Due 2020-09-14 4:45pm in gradescope.

Quantum Computing for Computer Scientists will be very useful.

Ok to work in groups of 3 and submit one solution.

1. (2 points) Compute (1+2i)/(3+4i).

2. (5 pts) Compute the eigenvalues of $\begin{vmatrix} 1&2\\3&4 \end{vmatrix}$.

3. (2 pts) Considering complex numbers as points in the 2D plane, what is the geometric effect of multiplying a complex number by (.6+.8i) ?

4. (5 pts) Do exercise 1.3.8. Let c = 1 − i. Convert it to polar coordinates, calculate its fifth power, and revert the answers to Cartesian coordinates.

5. (5 pts) Exercise 1.3.9 Find all the cube roots of c = 1 + i.

6. (5 pts) Find the inverse Mobius transformation to

   $$R_{a,b,c,d}(z) = \frac{az+b}{cz+d}$$

7. (5 pts) Invert the Hadamard matrix $\frac{1}{\sqrt{2}} \begin{vmatrix} 1&1\\1&-1\end{vmatrix}$.

8. (2 pts) Could a classical OR gate, with 2 inputs and 1 output, be a quantum gate? Why?

9. (2 pts) Could the following extension of a classical OR gate, with 2 inputs and 2 outputs, be a quantum gate? Why?:

   A' = A or B
   B' = B

10. (2 pts) What about this extension of a classical XOR gate, with 2 inputs and 2 outputs. Can it be a quantum gate? Why?:

    A' = A xor B
    B' = B

Total: 35