Quantum Error Correction and Fault Tolerance

Course number: QIC 890, cross-listed as CS 867
Instructors: Daniel Gottesman (dgottesman@perimeterinstitute.ca) and Beni Yoshida (byoshida@perimeterinstitute.ca)
Location: Perimeter Institute, room 405 (Bob)
Time: Thursdays 2:00 - 4:50
Term: Winter 2018
Office Hours: by appointment, at Perimeter Institute
Course web page: http://perimeterinstitute.ca/personal/dgottesman/QECC2018

Grading: 70% of the grade will come from problem sets (suitably processed) and 30% from the final project. The final project will consist of a paper and presentation of material from research papers on a topic related to the subject of the class.

• Lecture 1:
• Quantum channels, quantum error-correcting codes, the 9-qubit code, and the QECC conditions. (PIRSA:18010047)
• Problem Set 1 (pdf)
• Lecture 2:
• Stabilizer codes and CSS codes (PIRSA:18010048)
• Problem Set 2 (pdf)
• Lecture 3:
• Qudit stabilizer codes, bounds on QECCs, Clifford group (PIRSA:18010049)
• Problem Set 3 (pdf)
• Lecture 4:
• Clifford group continued, basic model of fault tolerance, transversal gates (PIRSA:18010050)
• Problem Set 4 (pdf)
• Lecture 5:
• Definition of fault tolerance, Shor error correction and measurement, Steane error correction and measurement (PIRSA:18020009)
• Problem Set 5 (pdf)
• Lecture 6:
• Magic state injection and distillation, level reduction, and the threshold theorem (PIRSA:18020010)
• Problem Set 6 (pdf)
• Lecture 7:
• Toric code (PIRSA:18020011)
• Problem Set 7 (pdf) -- Due Mar. 1
• Lecture 8:
• Topological phases, color codes (PIRSA:18030009)
• Problem Set 8 (pdf) -- Due Mar. 8
• Lecture 9:
• Bravyi-Konig bound, locally definable states (PIRSA:18030010)
• Problem Set 9 (pdf) -- Due Mar. 15
• Lecture 10:
• Scrambling and Hayden-Preskill (PIRSA:18030011)

Tentative Syllabus

• January: Quantum error correction
• 9-qubit code
• Quantum error correction conditions
• Stabilizer codes
• Clifford group
• Qudit codes
• Bounds on quantum error-correcting codes
• February: Fault tolerance
• Transversal gates
• Shor, Steane, and Knill quantum error correction
• Fault-tolerant state preparation
• Magic states
• Threshold theorem
• March: Other topics
• Toric and surface codes
• Color codes

Course Material

The textbook (a work in progress), Surviving as a Quantum Computer in a Classical World, will be available in class. Please send any comments you may have on it to me (DG). Also, please do not distribute copies of the book outside the class.

Final Project

I recommend that students tell one of us the subject of their final project by Mar. 1. Then one of us will discuss with you which research papers (usually 1-3, depending on size) to cover.

Due: March 29, written term paper plus a presentation to the class
Recommended length: about 20 pages
Length of presentation: roughly 20 minutes (subject to change, depending on the number of projects)

Possible paper topics

This is not an exhaustive list. If you want a subject not on this list, let me know and we can discuss if it is acceptable.

• Coding theory:
• Quantum Turbo codes
• Quantum polar codes
• Quantum LDPC codes
• Continuous variable codes
• Holographic codes
• Codeword-stabilized codes
• Subsystem codes
• Self-correcting quantum memories
• Decoding algorithms for surface codes
• Complexity of decoding quantum codes
• Fault tolerance:
• Knill FT scheme
• Comparison of thresholds from different codes
• Upper bounds on the threshold
• Magic state distillation protocols
• Variations on the threshold theorem (e.g., different assumptions)
• Fault tolerance with LDPC codes
• Alternatives to magic states
• Non-abelian anyons
• Other topics:
• Aspects of topological order and topological codes
• Experimental quantum error correction and fault tolerance
• Shor-Preskill QKD proof
• Extensions of efficient simulation of Clifford group circuits