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

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 This picture stands for Schroeding’s cat, by Univesity Pierre, French

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

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

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

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

Tensor 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 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

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

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

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

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

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 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.

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: http://search.eb.com/nobel/cap/omechan064a4.html Figure 7: Three stage cascade Stern-Gerlach measurements

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: http://search.eb.com/nobel/cap/omechan064a4.html Figure 8: Assignment of the qubit states

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: http://search.eb.com/nobel/cap/omechan064a4.html Figure 8: Assignment of the qubit states