HW4 - 1 + 2 along the same axis: R A ( 1 ) R A ( 1 ) = R A...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Department of Electrical Engineering and Computer Science University of Central Florida COT 6600– Quantum Computing Fall 2010 (dcm) Homework Assignment 4. – Due Monday November 8, 2010 Problem 1 (30 points) Show that the following relations exist among the identity matrix, σ I , the Hadamard matrix H , and the Pauli matrices σ x y z : x H = σ z , z H = σ x , y H = - σ y . Show that: σ 2 x = σ 2 y = σ 2 z = σ I , σ x = σ x , σ y = σ y , σ z = σ z . Show that the commutators of the Pauli matrices satisfy the following relations: [ σ y z ] = 2 x , [ σ z x ] = 2 y . Problem 2 (30 points) Show that any single qubit transformation given by a 2 × 2 matrix M can be represented as a linear combination of Pauli matrices: M = c 0 σ I + c 1 σ x + c 2 σ y + c 3 σ z , c 0 ,c 1 ,c 2 ,c 3 C . Consider a point A on the Bloch sphere with coordinates x A ,y A ,z A and the vector con- necting the origin of the sphere with A . Show that a rotation with an angle θ around this vector is described by: R A ( θ ) = cos θ 2 σ I + i sin θ 2 ( x A σ x + y A σ y + z A σ z ) Show that the composition of two rotation operations about the same axis with angles θ 1 and θ 2 respectively is a rotation with angle
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 1 + 2 along the same axis: R A ( 1 ) R A ( 1 ) = R A ( 1 + 2 ) Problem 3 (20 points) Show that the classical Fredkin and Tooli gates perform a fanout operation while their quantum counterparts do not violate the no cloning theorem. Hint: consider a control qubit in a superposition state 1 2 ( | i + | 1 i ). Problem 4 (30 points) Given a system with density operator and a set of orthonormal vectors | i i H n the spectral decomposition of the operator is: = X i i | i ih i | . The von Neumann entropy is S ( ) =-tr( log ) =-X i i log i . 1 Show that when = X i p i i then S ( ) =-X i p i log p i-X i p i S ( i ) or S ( ) = H ( p i )-X i p i S ( i ) with H ( p i ) the Shannon entropy of a random variable X with the probability density function p X = p X ( x = i i ) = p i . 2...
View Full Document

Page1 / 2

HW4 - 1 + 2 along the same axis: R A ( 1 ) R A ( 1 ) = R A...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online