FE8100 Quantum computation and quantum information, 2009

Code: FE8100 (PhD course).

Instructors: Johannes Skaar (johannes skaar krøllalfa iet ntnu no), Øystein Marøy (oystein maroy krøllalfa iet ntnu no).

Weekly hours: 4

Lecture time: Monday 12-14 in B333 OR Wednesday 12-14 in G138, Elektrobygget. (See detailed plan below)

Qualifications: Basic knowledge of linear algebra. We will try to adjust the course to varying knowledge of quantum mechanics

Date of exam: 11.December

Type of examination: Written

Material: M. A. Nielsen og I. L. Chuang: Quantum computation and quantum information. List of errors here.

Quantum mechanical description of linear optics (Sections I, II, III, V, VI).

Content:

Introduction to quantum mechanics: Linear algebra, postulates, evolution, measurements, density operators. The Einstein-Podolsky-Rosen paradox, teleportation and Bell's inequality. Classical and quantum circuits. Quantum algorithms. Quantum information theory: Schmidt decomposition and purification, fitelity, quantum operations, entropy, Holevo’s bound. Error-correcting codes, measurements of entanglement and distillation of entanglement. Quantum cryptography. Physical realizations of quantum circuits with focus on photonic realizations.

Tentative plan:

Tuesday Aug. 25: First meeting.

Wednesday Sept. 9: Measurements, composite systems, no-cloning (Chapter 2).

Wednesday Sept. 16: Superdense coding, density operators (Chapter 2).

Wednesday Sept. 23: Density operators and partial trace, teleportation (Chapter 2).

Wednesday Sept. 30: Bell's inequality (Chapter 2), quantum circuits (Chapter 4)

Monday Oct. 5: Quantum circuits.

Wednesday Oct. 14: Quantum circuits (Chapter 4), Deutsch-Jozsa algorithm (Chapter 1).

Monday Oct. 19: Quantum operations.

Wednesday Oct. 28: Quantum operations, error-correcting codes.

Monday Nov. 2: Physical implementation of quantum computers, linear optics.

Wednesday Nov. 11: Entropy.

Monday Nov. 16: Linear optics.

Wednesday Nov. 25: Quantum key distribution.

Reading list / syllabus:

Short introduction

Nielsen&Chuang:

Ch. 1: 13-36

Ch. 2: 60-117

Ch. 3: 129-142

Ch. 4: 171-194

Ch. 6: 248-255

Ch. 7: 277-283, 287-297

Ch. 8: 353-366, 375-379, 383-387

Ch. 9: 399-400, 403-406, 409-410

Ch. 10: 425-434

Ch. 11: 500-508, 510-511, 513-514

Ch. 12: 532, 582-583, 586-591

Quantum mechanical description of linear optics (Sections I, II, III, V, VI).

Exercises with solutions:
Exercise 1, Linear Algebra: 2.2, 2.5, 2.7, 2.9, 2.11, 2.17, 2.18, 2.24
Solution 1

Exercise 2, Tensor products and Trace: 2.26-2.33, 2.37-2.39
Solution 2
Exercise 3, Quantum Mechanics: 2.51-2.54, 2.57, 2.59, 2.63 and 2.66. A hint for 2.63, use singular value decomposition, p.73 inN&C
Exercise 4, Quantum Circuits: 4.1, 4.2, 4.7, 4.13, 4.16-4.18, 4.32, 4.39
Exercise 5, Grover’s Algorithm: 6.1-6.3, 6.6
Exercise 6, Quantum Computers: 7.1-7.7, 7.9
Exercise 7, Density operators: 2.71-2.72, 2.74, 2.79, 2.82
Exercise 8, Quantum Operations: 8.4, 8.5, 8.15, 8.18, 8.26
Exercise 9, Distance Measures: 9.2, 9.6, 9.7, 9.14 (also prove that the Trace distance is invariant under unitary transformations), Show that F(rho
,sigma)=1 <=> rho=sigma and that F(rho,sigma)=0 <=> rho and sigma has support on orthogonal subspaces.

Exercise 10, Error correcting codes: 10.1, 10.3, 10.6
Exercise 11, Entropy: 11.1, 11.3, 11.5, 11.11, 11.12, 11.13

Old exercises with some solutions (Norwegian)

Previous exams:

Eksamen med løsning, høsten 2002

Eksamen med løsning, høsten 2003 (med noen små endringer).

Eksamen med løsning, høsten 2004

Eksamen med løsning, høsten 2005

Eksamen med løsning, høsten 2006

Eksamen med løsning, høsten 2007

Eksamen med løsning, høsten 2008