Topological Quantum Computation from non-abelian ... - Paul Fendley

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A (hopefully not unique) procedure to construct such a quantum loop model is1. find a 2d classical loop model which has a critical point2. use each loop configuration as a basis element of the quantum Hilbert space3. use a Rokhsar-Kivelson Hamiltonian to make the quantum ground state a sum overloop configurations with the appropriate weighting.Kitaev; Moessner and Sondhi; FreedmanThe inner product must be topological, i.e. 〈a|b〉 depends only on topological data.Levin; <strong>Fendley</strong>30
To realize this form of braiding, each loop in the ground state gets a weight d (= τ forFibonacci)i.e. the ground state is the sum over all loop configurations|g.s.〉 = ∑ Ld n L |L〉where n L is the number of loops in configuration L.31
Page 1 and 2: Topological Quantum Computationfrom
Page 3: Outline:1. What are non-abelian sta
Page 7 and 8: Errors!In a classical computer, jus
Page 9: In three dimensions, the only stati
Page 12 and 13: Even more spectacular phenomena tha
Page 14 and 15: We need to understand fusion: what
Page 16 and 17: φ × φ = I + φPrecise meaning in
Page 18 and 19: To make a qubit requires 4 Fibonacc
Page 20 and 21: One can find the braiding and fusin
Page 22 and 23: A simple way of satisfying the cons
Page 24 and 25: These relations can all be treated
Page 26 and 27: The problem: find a quantum Hamilto
Page 28 and 29: The leading candidate wavefunction
Page 32 and 33: The excitations with non-abelian br
Page 34 and 35: These nets can be described using t
Page 36 and 37: Such loops satisfy a property under
Page 38 and 39: We find models with deconfined Fibo
Page 40 and 41: This allows construction of a quant
Page 42 and 43: 2. Level-rank duality: Here we expl

Topological Quantum Computation from non-abelian ... - Paul Fendley

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