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In topological quantum computation the geometric details of a particle trajectory become irrelevant; only the topology matters. This is one reason for the inherent fault tolerance of topological quantum computation. I will speak about a model in which this idea is taken one step further. Even the topology is irrelevant. The computation is determined solely by the permutation of the particles. Unlike topological quantum computation, which requires anyons confined to two dimensions, permutational quantum computations can in principle be performed by permuting a set of ordinary spin-1/2 particles with definite total angular momentum in three dimensions. The resulting model of computation appears to be intermediate in power between classical computation (P) and standard quantum computation (BQP). The model may be equivalently defined in terms of spin networks, which are an important concept in loop quantum gravity.

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