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Citation: Moussa, Jonathan E. "Quantum circuits for qubit fusion." arXiv preprint arXiv:1512.06132 (2015).
Tags: Abelian-anyons, Computer-scientific
While every finite abelian group itself gives a model of computation which can be efficiently classically simulated, the "limiting steps" are different from model to model. For instance, when G=Z8 the standard generalized Z gate gives the T-gate on the subspace spanned by |0> and |1>, which is exactly the hard part of the Z2 toric code. Hence, if you were allowed to have a system which jumped between different finite abelian groups you could get universal TQC.
This can be put very concretely. Namely, we can have gate which implements the natural binary-representation gate from n copies of C[Z2] to C[Z(2^n)]. If you do this from C[Z2] to C[Z(2^3)], the above argument shows that you can get universal TQC. Namely, you use C[Z8] to compute the T gate and you use C[Z2] to compute everything else.
This article shows that you don't even need to go all the way to Z8. Namely, the fusion gate going from two copies of C[Z2] to one copy of C[Z4] is powerful enough to implement universal quantum computation. This paper is the one which introduces this idea of qubit fusion, and they hope to use this as a basis for a model of fault tolerant quantum computation. Namely, you have one block of Z2 surface code and one block of Z4 surface code, with an interface between them which allows for universal quantum computation.