Preface
The factors of any integer can be found quickly by a quantum computer. Since
P. Shor discovered this efficient quantum factoring algorithm in 1994 [S], people
have started to work on building these new machines. As one of those people,
I joined Microsoft Station Q in Santa Barbara to pursue a topological approach
in 2005. My dream is to braid non-abelian anyons. So long hours are spent on
picturing quasiparticles in fractional quantum Hall liquids. From my office on
UCSB campus, I often see small sailboats sailing in the Pacific Ocean. Many times
I am lost in thought imagining that the small sailboats are anyons and the ocean
is an electron liquid. Then to carry out a topological quantum computation is as
much fun as jumping into such small sailboats and steering them around each other.
Will we benefit from such man-made quantum systems besides knowing factors
of large integers? A compelling reason for a yes comes from the original idea of
R. Feynman: a quantum computer is an efficient universal simulator of quantum
mechanics. This was suggested in his original paper [Fe82]. Later, an efficient sim-
ulation of topological quantum field theories was given by M. Freedman, A. Kitaev,
and the author [FKW]. These results provide support for the idea that quantum
computers can efficiently simulate quantum field theories, although rigorous results
depend on mathematical formulations of quantum field theories. So quantum com-
puting literally promises us a new world. More speculatively, while the telescope
and microscope have greatly extended the reach of our eyes, quantum computers
would enhance the power of our brains to perceive the quantum world. Would it
then be too bold to speculate that useful quantum computers, if built, would play
an essential role in the ontology of quantum reality?
Topological quantum computation is a paradigm to build a large-scale quantum
computer based on topological phases of matter. In this approach, information is
stored in the lowest energy states of many-anyon systems, and processed by braiding
non-abelian anyons. The computational answer is accessed by bringing anyons to-
gether and observing the result. Topological quantum computation stands uniquely
at the interface of quantum topology, quantum physics, and quantum computing,
enriching all three subjects with new problems. The inspiration comes from two
seemingly independent themes which appeared around 1997. One was Kitaev’s idea
of fault-tolerant quantum computation by anyons [Ki1]; the other was Freedman’s
program to understand the computational power of topological quantum field the-
ories [Fr1]. It turns out the two ideas are two sides of the same coin: the algebraic
theory of anyons and the algebraic data of a topological quantum field theory are
both modular tensor categories. The synthesis of the two ideas ushered in topo-
logical quantum computation. The topological quantum computational model is
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