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"A modular functor which is universal for quantum computation", Michael Freedman, Michael Larsen, Zhenghan Wang, 2002

Reviewed July 22, 2023

Citation: Freedman, Michael H., Michael Larsen, and Zhenghan Wang. "A modular functor which is universal for quantum computation." Communications in Mathematical Physics 227 (2002): 605-622.

Web: https://arxiv.org/abs/quant-ph/0001108

Tags: Foundational, Quantum-groups, TQFT, Universal-scheme

This paper shows that topological quantum field theories (TQFTs) can be used to simulate arbitrary algorithms in BQP, in the sense that TQFTs naturally encode a BQP-complete problem. This problem is the approximation of the Jones polynomial at a primitive 5th root of unity. Seeing as topological quantum computation (TQC) is exactly the study of quantum computation via TQFTs, this can be summarized as showing that good non-abelian TQC models are just as powerful as quantum circuit models. In concert with the earlier work

> Freedman, Michael H., Alexei Kitaev, and Zhenghan Wang. "Simulation of topological field theories¶ by quantum computers." Communications in Mathematical Physics 227 (2002): 587-603.

this shows that TQC and quantum circuit models are equivalent.

The bulk of this paper is spent on technical arguments about Jones polynomials and manifolds. For those interested in how the algorithm actually works, it is better to read Aharonov-Jones-Landau's simplification which removes the TQFTs:

> Aharonov, Dorit, Vaughan Jones, and Zeph Landau. "A polynomial quantum algorithm for approximating the Jones polynomial." Proceedings of the thirty-eighth annual ACM symposium on Theory of computing. 2006.

Or even better, a pedagogical introduction to this simplified algorithm:

> Lomonaco Jr, Samuel J., and Louis H. Kauffman. "Topological quantum computing and the Jones polynomial." Quantum Information and Computation IV. Vol. 6244. SPIE, 2006.

I'd say that this is a paper where you don't lose much from only remembering the summary: TQFTs approximate Jones polynomials, and that's universal. If you choose your anyonic theory right, you can get universal TQC.

The exact TQFT shown to be universal corresponds to the Fibonacci anyonic model. Another great reference showing how to make the Fibonacci anyonic model universal (following the algorithm demonstrated here) is given in the following very short article:

> Bonesteel, Nicholas E., et al. "Braid topologies for quantum computation." Physical review letters 95.14 (2005): 140503.