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.

Test make-ups: you can still improve your Test 1 and Test 2 scores by retaking one Part B problem from each test (a written replacement problem on Canvas if you scored 7 or higher on it, or a 15-minute in-person retake booked through the calendar link on the Canvas main page if you scored 6 or lower). Details in the October 16th post.

Today we met the quantum Fourier transform (QFT). Since your poster drafts are due Friday, I’ll keep this daily post short, but it would be good to do a bit of basic computation with the QFT.

Relevant Textbook Sections:

Chapter 6 (Quantum): roots of unity and the quantum Fourier transform.

To Do:

1. Apply the $4 \times 4$ QFT matrix to the state

   $$\frac{1}{\sqrt{2}} \left( |00\rangle + \frac{1+i}{2} |01\rangle + \frac{1-i}{2} |11\rangle \right).$$

2. Draw the $8$th roots of unity in the complex plane and label them. You can use the $8$th root of unity notation $\omega := \omega_8 = e^{i\pi/4}$, but simplify so that the labels are from the set $\{1, -1, i, -i, \omega, -\omega, i\omega, -i\omega\}$.

3. Write down the QFT matrix of dimension $8 \times 8$. Again, use the notation $\omega := \omega_8 = e^{i\pi/4}$, but simplify so that the entries are all from the set $\{1, -1, i, -i, \omega, -\omega, i\omega, -i\omega\}$. (Hang on to this; we’ll use it to run Shor’s algorithm by hand next time.)

Optional: the Visualizing Complex Numbers & QFT and QFT Amplitudes pages (under the Tools menu, Module 4) let you watch the roots of unity sum up in the QFT of a periodic state.