Quantum Computing

(1) Consider the 2x2 Hadamard Matrix H

Calculate its eigenvalues and corresponding eigenvectors. Normalize the eigenvectors.

(2) Give the outer product representation of the Hadamard matrix H given above using the basis formed

(3) Prove that the exponential operator containing the first spatial derivative plays the role of a displacement operator, i.e.,

(4) If the operators A and B both commute with the commutator (A,B), prove that the following is true [A,F(B)] = [A,B] F'(B),

Where F'(B) is the first derivate of the operator F(B) whose Taylor expansion is given by

(5) Find the expression of the 4x1 column vector obtained by calculating the following expression

[6] Calculate the output of the following circuit which implements the applications of two successive Toffoli gates in two different ways

(a) By using the truth table for the Toffoli gate as discussed in class.

(b) By using the fact that for a Toffoli gate,