Preliminaries Single-qubit Pauli operators are defined as ?_x = [0 1; 1 0], ?_y = [0 -i; i 0], ?_z = [1 0; 0 -1], and i = ?-1. Problem 1 Express the following vectors or matrices using Dirac notation (a.k.a. bra-ket notation) in the computational basis. (a) [1/?2; 1/?2] (b) [?2/3; 1/?3] (c) [1/?6; 1/?3; 0; 1/?2] (d) [1/?2 0 0 i/?2] (e) [a b; c d] Problem 2 A Hermitian matrix satisfies A = A†. Prove that Hermitian matrices can only have real eigenvalues. Problem 3 (a) Show a Taylor series of a unitary matrix exp(-i?A) (? ? [0,2?] and A is a Hermitian matrix) up to 4th order in A. (b) Given A^2 = I, express the unitary matrix in (a) as a function of cos(?) and sin(?) in a closed form (hint: think about Euler's formula). (c) For a three-dimensional unit vector n? and a Pauli vector ? = (?_x, ?_y, ?_z), show that (n?·?)^2 = I. (d) Express exp(i??_y)|0? as a function of ?. (e) Find the value of ? ? [0,2?) that transforms |0? to (|0?+|1?)/?2 under an application of a unitary matrix exp(i??_y). (f) Find the final state |?_f? = U|00?, where U = exp(-i ?/4 ?_x ? ?_x). Express the answer using Dirac notation. Problem 4 Write the eigenvectors of the single-qubit Pauli operators in the computational basis using Dirac notation. Problem 5 Let |?? = ?|00? + ?|11?, where ? ? C, ? ? C, and |?|^2 + |?|^2 = 1. Express the following expected value of an observable in terms of ? and ?. (a) ??_z ? I? (b) ?I ? ?_y? (c) ??_x ? ?_x?
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