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

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

3. 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

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

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

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

7. Quasi-particle spectrum There is a gap in the excitation spectrum D D T/Tc

8. j1 I j2 Josephson junction • 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 Energy of the ground state ~ -EJcosj

9. SQUID Loop F jR jL

10. X Dynamics of a Josephson junction + + + + + + + _ _ _ _ _ _ _ j1 j2 =

11. Mechanical analogy

12. Washboard potential U(f)

13. 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

14. 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 1 104 Major difference is in the form of the non-linearity

15. U(f) Phase qubit Current-biased Josephson junction The qubit is manipulated by varying the current

16. X Flux qubit (t) j2 j1 The qubit is manipulated by varying the flux through the loop f and the potential landscape (by changing EJ)

17. Cooper pair box tunable: - external (continuous) gate charge nx - EJ by means of a SQUID loop

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

19. Cooper pair box

20. 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

21. 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

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

23. Charge qubit coupling - 1 EJ1 C F nx Cx EJ2 C F EJ2 C Vx EJ1 C Vx Inductance nx Cx L

24. Charge qubit coupling - 2 EJ1 C F nx Cx EJ2 C F EJ2 C Capacitance EJ1 C nx Cx

25. Charge qubit coupling - 3 EJ1 C F nx Cx EJ C F EJ2 C Josephson Junction F

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

27. |m> |m+1> ~Ec qubit Ej |0> |1> 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 Two qubit gate Fidelity One qubit gate Fidelity

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

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

30. Reduced dynamics – weak coupling • q=0 ”Charge degeneracy” (e = 0 , W = EJ) no adiabatic term optimal point • q=p/2”Pure dephasing” (EJ =0 , W = e) no relaxation

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

32. g=v/g weak vs strongly coupled charges “Weakly coupled” charge Decoherenceonlydepends 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

33. EJ=0 – exact solution Constant of motion

34. ~ EJ=0 – exact solution 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

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

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

37. Initial defocusing due to 1/f noise z HQ x • Large Nfl central limit theorem → gaussian distributed Optimal point s 2 Paladino et al. 04 • Slow noise: x(t) random adiabatic drivegM <W →adiabatic approximation • Retain fluctuations of the length of the Hamiltonian → longitudinal noise • Static Path Approximation (SPA) variance • expand to second order in x→ quadratic noise see also Shnirman Makhlin, 04 Rabenstein et al 04

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