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

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 |  > along a closed circuit r(T)=r(0) is given by - M.V.Berry 1984

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

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 phaseGeometrical 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=  /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 CHARGE BASIS Charging Josephson tunneling n N      x n nnn 2 J E nnnn C E11 2 IJIJ CjCj V CxCx n

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

17. Asymmetric SQUID H = E ch (n -n x ) 2 -E J (  cos (   E J2 C C x E J1 C VxVx

18. From the SQUID-loop to a spin-1/2 In the {|0>, |1>} subspace H B = - (1/2) B.  B x = E J cos  B y = E J sin  B z = E ch (1-2n x )

19. “Geometric” interference in nanocircuits H B = - (1/2) B.  B x = E J cos  B y = E J sin  B z = E ch (1-2n x ) B C “ ” In order to make non-trivial loops in the parameter space need to have both n x and 

20. Berry phase in superconducting nanocircuits 1/2 nxnx MM Role of the asymmetry

21. Berry phase - How to measure Initial state Sudden switch to n x =1/2 Adiabatic loop |0> (1/2 ½ )[|+> + |->] (1/2 ½ )[e i  +i  |+> +e -i  -i  |->]

22. Berry phase - How to measure Swap the states Adiabatic loop with opposite orientation Measure the charge (1/2 ½ ) [e i  +i  |-> +e -i  -i  |+>] (1/2 ½ ) [e 2i  |-> +e -2i  |+>] P(2e)=sin 2 2 

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

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

25. Geometric phase shift Two Cooper pair boxes coupled via a capacitance H coupling = - E K  z1  z2 -G. Falci et al 2000

26. Aharonov-Anandan phase is sup. nanocircuits AA phase in symmetric SQUID No dynamical phase “Fast” manipulation for qubits H = -A. Blais and A.-M. S. Tremblay 2003

27. Non-abelian case 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. Control parameters: N degenerate Adiabatic assumption:

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

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

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

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

32. One-bit operations

33. The qubits are coupled by symmetric SQUIDS (can be switched off) Capacitive coupling can be neglected if the capacitances of the junctions are sufficiently small. No 1st-order contributions: only one Cooper pair is allowed in each qubit set-up Two-bit operations

34. No 1st-order contributions: only one Cooper pair is allowed in each qubit set-up The non-vanishing 2nd-order contributions: + unwanted diagonal 2nd-order contributions Two-bit operations

35. DEGENERATE EIGENSTATES WITH 0 ENERGY EIGENVALUE 2nd-order energy shifts can be compensated by adjusting gate voltages Conditional phase shift

36. Adiabatic pumping L R V 1 (t) V 2 (t) Open systems – modulation of the phase of the scattering matrix Closed systems – periodic lifting of the Coulomb Blockade Charge transport, in absence of an external bias, by changing system parameters Charge is transferred coherently Quantization of transferred charge -P.W. Brouwer 1998 -…. -H. Pothier et al 1992

37. 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

38. Cooper pair pumping vs geometric phases EXAMPLE |0> |1> e i  |1>

39. 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 

40. Cooper pair sluice t I right coil VgVg t t I pumped Can be generalized to pump 2Ne per cycle. (N = 1,2,…?) A. O. Niskanen, J. P. Pekola, and H. Seppä, (2003).

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