Braid Groups, Hecke Algebras, Representations, and Anyons
summary
Fibonacci anyons implement qubits: three anyons fusing to one give a two‑dimensional Hilbert space and braiding those anyons generates a dense subgroup of U(2), so generic braids approximate arbitrary single‑qubit gates. Six anyons give two qubits but the 5‑dimensional braid representation splits as a 4‑dim qubit subspace plus one junk direction, producing a persistent leakage problem. Analysis uses Artin braid generators, Formanek’s classification, and Hecke algebra quotients (yielding a 695‑dimensional kernel).
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chapters (12)
- 0:00Intro
- 1:33Twodimensional space
- 9:36Anyon Models
- 13:57Fusion Processes
- 16:39Transformations
- 17:21Braiding
- 20:53Phase Factors
- 24:58Basis Vector
- 32:05Conclusion
- 32:25Questions
- 34:27Leakage Problem
- 35:59Braid Groups
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