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## "Quantum computation with abelian anyons", Seth Lloyd, 2000

*Reviewed July 26, 2023*

*Citation:* Lloyd, Seth. "Quantum computation with abelian anyons." Quantum Information Processing 1 (2002): 13-18.

*Web:* https://arxiv.org/abs/quant-ph/0004010

*Tags:* Abelian-anyons, Universal-scheme, Physical

This paper shows that one can perform universal quantum computation with Fermions, or any other type of non-bosonic (quasi)particle.
There is a general paradigm of "move non-abelian => more computational power" in quantum double models.
This paper is solidly establishing one end of that paradigm, by looking at the least-complex most-abelian anyons: Fermions.

The idea for the model of computation is as follows.
You store your qubits in pairs of sites, exactly one of which contains a Fermion.
If the top site contains the Fermion you call the state |1>, if the bottom site contains the Fermion you call the state |0>.
The key insight is that if you braid the top particle of one site with the top particle of another site you get a -1 phase factor if and only if both sites are |1>.
That is, the braiding topologically enacts the controlled phase shift operator.

It is a well-known fact that one qubit gates along with controlled phase shift is enough to perform universal quantum computation.
The only issue, thus, is to perform one qubit gates. The Pauli X operator is easy: just swap the particle content of your two sites.
Pauli Z is also easy: create a Fermion and braid it with the top particle.

Performing other one qubit gates (like Hadamard) is more complicated, and cannot be done in a topologically protected fashion.
Here is what Lloyd has to say "They could be enacted by applying localized potentials via, e.g., nanofabricated electrodes or scanning tunneling microscopes...
Single qubit operations are not topological in nature and are less robust."

He then goes on to give a description of how one might do this with interferometry.