Quantum Computing Mathematics and Postulates Advanced topic seminar SS02 “Innovative Computer architecture and concepts” Examiner: Prof. Wunderlich Presented.

Quantum Computing Mathematics and Postulates

Advanced topic seminar SS02

“Innovative Computer architecture and concepts”

Examiner: Prof. Wunderlich

Presented byPresented byChensheng QiuChensheng Qiu

Supervised bySupervised by

Dplm. Ing. Gherman Dplm. Ing. Gherman

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

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

C ),(

kkk

kkk wvawav ,,

*),(),( wvwv

0),( vv

Hilbert SpaceHilbert Space

Inner product space: linear space equipped with inner productHilbert Space (finite dimensional): can be considered as inner product space of a quantum systemOrthogonality: Norm: Unit vector parallel to

0wv

vvv

v

vv :

Hilbert Space (Cont’d)

Orthonormal basis:

a basis set where

Can be found from an arbitrary basis set by Gram-Schmidt Orthogonalization

nvv ,...,1 ijji vv

Unitary Operator

An operator U is unitary, if

Preserves Inner product

IUUτ

Uofadjoint for the stands Uwhere

wvwUvU ,,

Tensor ProductTensor 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 QMPostulates 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

24242424

Physical Systems -Physical Systems - Quantum Mechanics Connections Quantum Mechanics Connections

Postulate 1Isolated physical

system Hilbert Space

Postulate 2Evolution of a

physical system

Unitary transformation

Postulate 3Measurements of a

physical system

Measurement operators

Postulate 4Composite physical

system

Tensor product of

components

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

Qubits Cont'd

We may rewrite as…

From a single measurement one obtains only a single bit of information about the state of the qubitThere is "hidden" quantum information and this information grows exponentially

0 1

cos 0 sin 12 2

i ie e

cos 0 sin 12 2

ie

We can ignore ei as it has no

observable effect

Bloch Sphere

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

Oven

Z

Z

Z

Spin-up

Spin-down

Figure 6: Abstract schematic of the Stern-Gerlach experiment.

Qubit in Stern-Gerlach Exp.

Oven

Z

Z

Z

X Z

Z

Z

Figure 7: Three stage cascade Stern-Gerlach measurements

X

X

Z

X

Z

Qubit in Stern-Gerlach Experiment

Figure 8: Assignment of the qubit states

Z

X

Z

2/10

2/10

1

0

X

X

Z

Z

Qubit in Stern-Gerlach Experiment

Figure 8: Assignment of the qubit states

Z

X

Z

2/)(0

2/)1 0 ( and 2/)1 0 (

basis nalcomputatio Gerlach -Stern

1, 0 basis nalcomputatio Gerlach -Stern

XX Z

X

Z