The Structure of Quantum Computation from Physical Principles

Transcription

1 The Structure of Quantum Computation from Physical Principles John H. Selby & Ciarán M. Lee arxiv: ,

2 The structure of quantum computation...

3 The structure of quantum computation... Quantum departures from classicality Non-locality Contextuality Computational speed-up What provides the quantum speed-up? Why is there not more of a speed-up? How can we design optimal algorithms?

4 The structure of quantum computation... Quantum departures from classicality Non-locality Contextuality Computational speed-up

5 The structure of quantum computation... Quantum departures from classicality Non-locality Contextuality Computational speed-up What provides the quantum speed-up?

6 The structure of quantum computation... Quantum departures from classicality Non-locality Contextuality Computational speed-up What provides the quantum speed-up? Why is there not more of a speed-up?

7 The structure of quantum computation... Quantum departures from classicality Non-locality Contextuality Computational speed-up What provides the quantum speed-up? Why is there not more of a speed-up? How can we design optimal algorithms?

8 ...from physical principles. Internal perspective:

9 ...from physical principles. Internal perspective: Standard quantum formalism Features of interest

10 ...from physical principles. Reconstruction perspective: Standard quantum formalism Physical principles Features of interest

11 ...from physical principles. External perspective: Standard quantum formalism Physical principles Features of interest

12 Analogy: Study of non-local correlations Why are quantum correlations non-local?

13 Analogy: Study of non-local correlations Why are quantum correlations non-local? Why are quantum correlations not more non-local?

14 Analogy: Study of non-local correlations Why are quantum correlations non-local? Why are quantum correlations not more non-local? Information causality and Tsirelson s bound

15 Analogy: Study of non-local correlations Why are quantum correlations non-local? Why are quantum correlations not more non-local? Information causality and Tsirelson s bound Device independent key distribution

16 Outline Physical principles Components of computation Grover s algorithm

17 Physical principles QPL /06/2016 John Selby

18 Physical principles Causality

19 Physical principles Causality Purification and purity preservation

20 Physical principles Causality Purification and purity preservation Strong symmetry and pure and perfectly distinguishable states

21 Physical principle 1: Causality f =

22 Physical principle 2: Purification and purity preservation ρ = ψ ρ

23 Physical principle 2: Purification and purity preservation = ψ ρ ρ = φ ρ

24 Physical principle 2: Purification and purity preservation = ψ ρ ρ = φ ρ = φ ρ = ψ ρ R

25 Physical principle 3: Strong symmetry and the existence of pure and perfectly distinguishable states There exists { i } n i=0 and { j } n j=0

26 Physical principle 3: Strong symmetry and the existence of pure and perfectly distinguishable states There exists { such that i } n i=0 and { j } n j=0 i j = δ ij

27 Physical principle 3: Strong symmetry and the existence of pure and perfectly distinguishable states There exists { such that i } n i=0 and { j } n j=0 i j = δ ij Where { i } n i=0 and { i } n i=0

28 Physical principle 3: Strong symmetry and the existence of pure and perfectly distinguishable states There exists { such that i } n i=0 and { j } n j=0 i j = δ ij Where { are related by i } n i=0 and { i } n i=0 i T = i

29 Do they restrict us to quantum theory?

30 Do they restrict us to quantum theory? Real and fermionic quantum theory

31 Do they restrict us to quantum theory? Real and fermionic quantum theory Higher order interference

32 Do they restrict us to quantum theory? Real and fermionic quantum theory Higher order interference +

33 Do they restrict us to quantum theory? Real and fermionic quantum theory Higher order interference = + +

34 Component 1: Reversible controlled transformations

35 Component 1: Reversible controlled transformations i C {T i } = i T i

36 i i T i

37 i i T = T i

38 i i T = T i

39 i i T = T i

40 Component 2: Reversible phase transformations P i = i

41 Component 3: Phase kick-back algorithm s C {T i } = s P

42 Grover s algorithm QPL /06/2016 John Selby

43 Grover s algorithm Important quantum algorithm

44 Grover s algorithm Important quantum algorithm Provable advantage

45 Grover s algorithm Important quantum algorithm Provable advantage Provably optimal

46 Grover s algorithm Important quantum algorithm Provable advantage Provably optimal Oracle problem

47 The search problem Given an N element unstructured list with a unknown marked item x. Then given an oracle O x how many queries of O x are needed to find x with high probability?

48 Quantum oracles QPL /06/2016 John Selby

49 Quantum oracles U x i j = i j δ ix

50 Quantum oracles C U x i j = i j δ ix U x = {X δ ix }

51 Quantum oracles C U x i j = i j δ ix U x = {X δ ix } O x i = ( 1) δ xi i

52 Quantum oracles C U x i j = i j δ ix U x = {X δ ix } O x i = ( 1) δ xi i O x := 1 U x

53 General oracles QPL /06/2016 John Selby

54 General oracles Controlled transformation

55 General oracles C Controlled transformation O x = {T x i }

56 General oracles C Controlled transformation O x = {T x i } Phase transformation

57 General oracles C Controlled transformation O x = {T x i } Phase transformation O x := s C {T x i }

58 The lower bound Classical computers: O(N) queries

59 The lower bound Classical computers: O(N) queries Quantum computers: O( N) queries Computers satisfying our principles: Ω( N) queries

60 The quantum speed up QPL /06/2016 John Selby

61 The quantum speed up Quantum interference provides the speed up?

62 The quantum speed up Quantum interference provides the speed up? More interference gives more of a speed up, e.g. O(N 1 h )?

63 The quantum speed up Quantum interference provides the speed up? More interference gives more of a speed up, e.g. O(N 1 h )? Interference and phases

64 The quantum speed up Quantum interference provides the speed up? More interference gives more of a speed up, e.g. O(N 1 h )? Interference and phases φ

65 Our result Classical computers: O(N) queries Quantum computers: O( N) queries

66 Our result Classical computers: O(N) queries Quantum computers: O( N) queries Computers satisfying our principles: Ω( N/h) queries

67 Conclusion QPL /06/2016 John Selby

68 Conclusion Physical principles

69 Conclusion Physical principles elementary components of computation

70 Conclusion Physical principles elementary components of computation the quantum lower bound

71 Conclusion Physical principles elementary components of computation the quantum lower bound Do we need all of these principles?

72 Conclusion Physical principles elementary components of computation the quantum lower bound Do we need all of these principles? Can we reach this lower bound?

73 Conclusion Physical principles elementary components of computation the quantum lower bound Do we need all of these principles? Can we reach this lower bound? Practical applications?