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Citation: Hutter, Adrian, James R. Wootton, and Daniel Loss. "Parafermions in a Kagome lattice of qubits for topological quantum computation." Physical Review X 5.4 (2015): 041040.
This is a follow-up paper to
> Hutter, Adrian, and Daniel Loss. "Quantum computing with parafermions." Physical Review B 93.12 (2016): 125105.
In the first paper, the authors demonstrated general properties of parafermions. Here, they delve deep into the simplest non-trivial case: d=4. They work out more details, and show how to implement gates more explicitly.
Most importantly for our purposes, the authors give a good treatment of the d=4 surface code model. They place it on the Kagome lattice, and give a simple qubit-based Hamiltonian which one could imagine realizing on a quantum processor. To quote the authors: "The joint Hilbert space of two qubits allows the 4-dimensional generalized Pauli operators to be expressed in terms of two-qubit operators. Using this, we show how Z4 parafermions can emerge in a lattice of qubits with nearest-neighbor interactions only. This allows the computational power of Z4 parafermions to be harnessed in a qubit system."