A topological quantum computer is a theoretical quantum computer that uses quasiparticles called anyons where their world lines form threads that cross over one another to form braids in a two-dimensional world. These braids form the logic gates that make up the computer. The advantage of a quantum computer using quantum braids over using trapped quantum particles is that the former is much more stable; where the smallest perturbations can cause a quantum particle to decohere, and create errors in the computation, such small perturbations do not change the topological properties of the braids, which are derived from knot theory. This is like the effort required to cut a string and attaching the ends to form a different knot, as opposed to a ball simply bumping into a nearby wall. While the elements of a topological quantum computer originate in a purely mathematical realm, recent experiments indicate these elements can be created in the real world using semiconductors made of gallium arsenide near absolute zero and subjected to strong magnetic fields. In mathematics and physics, an anyon is a type of projective representation of a Lie group. ... World line of the orbit of the Earth depicted in two spatial dimensions X and Y (the plane of the Earth orbit) and a time dimension —usually put as the vertical axis. ... The Bloch sphere is a representation of a qubit, the fundamental building block of quantum computers. ... Trefoil knot, the simplest non-trivial knot. ...

Introduction

Anyons are quasiparticles in a two-dimensional space. Anyons are not strictly fermions or bosons, but do share the characteristic of fermions in that they cannot occupy the same state. Thus, the world lines of two anyons cannot cross or merge. This allows braids to be made that make up a particular circuit. In the real world, anyons form from the excitations in an electron gas in a very strong magnetic field, and carry fractional units of magnetic flux in a particle like manner. This phenomenon is called the fractional quantum Hall effect. The electron "gas" is sandwiched between two flat plates of gallium arsenide, which create the two-dimensional space required for anyons, and is super-cooled and subjected to intense transverse magnetic fields. Dimension (from Latin measured out) is, in essence, the number of degrees of freedom available for movement in a space. ... Fermions, named after Enrico Fermi, are particles which form totally-antisymmetric composite quantum states. ... Boson (game) Bosons, named after Satyendra Nath Bose, are particles which form totally-symmetric composite quantum states. ... World line of the orbit of the Earth depicted in two spatial dimensions X and Y (the plane of the Earth orbit) and a time dimension —usually put as the vertical axis. ... The quantum Hall effect is a quantum mechanical version of the Hall effect, observed in two-dimensional systems of electrons subjected to low temperatures and strong magnetic fields, in which the Hall conductance σ takes on the quantized values where e is the elementary charge and h is Plancks...

In a key development for Topological quantum computers, in 2005 Vladimir J. Goldman, Fernando E. Camino, and Wei Zhou was said to have created the first experimental evidence for using fractional quantum Hall effect to create actual anyons, although others have suggested their results could be the product of phenomena not involving anyons. It should also be noted that nonabelian anyons, a species required for Topological quantum computers, have yet to be experimentally confirmed.

Topological vs. Standard Quantum Computer

Computations

An important requirement for the usefulness of a Topological quantum computer is whether it can actually provide the unique computation properties that hold promise with conventional quantum computers. Fortunately in 2002, Michael H. Freedman along with Zhenghan Wang, both with Microsoft, and Michael Larson of Indiana University proved that a Topological quantum computer can, in principle, perform any computation that a trapped quantum particle type quantum computer can do. The difference is that the result would have an analog type accuracy to it, rather than a discrete result like that of a digital computer. Any desired level of accuracy would be achieved by adding more braid twists in a simple linear relationship.

Error Correction & Control

Even though Quantum Braids are inherently more stable than trapped quantum particles, there is still a need to control for error inducing thermal fluctuations, which produce random stray pairs of anyons which interfere with ajoining braids. Controlling these errors is simply a matter of separating the anyons to a distance where the rate of interfering strays drops to near zero. It has been estimated that the error rate for a logical NOT operation of a qubit state could be as low as 10^-30 or less.

References

• "Computing with Quantum Knots" Graham P. Collins, Scientific American, April 2006.
• "Topological quantum computer" Zdzislaw Meglicki 4/5/2005. http://beige.ucs.indiana.edu/M743/node23.html