1. Quantum Algorithms Preliminaria Artur Ekert

2. Computation INPUT OUTPUT 0 1 0 1 0 1 1 1 0 0 1 0 Physics Inside (and outside) THIS IS WHAT OUR LECTURES WILL BE ABOUT

3. Classical deterministic computation Initial configuration (input) Final configuration (output) Configuration = complete specification of the state of the computer and data Physically allowed operations, computational steps Intermediate configurations

4. Classical deterministic computation 000 001 101 110 Computational steps – moves from one configuration to another – are performed by elementary operations on bits

5. Boolean Networks NOT OR AND OR 0 0 0 1 1 0

6. Basic operations = logic gates 0 1 0 1 0 1 1 0 AND 1 0 OR Wire, identity NOT 1 1 Logical AND 0 0 Logical OR Output 0 apart from the (1,1) input Output 1 apart from the (0,0) input Fan out X X X

7. Classical probabilistic computation Input Possible outputs

8. Quantum computation Constructive or destructive interference: enhance correct outputs suppress wrong outputs GOOD SIDE: extra computational power BAD SIDE: sensitive to decoherence

9. Quantum computation Initial configuration of the three qubits

10. Bits and Qubits BIT QUBIT

11. Quantum Boolean Networks H HH

12. Quantum operations H HH

13. Single qubit gates H Hadamard Continuous set of phase gates Discrete set of phase gates

14. Single qubit interference H H

15. Any single qubit interference H H INPUT OUTPUT in the matrix form

16. Any unitary operation on a qubit H H INPUT OUTPUT in the matrix form – the most general SU(2) operation on a single qubit

17. Possible implementations © ENS Paris

18. Two and more qubits Notation

19. Operations on two qubits Controlled-NOT Controlled-U U U

20. Quantum interferometry revisited H H H H U REMEMBER THIS TRICK !

21. Phases in a new way H H U

22. Entangled states H entangled separable

23. Bell & GHZ states H H

24. Useful decomposition of any U in SU(2) For any U in SU(2) A A -1 BB -1 Rotation by  around some axis a Rotation by  around some axis b Recall that  x represents rotation by  around axis x Rotation by twice the angle between axis a and b around the axis perpendicular to a and b

25. Building controlled-U operations A A -1 BB -1 U = A, A -1, B and B -1 are single qubit operations and can be constructed from the Hadamard and phase gates. Controlled-U can be constructed from single qubit operations and the controlled-NOT gates. Hence any controlled-U gate can be constructed from the Hadamard, the controlled-NOT and phase gates.

26. Toffoli Gate = H H

27. Controlled-controlled NOT Computes logical AND Quantum adder

28. Quantum Networks H H Quantum adder H H H H Quantum Hadamard transform

29. Quantum Hadamard Transform H H H H

30. H H H H H H H H

31. Is also known as the quantum Fourier transform on group group = the set with operation (addition mod 2) group = the set with operation (addition mod 2 bit by bit) example for n=15

32. Quantum Fourier Transform Quantum Fourier transform on group

33. Recall H Hadamard Discrete set of phase gates

34. Quantum Fourier Transform H H H H H H F1F1 F2F2 F3F3

35. H H H H Uniform family of networks n Hadamard gates and n(n-1)/2 phase shifts, the size of the network = n(n+1)/2

36. Quantum Fourier Transform H H H H

37. H H H F3F3 H H H Fy3Fy3

38. Quantum function evaluation f Boolean function

39. Quantum function evaluation can be viewed as m Boolean functions f m-1 f m-2 f0f0 …………………………………………

40. Quantum function evaluation Group X Group (Y, ) bit by bit addition – group modular addition – group