1. Phase in Quantum Computing

2. Main concepts of computing illustrated with simple examples

3. Quantum Theory Made Easy 0 1 Classical p0p0 p1p1 probabilities Quantum a0a0 a1a1 0 1 amplitudes p 0 +p 1 =1|a 0 | 2 +|a 1 | 2 =1 bit qubit p i is a real numbera i is a complex number Prob(i)=p i Prob(i)=|a i | 2

4. Quantum Theory Made Easy Classical Evolution 00 01 10 11 00 01 10 11 Quantum Evolution stochastic matrix 00 01 10 11 00 01 10 11 transition probabilitiestransition amplitudes unitary matrix

5. Interference 0 1 0 1 measure 0 50% 1 measure 0 50% 1 100% 0% 100% 0% qubit input

6. Interfering Pathways 100% H 50% 50% C 50% H 10% 90% 20% 80% 15% H 85% C 1.0 H 0.707 0.707 C 0.707 H 0.707 -0.707 0.707 0.0 H 1.0 C Always addition!Subtraction! Classical Quantum

7. Superposition Qubits a0a0 a1a1 0 1 amplitudes a i is a complex number 1 √2√2 ( |  + |  ) Schrödinger’s Cat

8. Classical versus quantum computers

9. Some differences between classical and quantum computers superposition Hidden properties of oracles

10. Randomised Classical Computation versus Quantum Computation Deterministic Turing machine Probabilistic Turing machine

11. Probabilities of reaching states

12. Formulas for reaching states

13. Relative phase, destructive and constructive inferences Destructive interference Constructive interference

14. Most quantum algorithms can be viewed as big interferometry experiments Equivalent circuits

15. The “eigenvalue kick-back” concept

16. There are also some other ways to introduce a relative phase

17. The “eigenvalue kick-back” concept Now we know that the eigenvalue is the same as relative phase

18. The “eigenvalue kick- back” concept illustrated for DEUTSCH

19. The “shift operation” as a generalization to Deutsch’s Tricks

20. Change of controlled gate in Deutsch with Controlled-Ushift gate

21. Now we deal with new types of eigenvalues and eigenvectors

22. The general concept of the answer encoded in phase

23. Shift operator allows to solve Deutsch’s problem with certainty

24. Controlling amplitude versus controlling phase

26. Exercise for students

28. Dave Bacon Lawrence Ioannou Sources used