# Quantum Computation and the Localization of Modular Functors

@article{Freedman2001QuantumCA, title={Quantum Computation and the Localization of Modular Functors}, author={Michael H. Freedman}, journal={Foundations of Computational Mathematics}, year={2001}, volume={1}, pages={183-204} }

The mathematical problem of localizing modular functors to neighborhoods of points is shown to be closely related to the physical problem of engineering a local Hamiltonian for a computationally universal quantum medium. For genus =0 surfaces, such a local Hamiltonian is mathematically defined. Braiding defects of this medium implements a representation associated to the Jones polynomial and this representation is known to be universal for quantum computation.

## 67 Citations

Quantum algorithms for colored Jones polynomials

- Mathematics, EngineeringDefense + Commercial Sensing
- 2009

We review the q-deformed spin networks and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. The simplest case of these models is the…

Spin Networks and Quantum Computation

- Mathematics
- 2008

We review the q-deformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representa- tions of the braid groups that are dense in the unitary…

Unitary Representations of the Artin Braid Groups and Quantum Algorithms for Colored Jones Polynomials and the Witten-Reshetikhin Invariant

- Mathematics
- 2008

We review the q-deformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups.…

Topological Quantum Computation

- Mathematics, Physics
- 2001

The theory of quantum computation can be constructed from the abstract study of anyonic systems. In mathematical terms, these are unitary topological modular functors. They underlie the Jones poly-…

Mathematics of Topological Quantum Computing

- Mathematics, Physics
- 2017

In topological quantum computing, information is encoded in "knotted" quantum states of topological phases of matter, thus being locked into topology to prevent decay. Topological precision has been…

$q$ - Deformed Spin Networks, Knot Polynomials and Anyonic Topological Quantum Computation

- Mathematics, Physics
- 2006

We review the q-deformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups.…

Unitary R-matrices for topological quantum computing

- Mathematics
- 2006

The main problem with current approaches to quantum computing is the difficulty of establishing and maintaining entanglement. A Topological Quantum Computer (TQC) aims to overcome this by using…

Spin networks and anyonic topological computing II

- Mathematics, EngineeringSPIE Defense + Commercial Sensing
- 2007

We review the q-deformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups.…

Braiding , Majorana Fermions and Topological Quantum Computing

- 2016

This paper is an introduction to relationships between topology, quantum computing and the properties of fermions. In particular we study the remarkable unitary braid group representations associated…

Braiding, Majorana fermions, Fibonacci particles and topological quantum computing

- Physics, Computer ScienceQuantum Inf. Process.
- 2018

The remarkable unitary braid group representations associated with Majorana fermions are studied to study the relationships between topology, quantum computing, and the properties of Fermions.

## References

SHOWING 1-10 OF 46 REFERENCES

A Modular Functor Which is Universal¶for Quantum Computation

- Mathematics, Physics
- 2000

Abstract:We show that the topological modular functor from Witten–Chern–Simons theory is universal for quantum computation in the sense that a quantum circuit computation can be efficiently…

Quantum field theory and the Jones polynomial

- Mathematics
- 1989

It is shown that 2+1 dimensional quantum Yang-Mills theory, with an action consisting purely of the Chern-Simons term, is exactly soluble and gives a natural framework for understanding the Jones…

Topological quantum field theory with corners based on the Kauffman bracket

- Mathematics
- 1996

Abstract. We describe the construction of a topological quantum field theory with corners based on the Kauffman bracket, that underlies the smooth theory of Lickorish, Blanchet, Habegger, Masbaum and…

Simulation of Topological Field Theories¶by Quantum Computers

- Physics, Mathematics
- 2002

Abstract: Quantum computers will work by evolving a high tensor power of a small (e.g. two) dimensional Hilbert space by local gates, which can be implemented by applying a local Hamiltonian H for a…

Higher algebraic structures and quantization

- Physics, Mathematics
- 1994

We derive (quasi-)quantum groups in 2+1 dimensional topological field theory directly from the classical action and the path integral. Detailed computations are carried out for the Chern-Simons…

Quantum Groups

- Mathematics
- 1994

Here is an introduction to the theory of quantum groups with emphasis on the spectacular connections with knot theory and Drinfeld's recent fundamental contributions. It presents the quantum groups…

Invariants of 3-manifolds via link polynomials and quantum groups

- Mathematics
- 1991

The aim of this paper is to construct new topological invariants of compact oriented 3-manifolds and of framed links in such manifolds. Our invariant of (a link in) a closed oriented 3-manifold is a…

The Geometry and Physics of Knots

- Physics, Mathematics
- 1990

Preface 1. History and background 2. Topological quantum field theories 3. Non-abelian moduli spaces 4. Symplectic quotients 5. The infinite-dimensional case 6. Projective flatness 7. The Feynman…

Quantum Groups

- 1993

This thesis consists of four papers. In the first paper we present methods and explicit formulas for describing simple weight modules over twisted generalized Weyl algebras. Under certain conditions…