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Quantum Computing Mathematics and Postulates
Advanced topic seminar SS02
“Innovative Computer architecture and concepts”
Examiner: Prof. Wunderlich
Presented byPresented byChensheng QiuChensheng Qiu
Supervised bySupervised by
Dplm. Ing. Gherman Dplm. Ing. Gherman
Examiner: Prof. Wunderlich
Requirements On Mathematics Apparatus
Physical states ⇔ Mathematic entities
Interference phenomena
Nondeterministic predictions
Model the effects of measurement
Distinction between evolution and
measurement
What’s Quantum Mechanics
A mathematical framework
Description of the world known
Rather simple rules
but counterintuitive
applications
Introduction to Linear Algebra
Quantum mechanics The basis for quantum computing and
quantum information
Why Linear Algebra? Prerequisities
What is Linear Algebra concerning? Vector spaces Linear operations
Basic linear algebra useful in QM
Complex numbers
Vector space
Linear operators
Inner products
Unitary operators
Tensor products
…
Dirac-notation
For the sake of simplification
“ket” stands for a vector in Hilbert
“bra” stands for the adjoint of
Named after the word “bracket”
Inner Products
Inner Product is a function combining two vectors
It yields a complex number
It obeys the following rules
C ),(
kkk
kkk wvawav ,,
*),(),( wvwv
0),( vv
Hilbert SpaceHilbert Space
Inner product space: linear space equipped with inner productHilbert Space (finite dimensional): can be considered as inner product space of a quantum systemOrthogonality: Norm: Unit vector parallel to
0wv
vvv
v
vv :
Hilbert Space (Cont’d)
Orthonormal basis:
a basis set where
Can be found from an arbitrary basis set by Gram-Schmidt Orthogonalization
nvv ,...,1 ijji vv
Unitary Operator
An operator U is unitary, if
Preserves Inner product
IUUτ
Uofadjoint for the stands Uwhere
wvwUvU ,,
Tensor ProductTensor Product
Larger vector space formed from two
smaller ones
Combining elements from each in all
possible ways
Preserves both linearity and scalar
multiplication
Postulates in QMPostulates in QM
Why are postulates important? … they provide the connections between
the physical, real, world and the quantum mechanics mathematics used to model these systems
- Isaak L. Chuang
24242424
Physical Systems -Physical Systems - Quantum Mechanics Connections Quantum Mechanics Connections
Postulate 1Isolated physical
system Hilbert Space
Postulate 2Evolution of a
physical system
Unitary transformation
Postulate 3Measurements of a
physical system
Measurement operators
Postulate 4Composite physical
system
Tensor product of
components
Mathematically, what is a qubit ? (1)
We can form linear combinations of
states
A qubit state is a unit vector in a two
dimensional complex vector space
Qubits Cont'd
We may rewrite as…
From a single measurement one obtains only a single bit of information about the state of the qubitThere is "hidden" quantum information and this information grows exponentially
0 1
cos 0 sin 12 2
i ie e
cos 0 sin 12 2
ie
We can ignore ei as it has no
observable effect
Bloch Sphere
How can a qubit be realized?
Two polarizations of a photon
Alignment of a nuclear spin in a uniform magnetic field
Two energy states of an electron
Qubit in Stern-Gerlach Experiment
Oven
Z
Z
Z
Spin-up
Spin-down
Figure 6: Abstract schematic of the Stern-Gerlach experiment.
Qubit in Stern-Gerlach Exp.
Oven
Z
Z
Z
X Z
Z
Z
Figure 7: Three stage cascade Stern-Gerlach measurements
X
X
Z
X
Z
Qubit in Stern-Gerlach Experiment
Figure 8: Assignment of the qubit states
Z
X
Z
2/10
2/10
1
0
X
X
Z
Z
Qubit in Stern-Gerlach Experiment
Figure 8: Assignment of the qubit states
Z
X
Z
2/)(0
2/)1 0 ( and 2/)1 0 (
basis nalcomputatio Gerlach -Stern
1, 0 basis nalcomputatio Gerlach -Stern
XX Z
X
Z
