary puncture model is shown to be fault-tolerant in both subspaces, offering resistance to collective dephasing noise and collective rotation noise. In addition, we propose and validate a quantum information masking scheme within the three-partite mixed boundary puncture model.
It is well known that the Abelian Z2{Z}_{2}Z2​ anyonic model (toric code) can be realized on a highly entangled two-dimensional spin lattice, where the anyons are quasiparticles located at the end points of stringlike concatenations of Pauli operators. Here we show that the same entangled states of the same lattice are capable of supporting the non-Abelian Ising model, where the concatenated operators are elements of the Clifford group. The Ising anyons are shown to be essentially superpositions of the Abelian toric code anyons, reproducing the required fusion, braiding, and statistical properties. We propose a string framing and ancillary qubits to implement the nontrivial chirality of this model.
We study the twist defects in the toric code model introduced by Bombin [Phys. Rev. Lett. 105, 030403 (2010)]. Using a generalized two-dimensional Jordan-Wigner transformation and a projective construction, we show explicitly that twist defects carry unpaired Majorana zero modes. In addition, we propose a quantum nondemolition measurement scheme of the parity of Majorana modes. Such a scheme provides an alternative avenue to demonstrate the non-Abelian statistics of Majorana fermions. The braiding operation is simulated by a nonrepetitive measurement-based approach which removes the uncertainty associated with forced measurements.
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There exists a close relationship between topological quantum error-correcting codes and topological order in condensed matter systems. Indeed, a topological stabilizer code can be regarded as the ground state of a suitable Hamiltonian model, so that "wrong" syndromes correspond to excitations. These excitations are anyons, quasiparticles that carry a topological charge and exhibit exotic statistics. Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We call these defects twists. Twists give rise to new topological degrees of freedom in the ground state, useful as a quantum memory. Moreover, twists can be braided to perform topologically protected gates on these topological qubits. Thus, twists provide a new way to encode and compute with topological codes through code deformations. Because the properties of twists depend on the anyon model, codes with different anyon content give rise to different computational capabilities. E.g., in the well-known toric code a process where suitable twists are braided and fused has the same outcome as if they were Ising anyons. These are non-abelian anyons: braiding produces non-trivial gates on encoded qubits.
Maximum likelihood (ML) decoders perform quantum error correction by measuring a set of syndromes, classically computing an optimal recovery based on these syndromes and the noise model, and then performing that recovery. ML decoders are well understood for the toric code and generalize straightforwardly to other surface codes with Abelian anyons, where they can saturate information-theoretic decoding thresholds. On the other hand, codes with non-Abelian anyons have become increasingly relevant because they can support transversal or constant-depth non-Clifford gates and admit efficient preparation protocols in modern quantum devices. However, they present new challenges for decoding: non-Abelian anyons cannot in general be deterministically annihilated, and the logical operators cannot be classified by Abelian homology. In this work, we show that refined syndrome sets, including measurements that resolve non-Abelian anyon fusion outcomes, systematically improve the performance of ML decoders. We demonstrate this concretely by constructing explicit decoders for the D4D_4D4​ quantum double model under charge noise, and obtaining numerical results for thresholds via mappings to classical statistical-mechanics models. We show that there is a nonzero noise threshold below which the entire logical qudit can be recovered with a single round of measurements.