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Quantum Information and Computation (2024-25)

PYL749Odd 2024-25IIT 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

  • Midterm exam
  • Quiz
  • Major exam

Lecture outline

  1. Course overview.
  2. Introduction.
  3. Bounded-error complexity classes: analysis of classical algorithms for the umpire problem, separation of complexity classes between classical and quantum algorithms — a teaser.
  4. Introduction to quantum computers, qubits and comparison with classical counterparts.
  5. More on quantum computers, qubits, and quantum gates.
  6. Mathematics of qubits: superposition, basis, observables, eigenvalues, postulates.
  7. Global vs relative phases, Bloch sphere for a single qubit, measurements.
  8. What is a wavefunction? Wave-particle duality; clearing misconceptions on the physical meaning of wavefunctions, superposition, and probabilistic vs deterministic nature.
  9. Stern-Gerlach experiment, measurements revisited, interpretations of QM, local hidden variable theory.
  10. Stern-Gerlach experiment (continued), measurement and projection operators.
  11. Multi-qubit states, tensor product spaces, entangled & separable states.
  12. Wavefunction vs state, entangled vs separable, measurement of two qubits, probabilities, physical interpretation of the multi-qubit wavefunction.
  13. Tensor product of vector spaces, direct sum decomposition, multi-qubit measurements.
  14. Multi-qubit measurements (continued), quantum key distribution (Ekert ‘91).
  15. Projection operators (single and multi-particle), projective measurements.
  16. Hermitian operators as observables, eigenspace decomposition, measurement of Hermitian operators.
  17. Tensor product of Hermitian operators, measurement of multi-qubit observables.
  18. Three-qubit states, entanglement, measurement, projectors.
  19. GHZM experiment (arXiv:2011.12671).
  20. GHZM experiment (continued), quantum teleportation.
  21. Quantum teleportation (continued), quantum circuits, quantum gates.
  22. Superdense coding, no-cloning theorem.
  23. Logic gates and their reversible implementations, universality of classical logic gates.
  24. Proof of reversibility, quantum oracles, proof of unitarity of the oracle.
  25. Quantum gates, Toffoli gate, quantum circuits.
  26. Quantum universality of gates, Solovay-Kitaev theorem, Gottesman-Knill theorem.
  27. Phase oracle, interference of amplitudes, parallelism.
  28. Deutsch’s algorithm, Deutsch-Jozsa algorithm.
  29. Bernstein-Vazirani algorithm.
  30. Quiz.
  31. Bernstein-Vazirani: parallelism or not? The algorithm of choice.
  32. Simon’s algorithm (period-finding).
  33. Fourier transform.
  34. Fast Fourier transform.
  35. Quantum Fourier transform circuit.
  36. Quantum phase estimation.
  37. Quantum order-finding algorithm.
  38. Shor’s algorithm, Grover’s search algorithm.