# 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…

## 425 Citations

### Quantum Computation and the Localization of Modular Functors

- MathematicsFound. Comput. Math.
- 2001

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…

### 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…

### A categorical semantics for topological quantum computation

- Mathematics
- 2004

The aim of this thesis is to develop an abstract categorical setup in order to show that C-colored manifolds (i.e. compact closed manifolds with boundary where each boundary component is colored with…

### 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…

## References

SHOWING 1-10 OF 33 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…

### A combinatorial approach to topological quantum field theories and invariants of graphs

- Mathematics
- 1993

The combinatorial state sum of Turaev and Viro for a compact 3-manifold in terms of quantum 6j-symbols is generalized by introducing observables in the form of coloured graphs. They satisfy braiding…

### 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…