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…
421 Citations
Large $k$ topological quantum computer
- Physics
- 2021
Chern-Simons topological quantum computer is a device that can be effectively described by the Chern-Simons topological quantum field theory and used for quantum computations. Quantum qudit gates of…
Braiding Operators are Universal Quantum
- Physics
- 2004
This paper is an exploration of the role of unitary braiding operators in quantum computing. We show that a single specific solution R of the Yang-Baxter Equation is a universal gate for quantum…
Braiding operators are universal quantum gates
- Physics
- 2004
This paper explores the role of unitary braiding operators in quantum computing. We show that a single specific solution R (the Bell basis change matrix) of the Yang–Baxter equation is a universal…
Ja n 20 04 Braiding Operators are Universal Quantum Gates
- Physics
- 2004
This paper is an exploration of the role of unitary braiding operators in quantum computing. We show that a single specific solution R of the Yang-Baxter Equation is a universal gate for quantum…
Combinatorial Framework for Topological Quantum Computing
- Computer Science
- 2012
In this chapter we describe a combinatorial framework for topological quantum computation, and illustrate a number of algorithmic questions in knot theory and in the theory of finitely presented…
Elementary Particles as Gates for Universal Quantum Computation
- Physics
- 2013
It is shown that there exists a mapping between the fermions of the Standard Model (SM) represented as braids in the Bilson-Thompson model, and a set of gates which can perform Universal Quantum…
Topological Quantum Computation
- 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-…
Simulation of Topological Field Theories¶by Quantum Computers
- Physics
- 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…
Contemporary Mathematics Quantum Computing and the Jones Polynomial
- Computer Science
- 2002
The structure of the Jones polynomial in relation to representations of the Temperley Lieb algebra is discussed, and an example of a unitary representation of the braid group is given.
References
SHOWING 1-10 OF 45 REFERENCES
Simulation of Topological Field Theories¶by Quantum Computers
- Physics
- 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…
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…
Quantum Circuit Complexity
- Computer ScienceFOCS
- 1993
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
- MathematicsExp. Math.
- 1997
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
- Physics
- 1994
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
- Physics
- 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…
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…
Quantum computational networks
- Physics, Computer ScienceProceedings of the Royal Society of London. A. Mathematical and Physical Sciences
- 1989
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
- Physics
- 2002
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
- Physics, Computer Science
- 2000
This chapter discusses quantum information theory, public-key cryptography and the RSA cryptosystem, and the proof of Lieb's theorem.