#### LTAT.00.014 3-6 ECTS

*Basqect*^{*} — Basics of Quantum Error Correction

Classical communication and computation devices are prone to errors, which makes (classical) error correcting codes necessary. For quantum devices, that problem is much larger: Not only are quantum errors more pervasive, but correcting them is more subtle, as inspecting the quantum state in order to fix it is not easily possible without destroying it.

While some low-hanging fruit in quantum communication and quantum computation can be reached on noisy quantum devices (e.g., simple Quantum Key Distribution in quantum communication, or simple simulations of condensed matter with quantum computing devices), for the second quantum revolution (arising from the ability to coherently manipulate quantum systems) to happen, quantum error must be corrected.

Luckily, based on Nobel-prize worthy ideas of Peter Shor from the late 1990s, correcting quantum error and "undoing" decoherence (due to interaction of the quantum device with the rest of the universe) is possible, and can be understood and applied by the math-capable student.

^{*}) Pronounce like "basket" 🙄

#### Required Background

- Fou-Math MTAT.05.008
- Qualms LTAT.04.008
- Quantum Crypto MTAT.07.024

Students with undergraduate degree in Math have additional options: Contact the instructors!

#### Content

The course will focus on the predominant proposal for quantum error correction, which is based on so-called *Stabilizer Codes*. As the target audience is computer science students, it is necessary, though, to go through some math first. At the end of the course, the successful student will have an understanding of stabilizer codes, and how to use them to correct Pauli quantum errors.

- Math: Basics of group theory (normal subgroups, isomorphism theorems, group actions, centralizer, normalizer, etc)
- Quantum: Groups of unitary operators, Pauli groups and Clifford groups
- Math: Linear algebra over the field with 2 elements
- Quantum: GF(2)-arithmetic for Pauli (sub-)groups
- Quantum: Review of density operators and quantum channels
- Quantum: Stabilizer codes and their stabilizer groups, dimension of the code space, logical qubits, logical operations, error syndromes and error correction
- Examples: 5-qubit code, Steane 7-qubit code, 9-qubit Shor-code....

There are rumors according to which it might be possible to teach the subject matter more in a "physics style". The present course (designed and taught by mathematicians) is strictly mathematical, though, involving lots of yummy rigorous proofs.

#### Organization

This is a "Book Course", i.e., students mostly learn independently, based on lecture notes handed out to them...

... But this is the first time that this "Book Course" is being taught, and we will have to figure out what the best organization is. Current plan:

- The course takes place in weeks 1-8 of the semester
- Two class meetings per week: One with the instructor (to learn from / discuss with them), and one without the instructor (for students to learn from / discuss with each other).
- Homework (reviewing, reading, simple exercises, understanding) is not handed in / marked.
- The lecture notes will be created in parallel with actual black-board lectures.
- Pass-fail evaluation by exam (written? oral?)

Instructors:

- Javier Gil Vidal (classes)
- Assoc Prof Dirk Oliver Theis (design & content)

#### Outlook

At the end of this 3-ECTS course, you can correct quantum errors — which is amazing! In terms of quantum communication, it already gets you somewhere.

The 6 ECTS reading course LTAT.00.0015 ** "Fatol Surf"** (FAult TOLerance with SURFace codes) subsumes the content of LTAT.00.0014, and then, in the second half of the semester, moves to fault tolerant quantum computing based on surface codes (a special type of stabilizer codes) — the real deal.

*Fatol Surf*is considerably more demanding though, than

*Basqect*: It has no lectures, and the reading consists of a couple of research papers. The number of participants in

*Fatol Surf*is limited; students specializing in quantum computing are preferred. The ultimate goal of

*Fatol Surf*is: (1) to understand what really happens on a quantum computing device (e.g., quantum repeater, quantum computer) with error correction when a quantum algorithm is executed; and (2) to become able to experiment with quantum codes (surface or not) on quantum communication or computing devices, and so contribute to the quantum revolution.

This course's page on Quantum Computing at the University of Tartu.

##### ECTS and the relationship between Basqect and Fatol Surf

- Basqect (LTAT.00.0014) keeps you busy busy for a half semester (= 3 ECTS).
- Fatol Surf (LTAT.00.0015) keeps you busy for the full semester (= 6 ECTS), the first half of which is the
*content*of Basqect.