1. Quantum computation with solid state devices - “Theoretical aspects of superconducting qubits” Quantum Computers, Algorithms and Chaos, Varenna 5-15 July 2005 Rosario Fazio

2. Outline Lecture 1 - Quantum effects in Josephson junctions - Josephson qubits (charge, flux and phase) - qubit-qubit coupling - mechanisms of decoherence - Leakage Lecture 2 - Geometric phases - Geometric quantum computation with Josephson qubits - Errors and decoherence Lecture 3 - Few qubits applications - Quantum state transfer - Quantum cloning

3.  Quantum information protocols without external control (less flexible but more stable) Choose a given model and use just the time evolution (less flexible but more stable) Easier to implement in solid state systems  Implementation of Quantum communication schemes in solid state devices Josephson arrays in quantum communication Motivations

4. Alice Bob Quantum channel Protocols Cloning Quantum state transfer

5. Alice Bob ~ |  > Quantum communications with spin chains |  >=a|0>+b|1> Quantum channel

6. Alice Bob... |  >  |  |0> |  > |0> J J J Initial state S. Bose (2002), M. Christandl et al (2003), F. Verstraete, M. Martin-Delgado and J.I. Cirac (2003), D. Burgarth and S. Bose (2004), D. Burgarth, V. Giovanetti and Bose (2005), V. Giovannetti and R. Fazio (2005), A. Romito, C. Bruder and R. Fazio (2005), G. De Chiara D. Rossini, S. Montangero and R. Fazio (2005), … Quantum communications with spin chains

7. Alice Bob... J J J Quantum communications with spin chains

8. Initial state Time evolution Sender at site 1 Receiver at site L Quantum communications with spin chains

9. Total magnetization Is a constant of motion where

10. Jt Quality of the channel – average fidelity

11. Fidelity ~ L -1/3 L Quantum communications with spin chains

12. Perfect communications with spin chains

13. No cloning theorem U|a>|0> |a>|a> U(|a>+|b>) |0> |aa>+|bb> ≠ (|a>+|b>) (|a>+|b>) Quantum cloning A quantum copying machine does not exist! W. Wootters and W. Zurek (1982) A quantum copying machine does not exist!

14. Although perfect cloning is not possible ….. … Imperfect cloning has been considered Quantum cloning V. Buzek and M. Hillery (1996), D. Bruβ et al (1998), D. Bruβ, A. Ekert and C. Macchiavello (1998), R. Werner (2000), … A.Lama-Linares et al (2002), De Martini et al (2004), J. Du et al (2004), …

15. Central quantity Fidelity for n m cloning n states to be copied  belonging to a portion of the Hilbert space the m cloned states are in in the mixed state  Independent on 

16. Universal Cloner Phase Covariant Cloner Phase Covariant Cloner Examples Fidelity at the equator

17. R y (  /2) Quantum circuits

18. XY Model Heisenberg Model Ising Model Start from the state to clone Wait for a time (independent on the state to be cloned) Cloning 1 m | > =|0> | > = cos  |0>+sin  e i  |1> G. De Chiara et al. (2004,2005) Spin network cloning

20.  Fidelity Heisenberg Ideal cloner coincides with the XY model Phase covariant cloner 

21. Best cloner Heisenberg XY m Fidelity

22. nmFJt 230.9481 250.8773 270.8169 350.97584 Cloning from n to m

23. F Cloning in the presence of noise  /J

24. Vg Josephson coupling realizes the XY model Quantum cloning with Josephson qubits

25. Vg Josephson coupling realizes the XY coupling F U1U1 U2U2 Quantum cloning with Josephson qubits

26. Josephson arrays as artificial 1D magnets Charge regime - C. Bruder R, Fazio, G. Schön, PRB 47, 342 (1993) Flux regime - L. Levitov, T.P. Orlando, J.B. Mayer, J.E. Mooij cond-mat 0108266 Quantum communication with JJA Bose-Hubbard = Quantum Phase Model = XXZ model

27. preparation measurement state propagation C. Bruder et al (1993) Quantum communication with JJAs

28. Example - N=3 t Fidelity E J /E C =0.1 C/C 0 =10 Fidelity ~ 0.999 Averaged over the initial state

29. F max N Fidelity vs N

30. F max C 0 /C Dependence on the electrostatic interaction

31. V t = t max |>|> ~ The current is proportional to the fidelity VgVg VgVg VgVg VgVg The charge state of the N island as a function of t p Current correlation Charge correlation

32. Fidelity ~ 0.95  E J /E J =10%  n x /n x =10% Imperfections – N=3

33. t

34. L. Levitov, T.P. Orlando, J.B. Mayer, J.E. Mooij cond-mat 0108266 x x x x x x State transfer with flux qubits

35. Alice Bob Quantum channel Entangled Entanglement sharing {{ ………………………………………..

36. singlet ● ● ● time -1 0 1 23 4site # ● ● ● Singlet propagation

37. Entanglement Singlet initial state

38. Entropy for symmetric sites

39. Entropy for sites (-6,7) Fidelity to the initial singlet

40. JJ arrays can be used in quantum communication  Entanglement sharing  Quantum Cloning  State transfer Experiments seems to be possible at present Conclusions