We (Hanmeng Zhan, Chris Godsil, Gabriel Coutinho) will hold an on-line course on quantum walks in Winter 2021. There will be two lectures each week, usually offered by Gabriel and Harmony. The intended audience is entering graduate students. The background requirement is some knowledge of graph theory and a thorough understanding of linear algebra. Many of the lectures will based on the following drafts of books:

  1. Graph Spectra and Continuous Quantum Walks, Coutinho and Godsil.
  2. Discrete Quantum Walks, Godsil and Zhan.
We will post exercises, but we are not offering to grade solutions. Some of the exercises will involve computation, we will make some related sage code available. We will assign reading. The bulk of the lectures will be given by Gabriel and Harmony, but there will very likely be some guest lectures as well.

It might be possible to get credit from your institution for taking this course. If they charge you a fee, they can pay for grading. :-)


Lectures are on zoom, Tuesdays and Thursdays 11:00am, starting January 12.

Recordings of the lectures: (GC= Gabriel, continuous walks. HZ=Harmony, discrete walks.)

  1. Jan 12 (GC)
  2. Jan 14 (HZ)


Continuous. Discrete.

Content and Outline

We will focus on the mathematical aspects of quantum walks. The course will cover both discrete and continuous walks, on a more or less equal footing. Possible topics:

  1. motivation
  2. Physics
  3. arc-reversal walks
  4. representation theory for two reflections
  5. Grover
  6. face-vertex walks
  7. bipartite walks and Hamiltonians
  8. graph isomorphism failures
  9. real states (periodicity, state transfer, ratio condition)
  10. pst on Laplacians of trees
  11. cubelike graphs
  12. strongly cospectral vertices and symmetries
  13. uniform mixing on cycles
  14. orthogonal polynomials
  15. pretty good state transfer
  16. fractional revival
  17. average mixing, average states


For further information, contact Chris Godsil.