Introduction to Quantum Computing

Transcription

1 Introduction to Quantum Computing Einar Pius University of Edinburgh

2 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

3 The Lecturers Einar Pius The guy talking in front of you Will give the first lectures. (Introduction)

4 The Lecturers Einar Pius The guy talking in front of you Vedran Dunjko Will join us on the second Week Will give the first lectures. (Introduction)

5 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

6 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]

7 What We Will Not Talk About Quantum Information Theory Error correction and fault tolerance Shor s algorithm Quantum key distribution Building quantum computers

8 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

9 Introduction to the Course

10 Motivation Simulating quantum physics (Feynman, 1982)

11 Motivation Simulating quantum physics (Feynman, 1982) Solving classically hard computational problems

12 Motivation Simulating quantum physics (Feynman, 1982) Solving classically hard computational problems Factorizing integers

13 Motivation Simulating quantum physics (Feynman, 1982) Solving classically hard computational problems Factorizing integers Computing discrete logarithms

14 Motivation Simulating quantum physics (Feynman, 1982) Solving classically hard computational problems Factorizing integers Computing discrete logarithms Approximating the Jones polynomial

15 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

16 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

17 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

18 Quantum Effects (The Stern-Gerlach Experiment)

19 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]

20 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

21 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

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23 ai = ~a = 0 a 1 1 a 2 B A bi = ~ b = a n 0 1 hb ai = b 1 b 2 b n a 1 0 a 2 B A = a n b 1 b 2 b n n X i=1 1 C A b i a i Linear Algebra

24 the Dirac Notation Vector. Also known as a ket. ai = ~a = 0 a 1 1 a 2 B A a n

25 the Dirac Notation Vector. Also known as a ket. ai = ~a = 0 a 1 1 a 2 B A a n Dual vector of ai. Also known as a bra. ha = a 1 a 2... a n z is the complex conjugate of the complex number (1 + i) =1 i z

26 The Inner Product The inner product of two complex vectors ai and bi is defined as: 0 1 hb ai = b 1 b 2 b n a 1 a 2 B A = a n n X i=1 b i a i such that: hb X i i a i i = nx i ihb a i i hb ai = ha bi ha ai 0 with equality only if ai =0

27 Operators A linear operator between vector spaces V and W is defined to be any function A : V! W which is linear in its inputs,! A X i a i v i i = X a i A( v i i) i Linear operators can be represented as matrices.

28 Hermitian Conjugate (Adjoint) In matrix representation, the Hermitian conjugate or adjoint of a matrix A is defined as its conjugate transpose: A =(A T )

29 Unitary and Hermitian operators Unitary operators Hermitian operators UU = I H = H

30 The Tensor Product

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