Introduction to Mathematical Quantum Error Correction

Intersession 2025
Illinois Mathematics and Science Academy

Instructor: Dheeran E. Wiggins
Email: dwiggins@imsa.edu or dheeran2@illinois.edu
Co-facilitator: Dr. Micah Fogel, Mathematics Faculty
Email: fogel@imsa.edu
Dates: January 13 -- January 17, 2025
Room: A151
Here are the Lecture Notes, the Homework, and the final Poster.

Welcome to Introduction to Mathematical Quantum Error Correction! Throughout the week we will study

  • the \(n\)-qubit Pauli group \(\mathcal{P}_n\).
  • error and recovery channels \(\Phi: \mathbb{B}(\mathcal{H}) \to \mathbb{B}(\mathcal{H})\).
  • the Knill-Laflamme subspace condition \(PE_a^{\dagger}E_bP = \lambda_{ab}P\).
  • the stabilizer formalism.
  • operator quantum error correction (OQEC) on a Hilbert space with an induced direct sum (biproduct) decomposition \[\mathcal{H} \simeq \bigoplus_{J} \mathcal{H}^{A}_J \otimes \mathcal{H}^B_J.\]
  • noiseless subsystems of a fixed partition \(\mathcal{H} \simeq (\mathcal{H}^A \otimes \mathcal{H}^B) \oplus \mathcal{C}^{\perp}\).
    • Notably, we will take a rigorous approach to the relevant mathematics, while supplementing the theorems with physical and information-theoretic intuition.

    Quantum error correction is riddled with beautiful mathematics. But, it is hard. My hope is that by the end of these lectures, you will be well-equipped to break into the literature, whether textbooks or papers. In particular, we will close out our discussion by working through some research papers ourselves.

    It is worth noting that while my QEC background has mostly been via mathematics, people from physics, computer science, and electrical engineering are working on these problems. I will do my best to incorporate as many of these perspectives as I can into lecture. If you are partial to some flavor of the subjects above, I will point you towards references. If you are worried about prerequisites, do not be. Ask questions and follow along as best you can. Intersession is a chance to learn something neat outside the stress of your standard coursework.

    Textbook

    No textbook is required. We will loosely follow my own lecture notes (see the top of the page), though we may deviate depending on time and interest. If you are interested in self-study beyond my notes, consider working through

  • Quantum Computation and Quantum Information "Mike and Ike" by Michael Nielsen and Isaac Chuang. This is a very standard text which does its best to balance all aspects of introductory quantum information. I think it does a good job (though I wish it was a bit more mathematically tailored).
  • An Introduction to Quantum Computing by Philip Kaye, Raymond Laflamme, and Michele Mosca. I have occasionally used this as a supplement to Mike and Ike. I like how it presents error correction.
  • Linear Algebra by Stephen Friedberg, Arnold Insel, and Lawrence Spence. This is my favorite of the "quintessential" linear algebra texts. The treatment is rigorous, while not trying too hard to be an abstract algebra book.
  • Abstract Algebra by David Dummit and Richard Foote. If you want to learn algebra, Dummit and Foote will carry you through the fundamentals, and further. I wish it was more categorical, but this is a minor complaint.

    • Assignments and Deliverables

    For obvious reasons, this minicourse cannot have regular problem sets. Still, I will include one to two exercises to be done as homework between lecture days. These will be pretty brief and should be easy after sitting through lecture. If not, I am happy to work through them with you!

    We will also develop an expositional \(\LaTeX\) poster together, to be presented at the Intersession Expo, summarizing memorable theorems from the week.

    Catalog Description

    Quantum computation (QC) and information (QI) are relatively young fields, conceived less than half a century ago. In the subsequent decades, those fluent in computer science, mathematics, or physics have seen an increased demand for QC and QI talent across industry and academia. Still, quantum computation retains an inherent constraint—"quantum noise"—a susceptibility to errors caused by quantum mechanical phenomena. Thus, the study of Quantum Error Correction (QEC) was born, giving way to a series of mathematical, algorithmic, and physical techniques used to identify and correct any such errors. After taking care of some group theory and linear algebra preliminaries, we develop some major tools of QEC, such as the Knill-Laflamme subspace condition and the stabilizer formalism, via an abstract, mathematical perspective. We then discuss some contemporary research results, arriving at Operator Quantum Error Correction (OQEC) developed by Poulin et al. in 2006.

    Acknowledgements

    This intersession would not exist without

  • Micah Fogel, whose advice steadied my hand while developing the course.
  • Anderson Trimm, whose category theory intersession while I was at IMSA inspired me to pursue higher mathematics.
  • Eric Chitambar and his wonderful QIP lectures.
  • Roy Araiza and Jihong Cai, from whom I learned all I know of error correction.