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Topics in Quantum Error-Correction and Holography

Ma 191bWinter 2021-22Caltech

The theory of quantum error-correcting codes and its relevance to the AdS/CFT correspondence (holography).

Overview

In this course, we will discuss the theory of quantum error-correcting codes and their relevance to one of the most mysterious conjectured dualities in theoretical physics — the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence (also called holography).

Broadly, this course will describe: quantum error-correcting codes based on finite fields, the AdS/CFT correspondence, and the connection between quantum error-correction and toy models of the AdS/CFT correspondence.

More specifically, topics to be covered include: finite dimensional quantum mechanics, density matrices, spectral theorem, Heisenberg group, Stone-von Neumann theorem, von Neumann entropy, entropy inequalities, bulk-boundary correspondence, subregion duality, bulk reconstruction, Ryu-Takayanagi formula, error-correction, tensor networks. If time permits, we’ll discuss stabilizer codes, CRSS algorithm, p-adic AdS/CFT, von Neumann algebras on finite dimensional Hilbert spaces.

Course prerequisites

  • linear algebra
  • abstract algebra
  • introductory quantum mechanics (Ph 12b)
  • elementary knowledge of finite fields (towards the end of the course)
  • No knowledge of quantum field theory or conformal field theory or holography is required

Meeting time and location

  • Tuesday and Thursday, 10:30 – 11:55 am
  • Offered online via Zoom

Instructor

  • Sarthak Parikh (he/him) · Office hours online via Zoom by appointment

Grading

  • Final presentation (30 minutes): 50%
  • Final write-up: 30%
  • Class participation: 20%

Lectures

  1. Introduction, general discussion about course structure, class expectations, assessments, review of basic quantum mechanics (Dirac notation, qubits, Pauli matrices, Stern-Gerlach).
  2. Review of basic quantum mechanics (GHZM experiment), classical and quantum error-correcting codes basic example, no-cloning theorem.
  3. Sphere packing, Shannon entropy, von Neumann entropy.
  4. Entropy basics -ctd-, Holography (partition function, dictionary, general remarks on conformal field theories and gravity).
  5. Invitation to holography -ctd-, CFTs.
  6. Density matrices: definitions, motivation, trace.
  7. Density matrices -ctd-: ensembles, pure vs mixed states, Bloch sphere, reduced density matrices and partial trace, Hilbert-Schmidt inner product.
  8. Schmidt decomposition, purification; Shannon entropy.
  9. Relative entropy, conditional entropy, mutual information, classical entropy inequalities.
  10. von Neumann entropy and its properties, inequalities.
  11. von Neumann entropy inequalities (SSA) -ctd-, entropy cone and inequalities.
  12. Holographic entropy cone, entropy in field theories.
  13. Holographic entanglement entropy, Ryu-Takayanagi formula.
  14. Holography, radial locality, bulk reconstruction.
  15. Quantum error-correction and holography: three-qutrit code, structure theorem.
  16. More general holographic codes, proof of structure theorem, quantum singleton bound, RT formula, modular Hamiltonian.
  17. Subsystem quantum encoding, complimentary recovery, subregion duality, Quantum Ryu-Takayanagi formula, JLMS.
  18. Finite dimension von Neumann algebras and holography, tensor networks.
  19. Tensor networks -ctd-.
  20. Student presentations.
  21. Student presentations.

Suggested books and reference material

  • Quantum information & error-correction
    • Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information
    • John Preskill’s notes (link), especially chapter 7
    • N. David Mermin. Quantum Computer Science
  • CFT
    • Slava Rychkov. Lightweight discussion on axioms of CFT. See also references 6, 7 & 8 in the article at arXiv:2007.14315
    • Graeme Segal. The Definition of a 2d CFT. Paper link
  • Entropy inequalities
    • Zhang and Yeung ‘98. On Characterization of Entropy Function via Information Inequalities. Paper link
    • Linden and Winter ‘04. A new inequality for the von Neumann entropy. arXiv:quant-ph/0406162
    • Bao, Nezami, Ooguri, Stoica, Sully, and Walter ‘15. The Holographic Entropy Cone. arXiv:1505.07839
  • Entanglement and Ryu-Takayanagi in field theories
    • Matthew Headrick ‘19. Lectures on entanglement entropy in field theory and holography. arXiv:1907.08126
    • Daniel Harlow ‘14. Jerusalem Lectures on Black Holes and Quantum Information. arXiv:1409.1231
  • Quantum error-correction and holography in the finite dimensional case
    • Daniel Harlow ‘18. TASI Lectures on the Emergence of Bulk Physics in AdS/CFT. arXiv:1802.01040
    • Daniel Harlow ‘16. The Ryu-Takayanagi Formula from Quantum Error Correction. arXiv:1607.03901
    • Pastawski, Yoshida, Harlow, and Preskill ‘15. Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence. arXiv:1503.06237