Answer Happy

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(2) 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=12 |)(| and I = -ZZG. 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 mo, cos(n + m) 0: = cos no cos mo sin ne sin mo. a. Find all the solutions to f(x, y, z, w) = 3. [2 marks]