CSCI 7000 - Quantum Computing - Fall 2012

Problem Set #1

Due: Sep 25th, 2012 at 2pm


1. For what values of $n$ is $i$ an $n$th root of 1?

2. Fix $n \geq 1.$ Let $\omega_0, \cdots, \omega_{n-1}$ be the $n$th roots of unity.

  1. Show that $(z-\omega_0)(z-\omega_1)\cdots(z-\omega_{n-1}) = z^n - 1$.
  2. In class we gave a hand-waving argument that for $n>1$, $ \sum_{i=0}^{n-1} \omega_i = 0 $ by arguing about the vectors forming an equilateral $n$-sided polygon. Give an algebraic proof using (a) above.
  3. What is $\omega_0\omega_1\cdots\omega_{n-1}$? Prove your answer.

3. Find the value of $(C_{ij})^2$, showing your derivation.

4. Give the matrix form for $S_{12}$ and compute its determinant.

5. Simplify $HYH$, showing your derivation. Don't use the matrix form; do this algebraically.

6. Simplify $C_{ij}S_{ij}C_{ij}$, showing your derivation.

7. Simplify $H_iH_j S_{ij} H_iH_j$, showing your derivation.

8. Characterize all 2x2 complex matrices $A$ where $A^2=1$ by saying what their eigenvalues must look like.