Kvantedatamaskiner og kvantekommunikasjon 2008

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 {E_m} 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.