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## "Anyons from non-solvable finite groups are sufficient for universal quantum computation", Carlos Mochon, 2003

*Reviewed July 26, 2023*

*Citation:* Mochon, Carlos. "Anyons from nonsolvable finite groups are sufficient for universal quantum computation." Physical Review A 67.2 (2003): 022315.

*Web:* https://arxiv.org/abs/quant-ph/0206128

*Tags:* Foundational, Kitaev-quantum-double, Universal-scheme

This is the first paper to prove universal quantum computation results for the quantum double model.
Namely, it is proved here that every non-solvable group gives rise to universal quantum computation by braiding and fusion alone.

A key warning: this paper does not rigorously follow as a sequel of Kitaev's work.
It does not define the Hilbert space for the quantum double system, or work explicitly with any sort of ribbon operator.
Instead, key properties of the quantum double model are asserted without proof, and are then used to demonstrate universal quantum computation.
This is not a problem in the sense that it leads to a very pedagogical discussion, but it does mean that there are some intermediate steps to work out more rigorously.

The proof is very nice. It introduces a gate set which is universal for quantum computation: measuring non-destructively with respect the the X and Y bases,
and performing universal classical computation. Classical computation here means any unitary which permutes the computational basis,
i.e. a gate which does not care about the "quantum aspects", phases/superposition.
It is a theorem that these classical computations are generated by the Toffoli gate,
and so that is the only gate which must be shown to be constructable.

The idea now is that (at least for simple groups) every non-solvable group can do any classical computation.
This is because any map G^n->G can be represented as the product of its inputs, inverses of its inputs, and other group elements.
Every function has a nice "product" representation. It is this fact that leads to universal quantum computation.

Another nice feature of this paper is that it says very explicitly what it needs from the physical system.
Namely: braiding, fusion, creation of quasiparticles with specified types but unspecifed internal states, and creation of ancillas.