# Topological Quantum Computation

@inproceedings{Freedman2001TopologicalQC, title={Topological Quantum Computation}, author={Michael H. Freedman and Alexei Y. Kitaev and Michael Larsen and Zhenghan Wang}, year={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- nomial and arise in Witten-Chern-Simons theory. The braiding and fusion of anyonic excitations in quantum Hall electron liquids and 2D-magnets are modeled by modular functors, opening a new possi- bility for the realization of quantum computers. The chief advantage of anyonic computation would be… Expand

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#### References

SHOWING 1-10 OF 36 REFERENCES

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… Expand

Fault-tolerant quantum computation

- Computer Science, Physics
- Proceedings of 37th Conference on Foundations of Computer Science
- 1996

For any quantum computation with t gates, it is shown how to build a polynomial size quantum circuit that tolerates O(1/log/sup c/t) amounts of inaccuracy and decoherence per gate, for some constant c; the previous bound was O( 1/t). Expand

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… Expand

P/NP, and the quantum field computer

- Computer Science, Medicine
- Proc. Natl. Acad. Sci. USA
- 1998

This work proposes that each physical theory supports computational models whose power is limited by the physical theory, and suggests that some physical system whose effective Lagrangian contains a non-Abelian topological term might be manipulated to serve as an analog computer capable of solving NP or even #P-hard problems in polynomial time. Expand

Quantum theory, the Church–Turing principle and the universal quantum computer

- Mathematics
- Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences
- 1985

It is argued that underlying the Church–Turing hypothesis there is an implicit physical assertion. Here, this assertion is presented explicitly as a physical principle: ‘every finitely realizible… Expand

Microscopic models of two-dimensional magnets with fractionalized excitations

- Physics
- 2001

We demonstrate that spin-charge separation can occur in two dimensions and note its confluence with superconductivity, topology, gauge theory, and fault-tolerant quantum computation. We construct a… Expand

Fault-tolerant quantum computation

- Physics, Mathematics
- 1997

The discovery of quantum error correction has greatly improved the long-term prospects for quantum computing technology. Encoded quantum information can be protected from errors that arise due to… Expand

Quantum computations: algorithms and error correction

- Mathematics
- 1997

Contents §0. Introduction §1. Abelian problem on the stabilizer §2. Classical models of computations2.1. Boolean schemes and sequences of operations2.2. Reversible computations §3. Quantum… Expand

2n-quasihole states realize 2n−1-dimensional spinor braiding statistics in paired quantum Hall states

- Physics
- 1996

By explicitly identifying a basis valid for any number of electrons, we demonstrate that simple multi-quasihole wave functions for the v = 12 Pfaffian paired Hall state exhibit an exponential… Expand

Quantum Computation and the Localization of Modular Functors

- Mathematics, Computer Science
- Found. 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… Expand