Efficient Quantum Algorithms and Circuit Complexity in Quantum Computing

1. Algorithms Artur Ekert

2. Our golden sequence H H

3. A B B B A B A B Circuit complexity n qubit circuit operation described by 2n x 2n unitary matrix size and depth of circuits grow with n n QUBITS Size 12 Depth 4 Fix your building units, a finite set of adequate gates A, B, C… # of gates (n) = size of the circuit (n) # of parallel units (n) = depth of the circuit (n)

4. Asymptotic notation for comparisons

5. Asymptotic notation

6. Asymptotic notation - example and hence it is is both

7. Quantifying growth cubic or polynomial quadratic or polynomial exponential linear root-n logarithmic

8. Efficient quantum algorithms B A B B A B A B A B A A B

9. Quantum Hadamard Transform H H H H H H H H

10. Insidious phases… Discrete set of phase gates Control phase gates =

11. Quantum Fourier Transform H F1 H F2 H H F3 H H

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

13. Aside – Hadamard is Fourier Quantum Fourier transform on group Quantum Fourier transform on group

14. Quantum function evaluation Boolean function f

15. Quantum function evaluation can be viewed as m Boolean functions … … … … … … … … fm-1 fm-2 f0

16. Query Scenario BLACK BOX, ORACLE very precious, you are charged fixed amount of money each time you use it INPUT: is a function f given as an ORACLE GOAL: is to determine some properties of f making as few queries to f as possible f

17. Quantum interferometry revisited H H H H U

18. Phases in a new way H H U

19. Deutsch’s Problem ? Given is f constant or balanced David Deutsch four possible oracles f CONSTANT BALANCED

20. Deutsch’s Problem Classical 2 queries + 1 auxiliary operation f f Quantum CONSTANT 1 query + 2 auxiliary operations H H BALANCED f

21. Deutsch’s Problem – The Guts H H f H

22. Deutsch’s Problem – The Guts H H f H But…

23. Deutsch’s Problem – The Guts H H f H So, it is now clear what happens if f(0) and f(1) are the same or different….

24. Deutsch’s Problem Generalised CLASSICAL COMPLEXITY: INPUT: queries either constant or balanced PROMISE: determine whether constant or balanced OUTPUT: H H 00000 CONSTANT H H H H any other output BALANCED H H H H f

25. Deutsch’s Problem Generalised H H f

26. Deutsch’s Problem Generalised H H f

27. Deutsch’s Problem Generalised H H f

28. Deutsch’s Problem Generalised What is the amplitude for finding the register in the |0> state? If f(x) constant, this has amplitude 1 i.e. it is guaranteed If f(x) balanced, this has amplitude 0 i.e. it will never happen

29. Deutsch’s Problem Generalised CLASSICAL COMPLEXITY: INPUT: queries either constant or balanced PROMISE: determine whether constant or balanced OUTPUT: H H 00000 CONSTANT H H H H any other output BALANCED H H H H f

30. classical probabilistic with error prob. : Fair comparison? classical deterministic: quantum : 1 FAIR COMPARISON Query in k places, if the queries had at least one 0 and one 1 then the function is balanced, otherwise assume it is constant. Probability that it is balanced when declared constant is

31. Bernstein-Vazirani Problem INPUT: is of the form PROMISE: binary string OUTPUT: H H H H H H H H H H f

32. Search Problem INPUT: Classical Complexity: PROMISE: binary string OUTPUT: Searching large and unsorted database containing 2n items • Example of a sorted database: • a phone book if you are given a name and looking for a telephone number • n lookups suffice • Example of an unsorted database: • a phone book if you are given a number and looking for a name • you need to check 2n items before you succeed with probability P=1 • you need to check 2n-1 items before you succeed with probability P=0.5

33. Grover’s algorithm It is easy to recognize a solution, although hard to find it.

34. Grover’s algorithm INPUT: Quantum Complexity: PROMISE: binary string OUTPUT: ITERATION 1 ITERATION 2 … … … … … … … … … … … H H H H H H H H H H H H H H H H H H H H f f0 f f0

35. Grover’s algorithm ITERATION H H H H H H H H f f0

36. Grover’s algorithm ITERATION H H H H H H H H f f0

37. Grover’s algorithm H H H H H H H H f f0

38. Grover’s algorithm H H H H H H H H is the state input at the start of the iterations

39. Grover’s algorithm Geometric Interpretation: Reflects a state about the axis orthogonal to So, we need to consider the composed, repeated actions of and

40. Grover’s algorithm sin = |<a|H|0>|

41. Grover’s algorithm Overall action: Rotation by angle 2

42. Grover’s algorithm H H H H H H H H

43. Grover’s algorithm ITERATION 1 ITERATION 2 … … … … … … … … … … … H H H H H H H H H H H H H H H H H H H H f f0 f f0

44. Grover’s algorithm After r iterations, the state is rotated by from the hyperplane for large n We iterate until

45. Query complexity classical probabilistic: quantum : Quadratic speedup compared to classical search algorithms Cryptanalysis: Attack on classical cryptographic schemes such as DES (the Data Encryption Standard) essentially requires a search among 256=7 £ 1016 possible keys. If these can be checked at a rate of, say, one million keys per second, a classical computer would need over a thousand years to discover the correct key while a quantum computer using Grover's algorithm would do it in less than four minutes.

46. Applications of Grover • Most common example is an unsorted database. Not a common scenario! • Finding most efficient route between two places on a map. • Brute-force code breaking (such as the DES example we’ve just seen). • Any classical algorithm with probabilistic outcome can be enhanced.

47. Simon’s Problem INPUT: Classical Complexity: PROMISE: period OUTPUT: Example: s=110 is the period (in the group) 000 001 010 011 100 101 110 111 111 010 100 110 100 110 111 010

48. Fields and vector spaces over them

49. Binary vectors binary vectors Inner product

50. Binary vectors vectors x vectors x a binary vector can be perpendicular to itself