Quantum Computing Mathematics and Postulates

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

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

14. Mathematically, what is a qubit ? (1) • We can form linear combinations of states • A qubit state is a unit vector in a two dimensionalcomplex vector space

15. Qubits Cont'd We can ignore eia as it has no observable effect • 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

16. Bloch Sphere

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. Spin-up Spin-down Qubit in Stern-Gerlach Experiment Oven Figure 6: Abstract schematic of the Stern-Gerlach experiment.

19. Qubit in Stern-Gerlach Exp. Oven Figure 7: Three stage cascade Stern-Gerlach measurements

20. Qubit in Stern-Gerlach Experiment Figure 8: Assignment of the qubit states

21. Qubit in Stern-Gerlach Experiment Figure 8: Assignment of the qubit states