CSCI 7000 - Quantum Computing - Fall 2012

Problem Set #2

Due: Oct 16th, 2012 at 2pm



1. Let $|\phi\rangle = \frac{1}{1+i}|0\rangle + \frac{1}{1-i}|1\rangle$. It this a valid superposition of a qubit?

2. Let $|\phi\rangle = \frac{1}{2}|00\rangle + \frac{x}{2\sqrt{2}}|01\rangle + \frac{1}{2\sqrt{2}} |10\rangle + \frac{1}{2}|11\rangle$. What values of $x$ would make this a valid superposition on a pair of qubits?

3. Let $|\Psi\rangle = \frac{1}{\sqrt{3}}(|00\rangle + |01\rangle +|10\rangle)$ and compute $|\Phi\rangle = (H \otimes H)|\Psi\rangle.$

4. Suppose we take $U_f$ from the Deutsch problem and compute $H^{\otimes 2} U_f H^{\otimes 2} |11\rangle$. What is the output? (Give the output for both the $f(0)=f(1)$ case and the $f(0)\neq f(1)$ case.)

5. Suppose I have the function $f(x)$ from the Bernstein-Vazirani problem, and its associated transform $U_f$. Then I compute $U_f(H^{\otimes (n+1)})(|0\rangle_n |1\rangle_1)$. If I measure the input register, what values are possible? And what is contained in the output register?

6. I want you to derive the two-bit state $|\Psi\rangle$ from problem 3. Here are the rules: you must start from $|00\rangle$. You can use any 1-qubit gates you want, and any 1-qubit unitary transforms you want. But the only 2-qubit transforms allowed are cNOTs.

7. Same as problem 6, but now you may use only a single cNOT. (Warning: this isn't an easy problem. You may want to first accomplish this by using a $C_H$ or "controlled Hadamard" 2-qubit gate first. Then try to implement $C_H$ with a single cNOT and some 1-qubit transforms.)