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Topological quantum computation. (English) Zbl 1019.81008
This is a survey paper describing quantum computation based on unitary topological modular functors. The chief advantage of topological computation is physical error correction, a consequence of topological stability. In contrast, other quantum computation models, which repair errors combinatorially, require an extremely low initial error rate.
The theoretical model considered by the authors is obtained by restricting to planar surfaces the Witten-Chern-Simons modular functor with gauge grop \(\text{SU}(2)\) at fifth roots of unity. Logical qubits are punctured disks with punctures colored by irreducible representations, and the gates (unitary transformations) are the operators of the Hecke algebra representation discovered by Jones. The authors conjecture that such a model could by implemented physically using the braiding and fusion of anyonic excitation in quantum Hall electron liquids with \(\nu={5\over 2}\).

MSC:
81P68 Quantum computation
57R56 Topological quantum field theories (aspects of differential topology)
81T45 Topological field theories in quantum mechanics
68Q05 Models of computation (Turing machines, etc.) (MSC2010)
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