1. Quantum 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

2. Requirements On Mathematics Apparatus
Physical states ⇔ Mathematic entities
Interference phenomena
Nondeterministic predictions
Model the effects of measurement
Distinction between evolution and measurement

3. What’s Quantum Mechanics
A mathematical framework
Description of the world known
Rather simple rules
but counterintuitive
applications
This picture stands for Schroeding’s cat, by Univesity Pierre, French

4. 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

5. Basic linear algebra useful in QM
Complex numbers
Vector space
Linear operators
Inner products
Unitary operators
Tensor products …

6. 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”

7. Inner Products Inner Product is a function combining two vectors
It yields a complex number
It obeys the following rules

8. Hilbert 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

9. Hilbert Space (Cont’d)
Orthonormal basis:
a basis set where
Can be found from an arbitrary basis set by Gram-Schmidt Orthogonalization

10. Unitary Operator
An operator U is unitary, if
Preserves Inner product

11. Tensor Product Larger vector space formed from two smaller ones
Combining elements from each in all possible ways
Preserves both linearity and scalar multiplication

12. Postulates 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

13. Physical Systems - Quantum Mechanics Connections
Postulate 1
Isolated physical system
 
Hilbert Space
Postulate 2
Evolution of a physical system
Unitary transformation
Postulate 3
Measurements of a physical system
Measurement operators
Postulate 4
Composite physical system
Tensor product of components
Picture slightly revised from
Diagram source: lecture notes for QC, University of Alberta, Canada

14. 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

15. We can ignore eia as it has no observable effect
Qubits 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

16. Bloch Sphere Slightly revised from
Original picture from lecture notes for QC, University of Alberta, Canada

17. 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

18. Qubit in Stern-Gerlach Experiment
Spin-up
Oven
Spin-down
The diagram is redrawn from the textbook “[7] Quantum Computation and Quantum Information, Michael A. Nielsen and Isaac L. Chuang”
The pictures Spin-up and Spin-down are from the source:
David M. Harrison, Department of Physics, University of Toronto
Figure 6: Abstract schematic of the Stern-Gerlach experiment.

19. Qubit in Stern-Gerlach Exp.
Oven
The diagram is redrawn from the textbook “Quantum Computation and Quantum Information, Michael A. Nielsen and Isaac L. Chuang”
The background picture is revised from the source:
Figure 7: Three stage cascade Stern-Gerlach measurements

20. Qubit in Stern-Gerlach Experiment
The diagram is redrawn from the textbook “Quantum Computation and Quantum Information, Michael A. Nielsen and Isaac L. Chuang”
The background picture is revised from the source:
Figure 8: Assignment of the qubit states

21. Qubit in Stern-Gerlach Experiment
The diagram is redrawn from the textbook “Quantum Computation and Quantum Information, Michael A. Nielsen and Isaac L. Chuang”
The background picture is revised from the source:
Figure 8: Assignment of the qubit states