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## "Parafermions in a Kagome lattice of qubits for topological quantum computation", Adrian Hutter, James Wootton, Daniel Loss, 2015

*Reviewed August 22, 2023*

*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.

*Web:* https://arxiv.org/abs/1505.01412

*Tags:* Abelian-anyons, Error-correcting-codes, Toric-code

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."