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# HW4 - θ 1 θ 2 along the same axis R A θ 1 R A θ 1 = R A...

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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

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Unformatted text preview: θ 1 + θ 2 along the same axis: R A ( θ 1 ) R A ( θ 1 ) = R A ( θ 1 + θ 2 ) Problem 3 (20 points) Show that the classical Fredkin and Toﬀoli 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 ρ and a set of orthonormal vectors | ϕ i i ∈ H n the spectral decomposition of the operator ρ is: ρ = X i λ i | ϕ i ih ϕ i | . The von Neumann entropy is S ( ρ ) =-tr( ρ log ρ ) =-X i λ i log λ i . 1 Show that when ρ = X i p i ρ i then S ( ρ ) =-X i p i log p i-X i p i S ( ρ i ) or S ( ρ ) = H ( p i )-X i p i S ( ρ 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...
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HW4 - θ 1 θ 2 along the same axis R A θ 1 R A θ 1 = R A...

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