Lecture 1, hands on 2

1. Create two-qubit functions that return the following quantum states:

• \(\left|\psi_+\right\rangle = \frac{1}{\sqrt{2}}\left(\left|00\right\rangle + \left|11\right\rangle\right)\)
• \(\left|\psi_-\right\rangle = \frac{1}{\sqrt{2}}\left(\left|00\right\rangle - \left|11\right\rangle\right)\)
• \(\left|\phi_+\right\rangle = \frac{1}{\sqrt{2}}\left(\left|01\right\rangle + \left|10\right\rangle\right)\)
• \(\left|\phi_-\right\rangle = \frac{1}{\sqrt{2}}\left(\left|01\right\rangle - \left|10\right\rangle\right)\)

2. Write a function that implements the SWAP gate, using only CNOT gates.  How many CNOT gates are needed?

3. Write a function which implements a Controlled-controlled-CNOT gate (left image) using only Toffoli gates (centre) and an auxiliary qubit (right).