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Quantum Information and Computation (2023-24)

PYL749Odd 2023-24IIT Delhi

Foundations of quantum information and quantum computing.

Overview

An introduction to the foundations of quantum information and quantum computing: qubits and their mathematics, the postulates of quantum mechanics, entanglement, quantum gates and circuits, and the major quantum algorithms.

Assessment

  • Minor exam
  • Quiz
  • Major exam
  • Term paper (due Nov 17th)
  • Class participation

Lecture outline

  1. Course overview; classical vs quantum and exponential speed-up.
  2. What is a quantum system? What are qubits, physically and mathematically?
  3. What are bits, physically and mathematically? How do classical and quantum computing work in practice; how the current status of quantum computing differs from classical.
  4. Mathematics of qubits (vector spaces, choice of basis).
  5. What is a vector space? What is a state space?
  6. State space as a projective vector space, inner products, finite-dimensional Hilbert space, Dirac bra-ket notation.
  7. Inner products in Dirac and matrix notation (orthogonality, orthonormality, norm, choice of basis).
  8. Superposition principle, relative vs global phase, measurement postulate, physical realization of qubits.
  9. Qubits, polarizations, polarizers, Malus’ law; different bases and their physical interpretation.
  10. Measurement postulate (choice of basis, Born rule), operators in a complex vector space.
  11. Action of operators (X, Y, Z, H), matrix and Dirac representations; interpretations of QM (local vs nonlocal, realism vs nonrealism).
  12. Postulates of QM (observables, measurements), Stern-Gerlach experiments.
  13. Stern-Gerlach experiments (quantum explanation, superposition principle), transformations using an operator.
  14. Hermitian vs unitary operators; how quantum computers function and sources of quantum noise.
  15. Measurement of Hermitian operators (projection operators), unitary evolution and “collapse” as a non-unitary process for the system but unitary for system+environment.
  16. Does the order of measurements matter? Commutation relations, explicit examples.
  17. Average value of measurements and statistical errors.
  18. Calculation of averages in quantum mechanics; two-qubit system, basis, probabilities (tensor products).
  19. Interactive general class discussion.
  20. Two-qubit system, basis, entangled qubits; two-qubit operators (observables & unitary transformations).
  21. CNOT (CX) gate: matrix and Dirac representations, general procedure to write an operator explicitly.
  22. Classical computation & logic (universal gates, AND, NOT, NAND, truth tables); quantum computation & gates (reversible implementations, XOR vs CNOT).
  23. AND/NAND gates as 3-(qu)bit reversible gates, Toffoli gate.
  24. Classical vs quantum gates, reversible vs irreversible gates, why reversibility matters; gates on superposition inputs, reversible gates as one-to-one maps, Z_N notation for bit-strings, Toffoli (CCNOT/CCX) gate.
  25. Toffoli gate revisited, NOT gate in terms of the Toffoli gate.
  26. OR gate: reversible implementation using AND/NOT and using Toffoli gates; NOR & XOR gates with 3 (qu)bits.
  27. Practice: matrix/Dirac representations of multi-qubit gates in quantum circuits.
  28. No-cloning theorem.
  29. No-cloning theorem with an expanded definition of the cloning process.
  30. Quantum teleportation.
  31. Quantum teleportation (continued); measurements in multi-qubit entangled systems.
  32. Superdense coding; quantum parallelism.
  33. Quiz and solutions.
  34. Bit- and phase-oracles; quantum algorithms.
  35. Deutsch’s algorithm, Deutsch-Jozsa algorithm.
  36. Deutsch-Jozsa algorithm (continued); balanced vs constant functions and their oracles.
  37. Bernstein-Vazirani algorithm (and its relation to Deutsch-Jozsa).
  38. Simon’s algorithm.
  39. Simon’s algorithm (continued).
  40. Quantum Fourier transform.
  41. Quantum Fourier transform; Shor’s algorithm.
  42. Shor’s algorithm.