1. IWEPNM 2007, Kirchberg Coworkers Principles of Quantum Computing Michael Mehring Physikalisches Institut, Univ. Stuttgart, Germany A.Heidebrecht, S. Krämer, O. Mangold J. Mende, W. Scherer 15 N@C 60 and 31 P@C 60 Cooperation Wolfgang Harneit FU Berlin

2. IWEPNM 2007, Kirchberg Richard Feynman was one of the first physicists, who contemplated about quantum computing already in 1982 R. P. Feynman, F. L. Vernon, Jr., and R. W. Hellwart, J. Appl. Phys. 28, 49 (1957).

3. IWEPNM 2007, Kirchberg The Quantum Bit (Qubit) Stern Gerlach Experiment (1922)

4. IWEPNM 2007, Kirchberg Quantumstates of a Qubit (Spin ½) The Spin Density Matrix SuperpositionPopulation Phase X Phase Y

5. IWEPNM 2007, Kirchberg The Single Qubit Basis: or x y z Bloch sphere Single qubit operations X X Y Z Y Z NOT ? 8:35 See: Quantum Computation and Quantum Information M. A. Nielsen and I. Chuang, Cambridge University Press 2000

6. IWEPNM 2007, Kirchberg More Single Qubit Gates   ST H Hadamard H H H H

7. IWEPNM 2007, Kirchberg A silicon-based nuclear spin quantum computer B.E.Kane Nature 393, 133 (May 1998) NMR-Quantum-Computing a là Kane

8. IWEPNM 2007, Kirchberg Array of Quantum Dots with Spins Single spin in Q-dot Daniel Loss proposal F. H. L. Koppens et. al. Nature, 442, 766 (2006)

9. IWEPNM 2007, Kirchberg Other Qubits: Atoms and ions in traps, photons, Q-dots, superconductors Photons as flying qubits Q algorithm Superconducting qubit: Nakamura et al. Spin echo with sc-qubit Bennet, Zeilinger, Zoller, Weinfurter et al. Two level system Fictitious spin 1/2 Ion traps: J. I. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091 (1995) 8:40

10. IWEPNM 2007, Kirchberg Two Qubits Basis: Bell states (entangled states): product state

11. IWEPNM 2007, Kirchberg The Greek LAOCOON and Quantum Entanglement H Hadamard U CNOT The EPR Pair Einstein, Podolsky and Rosen, Phys. Rev. 47, 777 (1935)

12. IWEPNM 2007, Kirchberg Two Qubit Gates Controlled NOT (CNOT) x CNOT How to generate an entangled state x CNOT H 8:45

13. IWEPNM 2007, Kirchberg Quantencomputing in Hilbertspace 8 qubits span a 256 dimensional Hilbertspace Quantum evolution in Hilbertspace Quantumcomputing is: Preparation+Superposition+Entanglement+Projection

14. IWEPNM 2007, Kirchberg Two Qubit Algorithms Superdense coding: AliceBob Alice is allowed to send only a single qubit. Can she transmit one of four different bits of information? Suppose Alice and Bob share the entangled state: Alice performs one of the following single qubit operations on her qubit: Alice sends her qubit to Bob. Bob performs a Bell state measurement. I, Z, X, iY 8:50

15. IWEPNM 2007, Kirchberg

16. Scenario of Quantum-Teleportation „Gentlemen beam me aboard“ Captain Kirk source of entangled EPR pairs qubit 2 qubit 3 Quantum channel AliceBob classical channel qubit 1 8:55

17. IWEPNM 2007, Kirchberg Quantum Teleportation x H M1M1 M2M2 X Z Alice Bob Alice receivesand wants to teleport it to Bob Initial state After CNOT After Hadamard(next page)is followed by Alices measurements

18. IWEPNM 2007, Kirchberg Procedure before and after Alices measurements The first two qubits belong to Alice the third one to Bob Alice performs the measurements and finds either Alice phones the result to Bob. Bob knows which operations he needs to perform Alice Bob

19. IWEPNM 2007, Kirchberg Quantum Teleportation with Photons Experiments performed by Bennet, DeMartini, Zeilinger, Weinfurter et al. Jian-Wei Pan et al. Phys. Rev. Lett. 80, 3892 (1998) D. Boschi et al., Phys. Rev. Lett. 80, 1121 (1998)

20. IWEPNM 2007, Kirchberg There is no danger of getting beamed off (right away) 100 Million Centuries with today‘s technology. S. Braunstein: The complete description of a human being would require about 1032 bits. The Teleportation of this information would take about

21. IWEPNM 2007, Kirchberg The Deutsch Jozsa Algorithm Mapping a single bit to a binary function f(x) with x  {0,1} and f(x)  {0,1} constant balanced Deutsch: Decision for f(x) being constant or balanced can be taken by a quantum computer in a single tep. D. Deutsch and R. Jozsa, Proc. Roy. Soc. London A 439, 553 (1992) 9:00

22. IWEPNM 2007, Kirchberg Deutsch Jozsa Algorithm U f(x) H H H H Frontside Backside ancilla

23. IWEPNM 2007, Kirchberg NMR Results of the Deutsch Jozsa Algorithm f 00 0 constant f 11 0 constant f 01 1 balanced f 10 1 balanced J.A.Jones and M. Mosca, J. Chem. Phys. 109, 1648 (1998), I.L. Chuang et al. Nature 393, 143 (1998) Sven Zülsdorff, master thesis, Stuttgart, 1999 M. S. Anwar et al. Phys. Rev. A 70, 032324 (2004) (parahydrogen)

24. IWEPNM 2007, Kirchberg Deutsch Collins Algorithm with three Qubits Example: x000001010011100101110111 f(x)00110110 There are 2 8 = 256 functions in total! Among these are 2 constant and balanced functions In the following we consider only the basic 1(constant) + 8 (balanced) functions f(x). The others follow from cyclic rotations and inversions of the 0 and 1 bits. HHH Oracle U f(x) HHH  out constant  000 balanced ≠  000

25. IWEPNM 2007, Kirchberg Oracle Transforms Entanglement Alert No entanglement involved output: A.H. Dorai and A. Kumar Paramana J. Phys. 56, 705 (2001) J. Kim, J. S. Lee and S. Lee, Phys. Rev. A 62, 0223204 (2000) O. Mangold, A. Heidebrecht and M.M. Phys. Rev. A 70, 042307 (2004)

26. IWEPNM 2007, Kirchberg Output of Oracle U 6 = U 00011110 experimental calculated Where is the entanglement?

27. IWEPNM 2007, Kirchberg Local Rotations of Bell States experimental calculated

28. IWEPNM 2007, Kirchberg Find the Prime Factors (Shor Algorithm) It is a simple task to build the product of two large prime numbers p und q. Calculating n = p x q is easy. But: 137703491 = ? 137703491 = 7919 x 17389 is extremely demanding and requires log(n)  steps with arbitrary large  superpolynomial Factoring a 400 digit number would take 10 10 years with today's fastest computers P. Shor(1994): Quantum computer requires only O[(ln n) 3 ] steps A quantum computer based on the current fastest clock rates would factor a 400 digit number in only about 3 years. 9:05

29. IWEPNM 2007, Kirchberg Implementing the Shor Algorithm on a Nuclear Spin Quantumcomputer |R1>|R2>=|x,f(x)> = |n3,n2,n1>|m4,m3,m2,m1> We need two quantum registers R1(3 qubit) und R2(4 qubit), which contain x and f(x): Superposition: L. M. K. Vandersypen et al., Nature 414, 883 (2001)

30. IWEPNM 2007, Kirchberg NMR experiment factors numbers with Gauss sums from number theory 157573 = M. Mehring, K. Müller, W. Merkel, I. S. Averbukh, W. P. Schleich, quant-ph/0609174 v1, Phys. Rev. Lett. 98, 120502 (2007)

31. IWEPNM 2007, Kirchberg 1 H NMR implementation with phase controlled pulse sequence phase controlled CPMG pulse sequence M. Mehring., K. Müller, W. Merkel, I. S. Averbukh, W. P. Schleich, Phys. Rev. Lett. 98, 120502 (2007)

32. IWEPNM 2007, Kirchberg Proposals for Quantumcomputing with N@C 60 Wolfgang Harneit, Phys. Rev. A 65, 0232222 (2002) D. Suter and K. Lim, Phys. Rev. A. 65, 052309 (2002) 9:10

33. IWEPNM 2007, Kirchberg The pseudo pure initial state 15 N@C 60 two qubit subspace

34. IWEPNM 2007, Kirchberg Preparation and tomography of entangled states in 15 N@C 60 M.M. and W. Scherer. Phys. Rev. Lett. 93, 206603-1 (2004) W. Scherer and M. Mehring, arXiv e-print, (quant-ph/0602201), 2006

35. IWEPNM 2007, Kirchberg Phase Encoding of Entangled States  = 1 MHz    = 6 MHz   phase frequencies 1 =2 MHz; 2 =1 MHz Application of phase rotations about the z-direction

36. IWEPNM 2007, Kirchberg Two Qubit Subsystems in 31 P@C 60 Planned experiments: Swap and qubit cloning, decoherence free subspaces Cooperation with W. Harneit, FU Berlin Preliminary results (J. Mende, B. Naydenov) Reduced symmetry Lifting of degeneracies 9:10

37. IWEPNM 2007, Kirchberg The S-Bus Concept M. M. and J. Mende Phys. Rev. A 73, 0520303 (2006)

38. IWEPNM 2007, Kirchberg Ce:CaF 2 single crystal: The qubyte+1 Mims ENDOR Energy 9:15

39. IWEPNM 2007, Kirchberg Implementation of the Collins version of the Deutsch-algorithm M. M. and J. Mende Phys. Rev. A 73, 0520303 (2006)

40. IWEPNM 2007, Kirchberg Two Qubit ( 19 F) Entanglement in the S-Bus Ce:CaF 2

41. IWEPNM 2007, Kirchberg A. Heidebrecht S. Krämer O. Mangold W. Scherer J. Mende