2. Let f be the function on four-bit strings defined by f(x, y, z, w) = x©y+y©z+z© w, where x, y, z, w = {0,1}, z denote

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2 Let F Be The Function On Four Bit Strings Defined By F X Y Z W X Y Y Z Z W Where X Y Z W 0 1 Z Denote 1 (378.5 KiB) Viewed 15 times

2. Let f be the function on four-bit strings defined by f(x, y, z, w) = x©y+y©z+z© w, where x, y, z, w = {0,1}, z denotes NOT(z) where we interpret z as a bit, denotes addition modulo 2 (equivalent to XOR on two bits), and + denotes standard integer addi- tion. Define the unitary Uƒ |x, y, z, w) |n) = |x, y, z, w) |n+ f(x, y, z, w) mod 4) acting on six qubits, where n = {0, 1}² is interpreted as the standard 2-bit binary representation of an integer in the range 0 ≤ n ≤ 3. Let G denote the subspace spanned by the states x, y, z, w) corresponding to the solutions to f(x, y, z, w) = 3. Let IIG denote the orthogonal projector onto G. Define ZG = 1-2IIG, Z = 1 − 2)(| and I = -Z ZG. Let ) be the four-qubit state: |) = (0001) + |0010) + |0100) + |1000)). In this question, you may use without proof the following trigonometric identities: sin(n + m)0: = sin ne cos me + cos no sin me, cos(n + m)0 = cos no cos me - sin ne sin me. a. Find all the solutions to f(x, y, z, w) = 3. [2 marks] S b. Decompose the state ) in the form y) = s|p) +c|p¹), where |p) is a state in the subspace G and p) is a state orthogonal to G. [4 marks] c. Show that I acts on the subspace spanned by y) and p) as a rotation matrix of the form (cos 20 - sin 20 ) and determine the value of 0. sin 20 cos 20 [10 marks]