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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. Adiabatic cyclic evolution • The Hamiltonian of a quantum system depends on a set of external parameters r • The external parameters are changed in time r(t) • Adiabatic approximation holds e.g. an external magnetic field B e.g. the direction of B If the system is in an eigenstate it will adjust to the instantaneous field

  4. What happens to the quantum state if r(0) = r(T) ????

  5. Parallel Transport e(T) ≠ e(0) After a cyclic change of r(t) the vector e(t) does NOT come back to the original direction a r(T)=r(0) The angle a depends on The circuit C on the sphere

  6. Quantum Parallel Transport Schroedinger’s equation implements phase parallel transport Schroedinger’s equation: Adiabatic approx: Instantaneous eigenstates Look for a solution:

  7. Berry phase The geometrical phase change of |y> along a closed circuit r(T)=r(0) is given by - M.V.Berry 1984

  8. Spin ½ in an external field C B Adiabatic condition wB << 1 The Berry phase is related to the solid angle that C subtends at the degeneracy

  9. Aharonov-Anandan phase Geometric phases are associated to the cyclic evolution of the quantum state (not of the Hamiltonian) Generalization to non-adiabatic evolutions Consider a state which evolves according to the Schrödiger equation such that Cyclic state - Y. Aharonov and J. Anandan 1987

  10. Aharonov-Anandan phase Introducing such that Dynamical phase Geometrical phase • Evolution does not need to be adiabatic • Adiabatic changes of the external parameters are a way to have a cyclic state • In the adiabatic limit

  11. Aharonov-Anandan phase (Example) The Hamiltonian Initial state evolves as The state is cyclic after T=p/B

  12. Experimental observations Geometrical phases have been observed in a variety of systems Aharonov-Bohm effect Quantum transport Nuclear Magnetic Resonance Molecular spectra … see “Geometric Phases in Physics”, A. Shapere and F. Wilczek Eds

  13. Is it possible to observe geometric phases in a macroscopic system?

  14. Geometric phases in superconducting nanocircuits Possible exp systems: Superconducting nanocircuits Implications: • “Macroscopic” geometric interference • Solid state quantum computation • Quantum pumping • G. Falci, R. Fazio, G.M. Palma, J. Siewert and V. Vedral 2000 • F. Wilhelm and J.E. Mooij 2001 • X. Wang and K. Matsumoto 2002 • L. Faoro, J. Siewert and R. Fazio 2003 • M.S. Choi 2003 • A. Blais and A.-M. S. Tremblay 2003 • M. Cholascinski 2004

  15. Cooper pair box V IJ Cx Cj E E C J 2 CHARGE BASIS n ( ) ( ) å å 2 - - + + + n n n n n n 1 n 1 n x n N Charging Josephson tunneling

  16. From the CPB to a spin-1/2 H = In the |0>, |1> subspace Hamiltonian of a spin In a magnetic field Magnetic field in the xz plane

  17. Asymmetric SQUID EJ1 C Cx F EJ2 C Vx H = Ech (n -nx)2 -EJ (F)cos (f - a)

  18. From the SQUID-loop to a spin-1/2 Bx = EJ cos a By = EJ sin a Bz = Ech (1-2nx) In the {|0>, |1>} subspace HB = - (1/2) B.s

  19. “Geometric” interference in nanocircuits Bx = EJ cos a(F) C By = EJ sin a(F) B Bz = Ech (1-2nx) In order to make non-trivial loops in the parameter space need to have both nx and F HB = - (1/2) B.s “ ”

  20. Berry phase in superconducting nanocircuits FM 1/2 nx Role of the asymmetry

  21. Berry phase - How to measure • Initial state • Sudden switch to nx=1/2 • Adiabatic loop |0> (1/2½)[|+> + |->] (1/2½)[eig+ib|+> +e-ig-ib|->]

  22. Berry phase - How to measure • Swap the states • Adiabatic loop with opposite orientation • Measure the charge (1/2½)[eig+ib|-> +e-ig-ib|+>] (1/2½)[e2ig|-> +e-2ig|+>] P(2e)=sin22g

  23. Quantum computation • Two-state system • Preparation of the state • Controlled time evolution • Low decoherence • Read-out Intrinsic fault-taulerant for area-preserving errors Phase shifts of geometric origin - J. Jones et al 2000 - P. Zanardi and M. Rasetti 1999

  24. Geometric phase shift Spin 1 Spin 2 sz- interaction Controlled phase gate

  25. Geometric phase shift Two Cooper pair boxes coupled via a capacitance Hcoupling = - EKsz1sz2 • G. Falci et al 2000

  26. Non-abelian case N degenerate Control parameters: When the state of the system is degenerate over the full course of its evolution, the system need not to return to the original eigenstate, but only to one of the degenerate states. Adiabatic assumption:

  27. Holonomic quantum computation Dynamical approach System S, with state space H , perform universal QC Geometric approach • able to control a set of parameters on which depend a iso-degenerate • family of Hamiltonian • information is encoded in an N degenerate eigenspace C of a distinguished • Hamiltonian • Universal QC over C obtained by adiabatically driving the control parameters • along suitable loops rooted at

  28. Josephson network for HQC L. Faoro, J. Siewert and R. Fazio, PRL 2003 One excess Cooper pair in the four-island set up There are four charges states |j> corresponding to the positionof the excess Cooper pair on island j L.M.Duan et al, Science 296,886 (2001)

  29. Josephson network for HQC control parameters DEGENERATE EIGENSTATES WITH 0-ENERGY EIGENVALUE

  30. One-bit operations In order to obtain all single qubit operations explicit realizations of : Rotation around the z-axis Rotation around the y-axis Intially we set so the eigenstates correspond to the logical states

  31. Adiabatic pumping R L V1(t) V2(t) Charge transport, in absence of an external bias, by changing system parameters • Open systems – modulation of the phase of the scattering matrix • Closed systems – periodic lifting of the Coulomb Blockade Charge is transferred coherently Quantization of transferred charge • P.W. Brouwer 1998 • …. • H. Pothier et al 1992

  32. Cooper pair pumping vs geometric phases Relation between the geometric phase and Cooper pair pumping • J.E. Avron et al 2000 • A. Bender, Y. Gefen, F. Hekking and G. Schoen 2004 • M. Aunola and J. Toppari 2003 • R. Fazio and F. Hekking 2004

  33. Cooper pair pumping vs geometric phases In order to relate the second term to the AA phase Take the derivative with respect to the external phase d

  34. Cooper pair sluice Iright coil t t Ipumped Vg A. O. Niskanen, J. P. Pekola, and H. Seppä, (2003). t Can be generalized to pump 2Ne per cycle. (N = 1,2,…?)

  35. Cooper pair sluice - exp The measured device Gate line Input coils Junctions SQUID loops

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