**Instructor**: Daniel Gottesman
(dgottesman@perimeterinstitute.ca, 519-569-7600x8581)
**Location**: Perimeter Institute, Bob room (405)
**Time**: Tuesdays 3:30 - 6:30
**Term**: Winter 2007
**Office Hours**: Perimeter Institute 458, Monday 3-4 PM,
Tuesday 2:30 - 3:30 PM
**Course web page**:
http://perimeterinstitute.ca/personal/dgottesman/QECC2007

The class is now complete. Recordings of all of the lectures are available on the PI website.

The class was cross-listed for credit in the UW Physics and C&O departments as CO 781 and PHYS 773.

**Lecture 1 (January 9, 2007)**:- Lecture 1A (PIRSA:07010014): Administrative introduction, quantum operations, examples of quantum channels, quantum code correcting bit flip errors, quantum code correcting phase errors
- Lecture 1B (PIRSA:07010015): 9-qubit Shor code, definition of a quantum error-correcting code, correcting linear combinations of errors, quantum error correction conditions, definition of distance
- Problem set 1: PDF (65K), PS (112K)
- Solution set 1: PDF (78K), PS (139K)

**Lecture 2 (January 16, 2007)**:- Lecture 2A (PIRSA:07010020): stabilizer codes (definition of stabilizer, basic properties of stabilizer, binary vector representation of stabilizer)
- Lecture 2B (PIRSA:07010021): 5-qubit code, logical Pauli group for stabilizer codes, classical linear codes (generator and parity check matrices, Hamming codes), CSS codes (definition, 7-qubit code)
- Problem set 2: PDF (53K), PS (92K)
- Solution set 2: PDF (71K), PS (124K)
- Introduction to group theory: PDF (96K)

**Lecture 3 (January 23, 2007)**:- Lecture 3A (PIRSA:07010022): Finite field GF(4), stabilizer codes as GF(4) codes, perfect quantum codes, definition of Clifford group, sample elements of Clifford group
- Lecture 3B (PIRSA:07010023): Clifford group as symplectic group, generators of the Clifford group & encoding circuits for stabilizer codes, efficient simulation of Clifford group circuits, efficient simulation of Pauli measurements
- Problem set 3: PDF (56K), PS (96K)
- Solution set 3: PDF (79K), PS (138K)

**Lecture 4 (January 30, 2007)**: Guest lectures- Lecture 4A
(PIRSA:07010024):
**Raymond Laflamme**, on experimental quantum error correction - Lecture 4B
(PIRSA:07010025):
**Robert Raussendorf**, on graph states

- Lecture 4A
(PIRSA:07010024):
**Lecture 5 (February 6, 2007)**:- Lecture 5A (PIRSA:07020017): Generators of symplectic group, quantum Gilbert-Varshamov bound, quantum Hamming bound, quantum Singleton bound
- Lecture 5B (PIRSA:07020018): Weight enumerators, quantum MacWilliams identity, quantum shadow enumerator, higher-dimensional Pauli group, stabilizer codes for qudits
- Problem set 4: PDF (38K), PS (69K)
- Solution set 4: PDF (61K), PS (107K)

**Lecture 6 (February 13, 2007)**:- Lecture 6A (PIRSA:07020021): Examples of qudit stabilizer codes, polynomial codes, Clifford group for qudits, introduction to fault-tolerance, definition of transversal gates, definition of fault-tolerant gates
- Lecture 6B (PIRSA:07020022): Transversal Pauli group, transversal Clifford group for 7-qubit code, transversal gates for 5-qubit code, overview of fault-tolerant protocols.
- Problem set 5: PDF (45K), PS (79K)
- Solution set 5: PDF (74K), PS (129K)

**Lecture 7 (February 20, 2007)**:- Lecture 7A (PIRSA:07020023): Definition of fault tolerance, Shor error correction, fault-tolerant measurement for stabilizer and CSS codes.
- Lecture 7B (PIRSA:07020024): Fault-tolerant stabilizer state preparation, Steane error correction, universal fault-tolerant set of gates through gate teleportation, magic state distillation.
- Problem set 6: PDF (45K), PS (79K)
- Solution set 6: PDF (65K), PS (111K)

**Lecture 8 (February 27, 2007)**:- Lecture 8A (PIRSA:07020028): Assumptions for fault tolerance, extended rectangles, good, bad, and correct rectangles.
- Lecture 8B (PIRSA:07020029): Equivalence of fault-tolerant circuit to less noisy unencoded circuits, threshold theorem, calculation of the threshold.
- Problem set 7: PDF (45K), PS (80K)
- Solution set 7: PDF (78K), PDF (135K)

**Lecture 9 (March 6, 2007)**:- Lecture 9A (PIRSA:07030016): Circuit assumptions for fault tolerance re-examined (other universal gate sets, local gates, fresh ancillas, no measurements, parallelism)
- Lecture 9B (PIRSA:07030017): Error assumptions for fault tolerance re-examined (Other error models, correlated errors, leakage errors, coherent and non-Markovian errors)
- Problem set 8: PDF (56K), PS (98K)
- Solution set 8: PDF (72K), PS (128K)

**Lecture 10 (March 13, 2007)**:- Lecture 10A (PIRSA:07030022): One- and two-way entanglement distillation protocols, twirling, stabilizer EDPs, definition of quantum channel capacity
- Lecture 10B (PIRSA:07030023): One- and two-way quantum capacities for the erasure channel, upper and lower bounds on the quantum capacities for the depolarizing channel, coherent information
- Problem set 9: PDF (56K), PS (97K)
- Solution set 9: PDF (84K), PS (143K)

**Lecture 11 (March 20, 2007)**:- Lecture 11A (PIRSA:07030035): Toric code (definition, fault-tolerance, particle model of errors), definition of qudit toric code
- Lecture 11B (PIRSA:07030036): Behavior of particles in qudit toric code, braid group, basic idea of fault tolerance with non-Abelian anyons

**Lecture 12 (March 27, 2007)**:- Lecture 12A (PIRSA:07030042): Non-Abelian anyons (charges, fusion rules, F and R matrices, pentagon and hexagon equations), Fibonacci anyons
- Lecture 12B (PIRSA:07030043): Universality of Fibonacci anyons, operator quantum error correction, Bacon-Shor codes

- Class evaluation form: Word (28K)

Students are encouraged to take the course on a credit/no credit basis.

- 2/3 of grade: Problem sets
- 1/3 of grade: Final paper

Late problem sets will be accepted for half credit, assuming they do not just copy the solution set.

The final paper required the student to read 2-3 research papers on a topic related to the subject of the course and write a paper presenting the results of the research papers. The students then had to give short presentations on the subjects of their papers on the last day of class.

The student presentations for the final project took place on April 3, 2007. They were not recorded.

A similar class was given in Winter 2004.