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VI Fast Workshop on Applied and Computational Mathematics

THE MATHEMATICS OF QUANTUM BITSBY

RENSSO CHUNG

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GOOD BYE BITS

The most basic piece ofinformation is called a bit.

Mathematically:

The field GF(2), the simplest GaloisField, is extremely useful for(classical) computer.

Physically:

we can implement a bit with anelectric circuit, e.g., zerovolts(binary 0) or +5 volts(binary

1), etc.

We are dealing with a two-statesystem.

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START THINKING ABOUT QUANTUM MECHANICSSTART THINKING ABOUT QUANTUM MECHANICSSTART THINKING ABOUT QUANTUM MECHANICSSTART THINKING ABOUT QUANTUM MECHANICS

Where

computersareconcerned,

mechanicsare the future.Raymond Laflamme ,

executive director of theInstitute for QuantumComputing (IQC) at theUniversity of Waterloo.

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QUANTUM MECHANICS: THIS tiny

WORLD IS STRANGE

Particles being describedby waves.

The position and momentumbeing represented asoperators.

e ynam cs e ng e neby Schrdinger equations .

The measurementscharacterized byprobabilities.

The entanglement

Schrdingers cat.

Uncertainty Principle.

Quantum superposition, etc.

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This basic unit of information inquantum computing is called the qubit,which is short for quantum bit.

The Quantum Bits (Qubits)

e a , a qu can a so e n oneof two states.

A qubit can exist in the state |0> orthe state |1>, but it can also exist inwhat we call a superposition state.

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This is a state that is a linear combinationof the states |0> and |1>. If we label this

state |>, a superposition state is written as

|>= |0> + |1>, (1)

The Quantum Bits (Qubits)

here , are complex numbers.

The laws of quantum mechanics tell us that

the modulus squared of , in (1) gives usthe probability of finding the qubit in state|0 > or |1 >, respectively.

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THE BLOCH SPHERE

It is useful to express a state of

a single qubit graphically.

Let us parameterize a one-qubitstate | > with and as

|(, ) > =

cos (/2)|0 > + exp(i) sin( /2)|1 >.

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PETER SHORS ALGORITHMA quantum algorithm forfactoring exponentiallyfaster than the best

currently-known algorithmrunning on a classicalcomputer.

S or s a gor t m so ves t ediscrete logarithm problemand the integer factorizationproblem in polynomial time.

(Shor, P. W. (1997). "Polynomial-Time Algorithmsfor Prime Factorization and Discrete Logarithmson a Quantum Computer". SIAM Journal onScientific and Statistical Computing 26: 1484.

arXiv:quant-ph/9508027)

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All things are numbers.Pythagoras

QUANTUM MECHANICS(MATH INSIDE)+ COMPUTER SCIENCE(MATH INSIDE)=

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AN INVITATION TO QUBIT PARTY

The seminal paper onquantum computing arein the web page:

http://arxiv.org/

(open access!)

John Preskills webpage:

http://theory.caltech.edu/people/preskill/ph229/