FE8100 Quantum computation and quantum information 2010 (7.5 credits)
UNIK942 Quantum computation and quantum information 2010 (10 credits)
Instructor: Johannes Skaar (johannes skaar krøllalfa iet ntnu no)
Lecture time: Every second Thursday 10-12 in B333, see detailed plan below. You are expected to read and to do some exercises before each lecture, see the plan below.
Qualifications: Basic knowledge of linear algebra. We will try to adjust the course to varying knowledge of quantum mechanics.
Date of exam: Thursday December 9, 09:00-13:00.
Type of examination: Written
Material: M. A. Nielsen og I. L. Chuang: Quantum computation and quantum information. Errata 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. No-cloning theorem. The Einstein-Podolsky-Rosen paradox, teleportation and Bell's inequality. Classical and quantum circuits. Quantum algorithms. Quantum information theory: Schmidt decomposition and purification, fidelity, quantum operations, entropy, Holevo’s bound. Error-correcting codes, measurements of entanglement and distillation of entanglement. Quantum cryptography. Physical realizations of quantum circuits with emphasis on photonic realizations.
Tentative plan:
The student is expected to read much of the material on his/her own.
Monday Aug. 30: First meeting/lecture. Postulates in quantum mechanics. Meausrements, composite systems, no-cloning (Ch 2).
Sept. 16: Before lecture: Read Ch. 1 and 2 up to the section on density operators. Do the exercises. Lecture: Density operators (Ch 2).
Sept. 30: Bell's inequality (Ch 2). Single qubit rotations (Ch 4).
Oct. 14: Quantum circuits, controlled operations, measurements (Ch 4).
Oct. 28: Quantum operations (Ch 8).
Thursday Nov. 4 at 16:05: A photonic implementation example, linear optics, etc., and perhaps also error correcting codes.
No lecture Nov. 11.
Friday Nov. 19, 15-17, room B343: Entropy and information, quantum key distribution.
Reading list / syllabus:
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:
There has
been some changes to Exercise 4 and 5. In Exercise 4 2.81 and 2.82(3)
has been added, and in Exercise 5 4.8 has been replaced with 4.9.
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 in N&C
Exercise with POVM measurement: Given three operators E1, E2 and E3 defined by Equations (2.118-2.120).
- Show that the operators are positive.
- We measure a qubit which is in one of the two states |y1> = |0> or |y2> = (|0>+|1>)/sqrt(2). Show that if the result of the measurement with {Em} is m = 1 then the inital state was |y2> and if the result is m = 2 the inital state was |y1>.
- Is it
possible to discriminate between two non-orthogonal states like |y1>
og |y2>
with this method?
Solution
3
Exercise 4,
Density
Operators: 2.69-2.72, 2.74, 2.79, 2.81,2.82.
Solution
4.
Exercise
5,
Quantum
Circuits: 4.1-4.3,
4.5-4.7,4.9,
4.13, 4.16-4.20, 4.32. Solution
5.
Exercise
6,
Grover’s Algorithm: 6.1-6.3, 6.6.
Exercise
7,
Quantum
Operations: 8.4, 8.5, 8.15, 8.17, 8.26. Solution
7.
Last
years exercises:
Exercise
6, Quantum Computers: 7.1-7.7, 7.9
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:
2002 (with solution)
2003 (with solution)
2004 (with solution)
2005 (with solution)
2006 (with solution)(English)
2007 (with solution)
2008 (with solution)(English)