Title: Quantum Computing Mathematics and Postulates
1Quantum Computing Mathematics and Postulates
Presented by Chensheng Qiu Supervised by Dplm.
Ing. Gherman Examiner Prof. Wunderlich
- Advanced topic seminar SS02
- Innovative Computer architecture and concepts
- Examiner Prof. Wunderlich
2Requirements On Mathematics Apparatus
- Physical states ? Mathematic entities
- Interference phenomena
- Nondeterministic predictions
- Model the effects of measurement
- Distinction between evolution and measurement
3Whats Quantum Mechanics
- A mathematical framework
- Description of the world known
- Rather simple rules
- but counterintuitive
- applications
4Introduction 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
5Basic linear algebra useful in QM
- Complex numbers
- Vector space
- Linear operators
- Inner products
- Unitary operators
- Tensor products
6Dirac-notation
- For the sake of simplification
- ket stands for a vector in Hilbert
- bra stands for the adjoint of
- Named after the word bracket
7Inner Products
- Inner Product is a function combining two vectors
- It yields a complex number
- It obeys the following rules
-
-
8Hilbert Space
- Inner product space linear space equipped with
inner product - Hilbert Space (finite dimensional) can be
considered as inner product space of a quantum
system - Orthogonality
- Norm
- Unit vector parallel to
9Hilbert Space (Contd)
- Orthonormal basis
- a basis set where
- Can be found from an arbitrary basis set by
Gram-Schmidt Orthogonalization
10Unitary Operator
- An operator U is unitary, if
- Preserves Inner product
11Tensor Product
- Larger vector space formed from two smaller ones
- Combining elements from each in all possible ways
- Preserves both linearity and scalar
multiplication
12Postulates 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
24
13Physical Systems - Quantum Mechanics Connections
14Mathematically, 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
15Qubits Cont'd
- We may rewrite as
- From a single measurement one obtains only a
single bit of information about the state of the
qubit - There is "hidden" quantum information and this
information grows exponentially
We can ignore eia as it has no observable effect
16Bloch Sphere
17How 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
18Qubit in Stern-Gerlach Experiment
Oven
Figure 6 Abstract schematic of the Stern-Gerlach
experiment.
19Qubit in Stern-Gerlach Exp.
Oven
Figure 7 Three stage cascade Stern-Gerlach
measurements
20Qubit in Stern-Gerlach Experiment
Figure 8 Assignment of the qubit states
21Qubit in Stern-Gerlach Experiment
Figure 8 Assignment of the qubit states