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
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Linear Algebra
Tuesday, 17 April 12
the Dirac Notation
• Vector. Also known as a ket.
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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
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Tuesday, 17 April 12
The Inner Product
• The inner product of two complex vectors and is defined as:
• such that:
•
•
• with equality only if
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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
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i
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=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