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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