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Quantum computation with solid state devices - “Theoretical aspects of superconducting qubits”

Quantum Computers, Algorithms and Chaos , Varenna 5-15 July 2005. Quantum computation with solid state devices - “Theoretical aspects of superconducting qubits”. Rosario Fazio. Scuola Normale Superiore - Pisa. Outline. Lecture 1 - Quantum effects in Josephson junctions

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Quantum computation with solid state devices - “Theoretical aspects of superconducting qubits”

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

  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. Motivations • Quantum information protocols without external control • 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

  4. Protocols Cloning Quantum state transfer Alice Quantum channel Bob

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

  6. Quantum communications with spin chains . . . Alice J J J Bob |y >qj |0 > |0> |0 > |0> 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), …

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

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

  9. Quantum communications with spin chains Total magnetization Is a constant of motion where

  10. Quality of the channel – average fidelity Jt

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

  12. Perfect communications with spin chains

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

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

  15. Central quantity Fidelity for n m cloning the m cloned states are in in the mixed state r n states to be copied |y> belonging to a portion of the Hilbert space Independent on |y>

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

  17. Quantum circuits Ry(p/2)

  18. Spin network cloning • XY Model • Heisenberg Model • Ising Model | > = cosq|0>+sinq eij |1> | > =|0> • Start from the state to clone • Wait for a time (independent on the state to be cloned) • Cloning 1 m G. De Chiara et al. (2004,2005)

  19. Phase covariant cloner Ideal cloner coincides with the XY model q Fidelity Heisenberg q

  20. XY Best cloner Fidelity Heisenberg m

  21. Cloning from n to m

  22. Cloning in the presence of noise F G/J

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

  24. Quantum cloning with Josephson qubits Vg Vg F Josephson coupling realizes the XY coupling U2 U1

  25. Quantum communication with JJA Josephson arrays as artificial 1D magnets Bose-Hubbard = Quantum Phase Model = XXZ model 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

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

  27. Averaged over the initial state Example - N=3 t Fidelity EJ /EC=0.1 C/C0=10 Fidelity ~ 0.999

  28. Fidelity vs N Fmax N

  29. Dependence on the electrostatic interaction Fmax C0/C

  30. ~ |y> Vg Vg Vg Vg • The charge state of the N island as a function of tp • Current correlation <I1(t)IN(t+tp)> • Charge correlation <n1(t)nN(t+tp)> t = tmax V The current is proportional to the fidelity

  31. Imperfections – N=3 dEJ/EJ=10% dnx/nx=10% Fidelity ~ 0.95

  32. t

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

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

  35. Singlet propagation singlet ● ● ● ● ● ● time ● ● ● ● ● ● ● ● ● ● ● ● site # -1 0 1 2 3 4

  36. Singlet initial state Entanglement

  37. Entropy for symmetric sites

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

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

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