CSS.427.1: A minicourse on Quantum Coding Theory - Monsoon Semester (2025-26)

Time: Wed & Friday 11:30-13:00
Location: A201
Instructors:
Homepage: https://www.tcs.tifr.res.in/~prahladh/teaching/2025-26/qecc/

Course Announcement

Lectures References

Toric Code, created by chatGPT

Lectures

  1. Introduction to Quantum Error Correction (29 Aug)
    Administrivia, why error correction (classical and quantum)?, brief history of quantum error correction and fault tolerance, Shor's 9-quibit code
    Ref: Got (Chap 1-2), Wri (Lecture 1-2)
  2. How is noise modeled? (3 Sep)
    Shor's 9-qubit code (correcting 1-bit X and Z-errors, general 1-quibit errors); quantum channel (CPTP operator, Krauss representation, examples); qecc -- definition
    Ref: Got (Chap 1, 2.3-2.4), Wri (Lecture 2-3)
  3. Quantum error correcting codes (5 Sep)
    QECC - definition, linearity of errors, necessary and sufficient conditions for qecc: Knill-Laflamme Conditions
    Ref: Got (Chap 2.4-2.5), Wri (Lecture 3-4)

    10-19 Sep: No lectures
  4. Knill-Laflamme Conditions (24 Sep)
    Review QECC definition, Knill-Laflamme Conditions: necessary and sufficient conditions for QECC, t-weight errors
    Ref: Got (Chap 2.4-2.5), Wri (Lecture 4-5)
  5. Knill-Laflamme Conditions (Contd) (26 Sep)
    Consequences of Knill-Laflamme Conditions: distance, degeneracy, error detection, classical linear codes, introduction to stabilizer codes
    Ref: Got (Chap 2.5, 3.1-3.3), Wri (Lecture 5-6)
  6. Stabilizer Codes (1 Oct)
    Stabilizer of a code, Centralizer, distance of a stabilizer code
    Ref: Got (Chap 3.1-3.3), Wri (Lecture 6-7)
  7. Stabilizer Codes (contd) (3 Oct)
    Stabiizer codes (recap of last lecture), distance and degeneracy of stabilizer codes, binary symplectic representation
    Ref: Got (Chap 3), Wri (Lecture 6-7)
  8. Binary Symplectic Representation and GF(4)-codes (8 Oct)
    Binary Symplectic representation of stabilizer codes, equivalence with weakly self-dual codes over (F_2)^2, GF($)-codes
    Ref: Got (Chap 3.5 and 5.2)
  9. GF(4) and CSS Codes (10 Oct)
    ` Binary Symplectic viewpoint of stabilizer codes, GF(4)-codes, CSS codes, examples
    Ref: Got (Chap 3.5 and 5), Wri (Lecture 8)
  10. Qudit Codes and Polynomial Codes (15 Oct)
    CSS Codes: explicit description of codespace; Qudit codes: Pauli group over larger alphabet, polynomial codes
    Ref: Got (Chap 5.1.3, 8.1-8.3), Wri (Lecture 9-10)
  11. Bounds on QECCs (17 Oct)
    Bounds on QECCs: quantum GV bound, quantum Hamming bound for non-degenerate codes; Polynomial codes; quantum Singleton bound
    Ref: Got (Chap 7.1-7.3, 8.3.2), Wri (Lecture 10)
  12. The toric code (22 Oct)
    Kitaev's toric code, topological codes, chain complexes
    Ref: Wri (Lecture 12-14)
  13. Clifford Group (24 Oct)
    Clifford group: definition, properties; classical simulation of Clifford circuits, stabilizer codes and Clifford gates, complete gate set for Clifford circuits
    Ref: Got (Chap 6), Wri (Lecture 23-24)

    29-31-19 Sep: No lectures

Tentative Schedule (for remaining lectures)

  1. 5 Nov: fault tolerance
  2. 7 Nov: other topics

Requirements

Students taking the course for credit will be expected to:

References

[Got]
Daniel Gottesman, "Surviving as a Quantum Computer in a Classical World", (draft of book), 2024.
[Wri]
John Wright, "CS 294: Quantum Coding Theory", Berkeley, Spring 2024.

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Prahladh Harsha