Introduction to Quantum Computing

Einar PiusUniversity of Edinburgh

Tuesday, 17 April 12

Why this Course

• To raise interest in quantum computing

• To show how quantum computers could be useful

• To talk about concepts not found in the textbooks

Tuesday, 17 April 12

The Lecturers

Einar Pius

• The guy talking in front of you

• Will give the first lectures. (Introduction)

Tuesday, 17 April 12

The Lecturers

Einar Pius

• The guy talking in front of you

• Will give the first lectures. (Introduction)

Vedran Dunjko

• Will join us on the second Week

Tuesday, 17 April 12

About the Course

• Course language: English vs Estonian

• Target audience (Computer Scientist vs Physicist)

• We expect basic knowledge of linear algebra

• We do not expect any knowledge of physics

• Basis for the new Quantum Computing course given at the University of Edinburgh next semester

Tuesday, 17 April 12

What You Will Learn

• The framework of quantum mechanics [Einar]

• The quantum circuits model [Einar]

• A few quantum algorithms [Einar]

• Quantum depth complexity [Einar]

• The Measurement Based Quantum Computing model [Vedran]

• Universal Blind Quantum Computing [Vedran]

Tuesday, 17 April 12

What We Will Not Talk About

• Quantum Information Theory

• Error correction and fault tolerance

• Shor’s algorithm

• Quantum key distribution

• Building quantum computers

Tuesday, 17 April 12

Today

• Introduction

• Motivation for quantum computers

• The Stern-Gerlach experiment

• Course structure

• How to pass the course

• Linear algebra

• Dirac notation

• Inner products

• Tensor products

• Operators

Tuesday, 17 April 12

Introduction to the Course

Tuesday, 17 April 12

Motivation

• Simulating quantum physics (Feynman, 1982)

Tuesday, 17 April 12

Motivation

• Simulating quantum physics (Feynman, 1982)

• Solving classically hard computational problems

Tuesday, 17 April 12

Motivation

• Simulating quantum physics (Feynman, 1982)

• Solving classically hard computational problems

• Factorizing integers

Tuesday, 17 April 12

Motivation

• Simulating quantum physics (Feynman, 1982)

• Solving classically hard computational problems

• Factorizing integers

• Computing discrete logarithms

Tuesday, 17 April 12

Motivation

• Simulating quantum physics (Feynman, 1982)

• Solving classically hard computational problems

• Factorizing integers

• Computing discrete logarithms

• Approximating the Jones polynomial

Tuesday, 17 April 12

Motivation

• Simulating quantum physics (Feynman, 1982)

• Solving classically hard computational problems

• Factorizing integers

• Computing discrete logarithms

• Approximating the Jones polynomial

• Solving problems faster than on classical computers

Tuesday, 17 April 12

Motivation

• Simulating quantum physics (Feynman, 1982)

• Solving classically hard computational problems

• Factorizing integers

• Computing discrete logarithms

• Approximating the Jones polynomial

• Solving problems faster than on classical computers

• Grover’s algorithm

Tuesday, 17 April 12

Motivation

• Simulating quantum physics (Feynman, 1982)

• Solving classically hard computational problems

• Factorizing integers

• Computing discrete logarithms

• Approximating the Jones polynomial

• Solving problems faster than on classical computers

• Grover’s algorithm

• Unconditionally secure quantum cloud computing

Tuesday, 17 April 12

Quantum Effects(The Stern-Gerlach Experiment)

Tuesday, 17 April 12

Timetable

• Quantum mechanics [Wednesday, April 18]

• Quantum cicuits [Thursday, April 19]

• Grover’s algorithm [Friday, April 20]

• Quantum Fourier Transform [Monday, April 23]

• Simulating Clifford circuits [Tuesday, April 24]

• Quantum depth complexity [Wednesday, April 25]

• The Measurement Based Quantum Computing model [Thursday, April 26]

• Universal Blind Quantum Computing [Friday, April 27]

• Lecture chosen by students [Monday, April 30]

Tuesday, 17 April 12

Passing the Course

• 40% is given for attendance

• 30% for discussion in the lectures

• 30% for homework

• Homework is given on Monday, April 30

• Answers can be found in text books and/or research papers

• Working in groups is allowed

• Individual answers from everyone

Tuesday, 17 April 12

Some Books

• Quantum Computation and Quantum Information (2000) Michael A. Nielsen & Isaac L. Chuang

• An Introduction to Quantum Computing (2007) P. Kaye, R. Laflamme, M. Mosca

Tuesday, 17 April 12

Tuesday, 17 April 12

|ai = ~a =

0

BBB@

a1a2...an

1

CCCA|bi = ~b =

0

BBB@

b1b2...bn

1

CCCA

hb|ai =�b⇤1 b⇤2 · · · b⇤n

�·

0

BBB@

a1a2...an

1

CCCA=

nX

i=1

b⇤i ai

Linear Algebra

Tuesday, 17 April 12

the Dirac Notation

• Vector. Also known as a ket.

|ai = ~a =

0

BBB@

a1a2...an

1

CCCA

Tuesday, 17 April 12

the Dirac Notation

• Vector. Also known as a ket.

• Dual vector of . Also known as a bra.

• is the complex conjugate of the complex number

|ai = ~a =

0

BBB@

a1a2...an

1

CCCA

|ai

ha| =�a⇤1 a⇤2 . . . a⇤n

�

(1 + i)⇤ = 1� i

zz⇤

Tuesday, 17 April 12

The Inner Product

• The inner product of two complex vectors and is defined as:

• such that:

•

•

• with equality only if

|ai |bi

hb|ai =�b⇤1 b⇤2 · · · b⇤n

�·

0

BBB@

a1a2...an

1

CCCA=

nX

i=1

b⇤i ai

hb|X

i

�i|aii =nX

i

�ihb|aii

hb|ai = ha|bi⇤

ha|ai � 0 |ai = 0

Tuesday, 17 April 12

Operators

• A linear operator between vector spaces and is defined to be any function which is linear in its inputs,

• Linear operators can be represented as matrices.

A : V ! W

V W

A

X

i

ai|vii!

=X

i

aiA(|vii)

Tuesday, 17 April 12

Hermitian Conjugate(Adjoint)

• In matrix representation, the Hermitian conjugate or adjoint of a matrix A is defined as its conjugate transpose:

A† = (AT )⇤

Tuesday, 17 April 12

Unitary and Hermitian operators

• Unitary operators

• Hermitian operatorsUU† = I

H = H†

Tuesday, 17 April 12

The Tensor Product

Tuesday, 17 April 12

Tuesday, 17 April 12