Preface

Years ago, when I was a beginning and naive graduate student, eagerly studying

mathematics at Princeton, I wanted to know the secret to success in mathematics.

I actually did believe that there must be just one secret that would lead to fame

and fortune in this ﬁeld. So during tea time in Old Fine Hall, I approached my

all-wise thesis advisor (then reverently called Professor Fox), and naively posed the

question:

What is the secret to success in mathematics?

Ralph Fox, without a moment’s hesitation, ﬁred back:

Work where two research ﬁelds are merging.

At that time, I did not fully comprehend the meaning and the signiﬁcance of his

prompt, laconic answer. But as the years have passed, I have begun to understand

more fully the wisdom behind his terse one-line response. His words of wisdom have

indelibly left an impression that has shaped the many choices made throughout my

research career.

Perhaps, Ralph Fox was referring to his student John Milnor’s success in cre-

ating the ﬁeld of diﬀerential topology, a beautiful merger of diﬀerential geometry

and topology. Or perhaps, he was referring to Stephen Smale’s proof of the higher

dimensional Poincare conjecture? Or was it his student John Stallings’ creative

merger of group theory and topology? In any case, throughout my career I have

come to see Ralph Fox’s prophecy repeatedly come true, over and over again.

Now, Ralph Fox’s prophecy is becoming true once again in the newly emerging

ﬁeld of quantum computation and quantum information, i.e., quantum information

science (QIS). Never before have I found such a convergence of so many research

ﬁelds that are currently shaping the development of QIS,...and yes, of mathematics,

itself. Never before has there been such a rich and immense research opportunity

for the mathematical community. In particular, mathematics is now shaping QIS,

and in turn, QIS is now shaping the development of mathematics.

For that reason, I organized and gave an AMS Short Course on Quantum

Computation at the Annual meeting of the American Mathematical Society held

in Washington, DC in January of 2000. This past short course is now recorded and

encapsulated in the AMS book

“Quantum Computation: A Grand Mathematical Challenge for the

Twenty-First Century and the Millenium,” PSAPM, vol. 58, Providence,

RI, (2002).

For the same reason, I also organized at the same AMS meeting an AMS Special

Session on Quantum Computation and Information which has been recorded in a

second AMS book

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