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Universal Quantum Computation with the $S_3$ Quantum Double: A Pedagogical Exposition
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abstract
Non-Abelian topological order (TO) enables topologically protected quantum computation with its anyonic quasiparticles. Recently, TO with $S_3$ gauge symmetry was identified as a sweet spot -- simple enough to emerge from finite-depth adaptive circuits yet powerful enough to support a universal topological gate-set. In these notes, we review how anyon braiding and measurement in $S_3$ TO are primitives for topological quantum computation and we explicitly demonstrate universality. These topological operations are made concrete in the $S_3$ quantum double lattice model, aided by the introduction of a generalized ribbon operator. This provides a roadmap for near-term quantum platforms.
Forward citations
Cited by 3 Pith papers
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Tensor Network Representations for Intrinsically Mixed-State Topological Orders
A Choi-state anyon condensation protocol builds fixed-point tensor networks for intrinsic mixed-state topological orders from decohered pure states.
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Bridging Microscopic Constructions and Continuum Topological Field Theory of Three-Dimensional Non-Abelian Topological Order
The D4 quantum double model realizes the BF+AAB field theory with (Z2)^3 gauge group at the level of its 22 excitations, fusion rules, and shrinking rules.
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Spectral properties and coding transitions of Haar-random quantum codes
Haar-random quantum codes lose correctability exactly at the hashing bound, and the spectral band structure predicts a higher detection threshold for postselected error correction.
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