Homework 3 ECE 455/555: Introduction to Quantum Mechanics and Quantum Information Processing Spring 2024 Date Assigned: Wednesday, February 21, 7:00 am MST Due Date and Time: Friday, March 1, 11:59 pm MST (in D2L) Maximum Credit: 100 points 1. [30 points] Circuit identities: (a) (10 points) In the lecture, we derived HZH X and HXH = Z by investigating the action of H on the eigenstates of Z and X. Similarly, compute the action of S on the eigenstates of X and show that it is the change-of-basis operator from X to Y, i.e., SXS Y. Also show that S fixes the Z-eigenstates, i.e., SZS = Z. (b) (10 points) We discussed the following circuit identities/Pauli conjugations/Pauli tracking for the CNOT gate in the class: X = X = = CNOT(XI) = (XX)CNOT CNOT(IX)=(IX)CNOT Z CNOT(Z)=(ZI)CNOT Z Z => Z CNOT(IZ)=(ZZ)CNOT Derive the analogous identities for the CZ gate using its definition: CZ = 0) (0 + 1) (1| Z. (c) (10 points) Recall the relationship between the CNOT and CZ. Use this to prove that: H H = H H