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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/