HW1 - Bonus Problems

Problem 1

Draw the following complex numbers and their complex conjugates in the complex plane. If they are in the standard representation, convert them to the phase representation and vice versa.

• \(w = 1/2 + i\)
• \(z = \frac{5}{2} - i\frac{5\sqrt{3}}{2}\)
• \(\beta = 6(\cos(\pi/4) + i\sin(\pi/4))\)
• \(\xi = 0.01(\cos(4\pi/3) + i\sin(4\pi/3))\)

Problem 2

For each pair of vectors below, compute their tensor product, \(\left<\phi|\psi\right>\) and \(\left<\psi|\phi\right>\).

• \(\left|\psi\right> = \begin{bmatrix}7 \\ 4i\end{bmatrix}\) and \(\left|\phi\right> = \begin{bmatrix}3i \\ 2\end{bmatrix}\)

\(\left|\psi\right> = \begin{bmatrix}3 + 2i \\ -2\end{bmatrix}\) and \(\left|\phi\right> = \begin{bmatrix}1 - 3i \\ 4 + i\end{bmatrix}\)?

Problem 3

Write down a function is_qubit() in a language of choice that takes as input a list of length 2 and returns True if the input represents a qubit, and False otherwise.

Problem 4

Write down a function random_qubit() in a language of choice that randomly generates a qubit state. The output of this should pass the is_qubit() test from the previous problem.

Problem 5

Write down a function inner_product(q1, q2) in a language of choice that takes two qubit states and returns their inner product. Try running this function with the vectors in problem 2 and compare your solution against the output of the code.