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. In collaboration with: G. Falci (DMFCI-Catania) E. Paladino(DMFCI-Catania) J. Siewert(DMFCI-Catania) D.V. Averin(Stony Brook) C. Bruder(U. Basel) L. Faoro(Rutgers U.) G.M. Palma(U. Milano) F. Plastina(U. Cosenza) C. Macchiavello(U. Pavia) A. Romito(Weizman Inst.) V. Vedral(U. Leeds) G. De Chiara (SNS-Pisa) D. Rossini(SNS-Pisa) S. Montangero(SNS-Pisa)

3. References M. Tinkham Introduction to Superconductivity (MacGraw Hill) D.V. Averin For. der Physik, 48, 1055 (2000) Yu. Makhlin, G. Schön, and A. Shnirman Rev. Mod. Phys. 73, 357 (2001) M. H. Devoret, A. Wallraff, and J.M. Martinis, cond-mat/0411174 D. Esteve and D. Vion cond-mat/0505676

4. Two-state system Preparation of the state Controlled time evolution Low decoherence Read-out “DiVincenzo list” (Esteve) (Averin) Geometric quantum computation Applications

5. 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

6. Solid state qubits Advantages - Scalability - Flexibility in the design Disadvantages - Static errors - Environment

7. Qubit = two state system How to go from N-dimensional Hilbert space (N >> 1) to a two-dimensional one?

8. All Cooper pairs are ``locked'' into the same quantum state

9.  There is a gap in the excitation spectrum  T/Tc Quasi-particle spectrum

10. Cooper pairs also tunnel through a tunnel barrier a dc current can flow when no voltage is applied A small applied voltage results in an alternating current Josephson junction     Energy of the ground state ~ -E J cos 

11. SQUID Loop  LL RR

12. Dynamics of a Josephson junction     + + + + + + + _ _ _ _ _ _ _ X =

13. Mechanical analogy

14. U  Washboard potential

15. Quantum mechanical behaviour The charge and the phase are Canonically conjugated variable From a many-body wavefunction to a one (continous) quantum mechanical degree of freedom Two state system

16. Josephson qubits Josephson qubits are realized by a proper embedding of the Josephson junction in a superconducting nanocircuit Charge qubit Charge-Phase qubit Flux qubit Phase qubit 1 10 4 Major difference is in the form of the non-linearity

17. Phase qubit Current-biased Josephson junction U  The qubit is manipulated by varying the current

18. Flux qubit The qubit is manipulated by varying the flux through the loop f and the potential landscape (by changing E J ) X (t) 11 22

19. Cooper pair box tunable: tunable: - external (continuous) gate charge n x - E J by means of a SQUID loop

20. Cooper pair box phase difference Cooper pair number, voltage across junction current through junction

21. Cooper pair box

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

23. 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

24. Coherent dynamics - experiments Nakamura et al 1999 Vion et al 2002 Chiorescu et al 2003 Chalmers group NTT group … See also exps by Schoelkopf et al, Yale NIST

25. Inductance  E J2 C nxCxnxCx E J1 C  E J2 C nxCxnxCx E J1 C L VxVx VxVx Charge qubit coupling - 1

26. Capacitance  E J2 C nxCxnxCx E J1 C  E J2 C nxCxnxCx E J1 C Charge qubit coupling - 2

27. Josephson Junction  E J C nxCxnxCx E J1 C  E J2 C  Charge qubit coupling - 3

28. Tunable coupling Untunable couplings = more complicated gating Variable electrostatic transformer The coupling can be switched off even in the presence of parasitic capacitances The effective coupling is due to the (non-linear) Josephson element Averin & Bruder 03

29. Leakage The Hilbert space is larger than the computational space Consequences: a) gate operations differ from ideal ones (fidelity) b) the system can leak out from the computational space (leakage) Leakage One qubit gate Fidelity Two qubit gate Fidelity |m+1> |m> ~E c EjEj |1> |0> qubit

30. Sources of decoherence in charge qubits Z electromagnetic fluctuations of the circuit (gaussian) discrete noise due to fluctuating background charges (BC) trapped in the substrate or in the junction Quasi-particle tunneling

31. Full density matrix TRACE OUT the environment RDM for the qubit: populations and coherences Reduced dynamics – weak coupling

32.        ”Charge degeneracy” (  = 0,  = E J ) no adiabatic term optimal point     ”Pure dephasing” (E J =0,  =  ) no relaxation Reduced dynamics – weak coupling

33. Background charges in charge qubits HQHQ z x Fluctuations due to the environment E E is a stray voltage or current or charge polarizing the qubit electrostatic coupling Charged switching impurities close to a solid state qubit charged impurities Electronic band E di+didi+di

34. “Weakly coupled” charge Decoherence only depends on = oscillator environment “Strongly coupled” charge large correlation times of environment discrete nature keeps memory of initial conditions saturation effects for g >>1 information beyond needed g=v/  weak vs strongly coupled charges

35. E J =0 – exact solution Constant of motion

36. In the long time behavior for a single Background Charge The contribution to dephasing due to “strongly coupled” charges (slow charges) saturates in favour of an almost static energy shift ~ ~ E J =0 – exact solution

37. Standard model: BCs distributed according to with yield the 1/f power spectrum from experiments Experiments: BCs are responsibe for 1/f noise in SET devices. Warning: an environment with strong memory effects due to the presence of MANY slow BCs Background charges and 1/f noise

38. “Fast” noise in general quantum noise fast gaussian noise fast or resonant impurities Split Two-stage elimination Slow noise ≈ classical noise slow 1/f noise Slow vs fast noise

39. 2  expand to second order in  → quadratic noise Large N fl central limit theorem → gaussian distributed Slow noise:  (t) random adiabatic drive  M  →  adiabatic approximation Retain fluctuations of the length of the Hamiltonian → longitudinal noise Paladino et al. 04 see also Shnirman Makhlin, 04 Rabenstein et al 04 Static Path Approximation (SPA) variance HQHQ z x Optimal point Initial defocusing due to 1/f noise

40. Standard measurements no recalibration with recalibration Falci, D’Arrigo, Mastellone, Paladino, PRL 2005, cond-mat/0409522 SPA HQHQ z x Optimal point Initial suppression of the signal due essentially to inhomogeneuos broadening (no recalibration) Initial defocusing due to 1/f noise