2 Unitary Transforms and Quantum Gates 4. (15 points) Show that the magic gate below is an unitary gate M = 1 1 0 0 2 00 2 1 200 2-1 100 (7) 5. (20 points) Show that if we apply the quantum Fourier transform 2"-12"-1 1 UQFT e 2 127kj /2" (k) (il j=0 k=0 to the following state in Hilbert space C (n = 3) 120)= j=0 cos(2j/8)\j), the resulting state will be given by (8) (9) UQFT 4) = (11) +17)). (10) 4. Write 75x7y in the simplest radical form. No decimal answers. 5. Solve the following equation. 2x-3-7=0 6. Three computer disks and two notebooks cost $17. Four computer disks and five notebooks cost $32. Let x be the price of a computer disk and y be the price of a notebook. Write a system of equations which model the situation. Solve the system of equations to find the price of a computer disk and the price of a notebook. 7. Simply the following. Write the final answer using positive exponents. (a2b-5)-3