Poster reminders: the first draft (complete, in LaTeX) is due Friday, November 6th, and the final version and report of work are due Friday, November 13th. Details on the Posters tab.

Solutions: solutions to the daily posts due October 23rd (qubits and the $L$/$V$ bases), October 26th (BB84 on paper, key length, catching Eve), October 30th (two-qubit states and entanglement), and November 2nd (multi-qubit measurement and single-qubit gates) are posted. All daily post solutions are collected on the Archive page.

Today we built gates on two and more qubits (CNOT, Hadamard on one qubit of a system, the reversible AND), and saw the general principle that any classical function can be computed quantumly, as long as we make it reversible. Here are some problems in those directions.

Relevant Textbook Sections:

Chapter 6 (Quantum): multi-qubit gates, CNOT, the reversible AND (Toffoli) gate, and computing classical functions reversibly.

To Do:

Unitary matrices.

1. Verify that the product of two unitary matrices is unitary. (This may require reviewing the transpose of a product.)
2. Verify that a unitary matrix is invertible and its inverse is unitary.

Two-qubit gates.

3. Verify that the CNOT gate $\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{pmatrix}$ is unitary.
4. Apply the CNOT gate to the state $\frac{1+2i}{\sqrt{6}}|01\rangle + \frac{1}{\sqrt{6}}|10\rangle$.
5. Consider “applying the Hadamard gate to the second qubit” as a two-qubit gate.
   1. What does this do to the state $|00\rangle$?
   2. What does this do to the state $|01\rangle$?
   3. Write out the two-qubit $4 \times 4$ matrix that does this, and verify that it is unitary.
6. Begin with the 2-qubit state $|00\rangle$.
   1. Apply the CNOT gate to the 2-qubit system and then the Hadamard gate to the 2nd qubit.
   2. Start over with $|00\rangle$. Now apply the Hadamard gate to the 2nd qubit and then the CNOT gate to the system. Do you get the same answer?
   3. Repeat the last two parts with the Hadamard gate applied to the 1st qubit instead of the 2nd. Do you get the same answer this time? Is either output entangled?
7. Determine a $2$-qubit quantum circuit that will, on input $|00\rangle$, produce output $\frac{1}{2}|00\rangle + \frac{1}{2}|01\rangle + \frac{1}{2}|10\rangle + \frac{1}{2}|11\rangle$. Hint: combine Hadamard gates.

Classical functions, quantumly.

8. Apply the reversible AND gate (as demonstrated in class) to the state $\frac{1}{2}|000\rangle + \frac{1}{2}|011\rangle + \frac{1}{2}|101\rangle + \frac{1}{2}|111\rangle$.
9. Make an $8 \times 8$ unitary matrix that implements a reversible OR gate.
10. Let $f(x_1, x_2) = x_1 \oplus x_2$ (XOR), and let $U_f$ be the three-qubit gate $U_f |x_1 x_2\rangle |y\rangle = |x_1 x_2\rangle |y \oplus f(x_1,x_2)\rangle$.
    1. Describe what $U_f$ does to each of the eight basis states. Explain why this guarantees $U_f$ is unitary.
    2. Apply $U_f$ to $\frac{1}{2}\left( |00\rangle + |01\rangle + |10\rangle + |11\rangle \right)|0\rangle$. Notice that you have evaluated $f$ on all four inputs at once.
    3. Express $U_f$ as a composition of two CNOT gates.