1. Quantum Computing: An Overview for non-specialists Mikio Nakahara Department of Physics & Research Centre for Quantum Computing Kinki University, Japan Financial supports from Kinki Univ., MEXT and JSPS

2. Overview @ Tehran 2009 Plan of lecture 1. Introduction 2. Qubits 3. Quantum Gates, Quantum Circuits and Quantum Computer 4. Simple Quantum Algorithms 5. DiVincenzo Criteria & Physical Realizations 6. Shor’s Factorization Algorithm

3. Overview @ Tehran 2009 I. Introduction

4. Overview @ Tehran 2009 More complicated Example

5. Overview @ Tehran 2009 Quantum Computing/Information Processing Quantum computation & information processing make use of quantum systems to store and process information. Exponentially fast computation, totally safe cryptosystem, teleporting a quantum state are possible by making use of states & operations which do not exist in the classical world.

6. Overview @ Tehran 2009 Plan of lectures 1. Introduction 2. Qubits 3. Quantum Gates, Quantum Circuits and Quantum Computer 4. Simple Quantum Algorithms 5. DiVincenzo Criteria & Physical Realizations 6. Shor’s Factorization Algorithm

7. Overview @ Tehran 2009 2. Qubits

8. Overview @ Tehran 2009 2.1 One Qubit

9. Overview @ Tehran 2009 Candidates of qubits ： Electron, Spin 1/2 Nucleus Photon Grand State and Excited State of Atom or Ion

10. Overview @ Tehran 2009 2.2 Two-Qubit System

11. Overview @ Tehran 2009 2.3 Multi-qubit systems and entangled states

12. Overview @ Tehran 2009 2.4 Algorithm = Unitary Matrix

13. Overview @ Tehran 2009 Physical Implementation of U

14. Overview @ Tehran 2009 Plan of lectures 1. Introduction 2. Qubits 3. Quantum Gates, Quantum Circuits and Quantum Computer 4. Simple Quantum Algorithms 5. DiVincenzo Criteria & Physical Realizations 6. Shor’s Factorization Algorithm

15. Overview @ Tehran 2009 3. Quantum Gates, Quantum Circuit and Quantum Computer

16. Overview @ Tehran 2009

17. 3.2 Quantum Gates

18. Overview @ Tehran 2009 Hadamard transform

19. Overview @ Tehran 2009

20. n-qubit Operations

21. Overview @ Tehran 2009 Quantum Mechanics

22. Overview @ Tehran 2009 3.3 Universal Quantum Gates

23. Overview @ Tehran 2009 3.4 Quantum Parallelism and Entanglement

24. Overview @ Tehran 2009 Power of Entanglement

25. Overview @ Tehran 2009 Plan of lectures 1. Introduction 2. Qubits 3. Quantum Gates, Quantum Circuits and Quantum Computer 4. Simple Quantum Algorithms 5. DiVincenzo Criteria & Physical Realizations 6. Shor’s Factorization Algorithm

26. Overview @ Tehran 2009 4. Simple Quantum Algorithms 4.1 Deutsch’s Algorithm

27. Overview @ Tehran 2009

29. Plan of lectures 1. Introduction 2. Qubits 3. Quantum Gates, Quantum Circuits and Quantum Computer 4. Simple Quantum Algorithms 5. DiVincenzo Criteria & Physical Realizations 6. Shor’s Factorization Algorithm

30. Overview @ Tehran 2009 Necessary Conditions for a PC to Work Properly Hardware (Memory, CPU etc), Able to reset all the memories to 0, The PC lasts till a computation stops (maybe a problem if it takes more than 10 years to finish the computation.) Able to carry out any logic operations Able to output the results (display, printer, …)

31. Overview @ Tehran 2009 Necessary Conditions for a Quantum Computer to Work Properly (DiVincenzo Criteria) Hardware (Memory, CPU etc) Able to reset all the memories to 0, The PC lasts till a computation stops. Able to carry out any logic operations Able to output the results (display, printer, ) A scalable physical system with well characterized qubits. A scalable physical system with well characterized qubits. The ability to initialize the state of the qubits to a simple fiducial state, such as |00…0>. The ability to initialize the state of the qubits to a simple fiducial state, such as |00…0>. Long decoherence times, much longer than the gate operation time. Long decoherence times, much longer than the gate operation time. A “universal” set of quantum gates. A “universal” set of quantum gates. A qubit-specific measurement capability. A qubit-specific measurement capability.

32. Overview @ Tehran 2009 DiVincenzo 2004@Kinki Univ.

33. Overview @ Tehran 2009 Physical Realization: NMR

34. Overview @ Tehran 2009 Physical Realization: Trapped Ions

35. Overview @ Tehran 2009 Physical Realization: Josephson Junction Qubits

36. Overview @ Tehran 2009 Tunable coupling (interaction on demand)

37. Overview @ Tehran 2009 Physical Realization: Neutral Atoms

38. Overview @ Tehran 2009 Physical Realization: Quantum Dots

39. Overview @ Tehran 2009 Plan of lectures 1. Introduction 2. Qubits 3. Quantum Gates, Quantum Circuits and Quantum Computer 4. Simple Quantum Algorithms 5. DiVincenzo Criteria & Physical Realizations 6. Shor’s Factorization Algorithm

40. Overview @ Tehran 2009 Difficulty of Prime Number Facotrization Factorization of N=890208368187479079568319892720916003 03613264603794247032637647625631554961 638351 is difficult. It is easy, in principle, to show the product of p=928101320540413151847590244727697333 8969 and q =9591715349237194999547 050068718930514279 is N. This fact is used in RSA (Rivest-Shamir- Adleman) cryptosystem.

41. Overview @ Tehran 2009 Factorization algorithm

42. Overview @ Tehran 2009 Realization using NMR (15=3×5) L. M. K. Vandersypen et al (Nature 2001)

43. Overview @ Tehran 2009 NMR molecule and pulse sequence (~300 pulses) perfluorobutadienyl iron complex with the two 13C-labelled inner carbons

44. Overview @ Tehran 2009

45. Foolproof realization is discouraging … ? Vartiainen, Niskanen, Nakahara, Salomaa (2004) Foolproof implementation of the factorization 21=3 X 7 using Shor’s algorithm requires at least 22 qubits and approx. 82,000 steps!

46. Overview @ Tehran 2009 Summary Quantum information and computation are interesting field to study. (Job opportunities at industry/academia/military). It is a new branch of science and technology covering physics, mathematics, information science, chemistry and more. Thank you very much for your attention!

47. Overview @ Tehran 2009