A Modular Functor Which is Universal¶for Quantum Computation

@article{Freedman2000AMF,
  title={A Modular Functor Which is Universal¶for Quantum Computation},
  author={Michael H. Freedman and Michael Larsen and Zhenghan Wang},
  journal={Communications in Mathematical Physics},
  year={2000},
  volume={227},
  pages={605-622}
}
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 approximated by an intertwining action of a braid on the functor's state space. A computational model based on Chern–Simons theory at a fifth root of unity is defined and shown to be polynomially equivalent to the quantum circuit model. The chief technical advance: the density of the irreducible sectors of… 
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References

SHOWING 1-10 OF 45 REFERENCES
Simulation of Topological Field Theories¶by Quantum Computers
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…
Quantum field theory and the Jones polynomial
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…
Quantum Circuit Complexity
  • A. Yao
  • Computer Science
    FOCS
  • 1993
TLDR
It is shown that any function computable in polynomial time by a quantum Turing machine has aPolynomial-size quantum circuit, and this result enables us to construct a universal quantum computer which can simulate a broader class of quantum machines than that considered by E. Bernstein and U. Vazirani (1993), thus answering an open question raised by them.
Invariants of Three-Manifolds, Unitary Representations of the Mapping Class Group, and Numerical Calculations
TLDR
Numerical agreement is found concerning the values of the Chern–Simons invariants for the flat SU(2)-connections as predicted by the asymptotic expansion of t(2) at roots of unity.
Quantum Invariants of Knots and 3-Manifolds
This monograph, now in its second revised edition, provides a systematic treatment of topological quantum field theories in three dimensions, inspired by the discovery of the Jones polynomial of…
Topological quantum field theory with corners based on the Kauffman bracket
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…
Invariants of 3-manifolds via link polynomials and quantum groups
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…
Quantum computational networks
  • D. Deutsch
  • Physics, Computer Science
    Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences
  • 1989
TLDR
The theory of quantum computational networks is the quantum generalization of the theory of logic circuits used in classical computing machines, and a single type of gate, the universal quantum gate, together with quantum ‘unit wires' is adequate for constructing networks with any possible quantum computational property.
Dynamical description of quantum computing: Generic nonlocality of quantum noise
We develop a dynamical non-Markovian description of quantum computing in the weak-coupling limit, in the lowest-order approximation. We show that the long-range memory of the quantum reservoir (such…
Quantum computation and quantum information
TLDR
This chapter discusses quantum information theory, public-key cryptography and the RSA cryptosystem, and the proof of Lieb's theorem.
...
...