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**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

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**Requirements On Mathematics Apparatus**

Physical states ⇔ Mathematic entities Interference phenomena Nondeterministic predictions Model the effects of measurement Distinction between evolution and measurement

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**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

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**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

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**Basic linear algebra useful in QM**

Complex numbers Vector space Linear operators Inner products Unitary operators Tensor products …

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**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”

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**Inner Products Inner Product is a function combining two vectors**

It yields a complex number It obeys the following rules

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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

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**Hilbert Space (Cont’d)**

Orthonormal basis: a basis set where Can be found from an arbitrary basis set by Gram-Schmidt Orthogonalization

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

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**Tensor Product Larger vector space formed from two smaller ones**

Combining elements from each in all possible ways Preserves both linearity and scalar multiplication

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**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

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**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

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**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

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**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

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**Bloch Sphere Slightly revised from**

Original picture from lecture notes for QC, University of Alberta, Canada

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**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

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**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.

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**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

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**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

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**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

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