Superconducting Flux Qubits: Coherence, Readout, and Coupling

1. Superconducting Flux Qubits: Coherence, Readout, and Coupling Britton L. T. Plourde Syracuse University with: T. L. Robertson, T. Hime, S. Linzen, P. A. Reichardt, C.-E. Wu, John Clarke, K. Birgitta Whaley, J. Zhang, Frank Wilhelm (LMU München) University of California, Berkeley Thank you: Patrice Bertet, Michel Devoret, Daniel Esteve, Kees Harmans, John Martinis, Robert McDermott, Hans Mooij, Rob Schoelkopf, Dale Van Harlingen, Denis Vion ISEC September 6, 2005

2. Quantum computing • Quantum Computer: composed of quantum bits = qubits • Potentially able to solve problems intractable or slow on classical computer • Factoring large numbers • Fast database searches • Simulation of quantum systems • To build a quantum computer, need many qubits with long coherence times • Architectures with solid-state qubits on a chip provide a route to scalability • Need interactions between qubits to generate entanglement Quantum Computation and Quantum Information, Nielsen and Chuang, 2000

3. Roadmap • Introduction • Variety of superconducting qubits • Principle of flux qubit • Implementation • Fabrication and measurement techniques • Spectroscopy and quantum coherent manipulation • Decoherence • Relaxation and dephasing • Sources of decoherence • Optimization of readout to reduce decoherence • Improving coherence at symmetry points • Scalability • Coupling • Direct coupling with fixed interaction • SQUID-based controllable coupling scheme

4. Superconducting qubits Phase Charge Flux • Low intrinsic dissipation in superconductor • Josephson junctions provide nonlinearity Berkeley, Delft, Jena, Kansas, MIT, NTT, Rome, Stony Brook, Syracuse Kansas, Maryland, NIST, UCSB Chalmers, NEC, Saclay, Yale

5. Flux qubit Consider superconducting loop interrupted by Josephson junction • Total flux

6. One junction flux qubit • Inductive energy of qubit loop • Josephson energy of qubit junction • Charging energy of qubit junction capacitance *Lowest two energy levels separated by energy

7. Flux qubit with dc SQUID readout • Three-junction flux qubit [Mooij et al., Science 285, 1036 (1999)] • Reduced sensitivity to fabrication asymmetries; allows arbitrarily small loop inductances • Microwave pulses to drive transitions • Readout with switching dc SQUID • Switching level of dc SQUID depends on total flux coupled to SQUID

8. Fabrication of qubit and SQUID Delft design Berkeley design 2 qubits with on-chip flux lines: Lq = 150 pH, Mqf ≈ 3 pH • E-beam lithography, Al-AlOx-Al double-angle evaporation Qubit junctions Qubit junctions SQUID junction AuCu quasiparticle traps Qubit 2 Two independently-controlled flux lines for biasing SQUID and qubits Qubit 1 200 nm SQUID junctions 175 x 200 nm2, I0 ≈ 0.23 μA, Cj ≈ 6.4 fF 35 μm *Microwaves applied with superconducting coax with φ ~ 1 mm short at end, ~ 3 mm above chip

9. Measurement configuration *Dilution refrigerator *Extensive filtering and shielding SQUID/qubit chip 87 mm Microwave coax Pb plating, also in chamber lid *Expect transverse cavity resonance around 6.7 GHz

10. Qubit excitation and state readout

11. Spectroscopy

12. Coherent manipulation of qubit state • Need ability to generate any arbitrary superposition. • Visualize state of spin-1/2 particle in static magnetic field B0 on Bloch sphere:

13. Roadmap • Introduction • Variety of superconducting qubits • Principle of flux qubit • Implementation • Fabrication and measurement techniques • Spectroscopy and quantum coherent manipulation • Decoherence • Relaxation and dephasing • Sources of decoherence • Optimization of readout to reduce decoherence • Improving coherence at symmetry points • Scalability • Coupling • Direct coupling with fixed interaction • SQUID-based controllable coupling scheme

14. Estimates of decoherence Qubit decoherence can be related to noise in the environment coupled to qubit. • Relaxation of non-thermal distribution. • Decay rate of resonance peaks • Dephasing caused by impedance both at level splitting and zero frequency. • Width of resonance peaks

15. Ramsey fringe measurement of dephasing

16. Spin echo sequence *Fit echo envelope for each *Extract echo fringe amplitude and plot against corresponding pulse separation of echo peak

17. Sources of decoherence in flux qubits Source Remedy *with careful thermalization of coax, = not a problem (T1, T2 > 1 ms) *weaken coupling to qubit (need larger critical currents for flux bias traces) *operate at qubit symmetry point • Microwave circuit • Flux bias • SQUID bias circuitry • Junction 1/f noise & defect states • Local flux noise, e.g. motion of vortices in nearby traces *weaken coupling to qubit (need to compensate with enhanced readout sensitivity) *alternative readout techniques *operate at SQUID symmetry point (Delft, Saclay, Yale) *For useful qubit, want *improvements in materials for junction tunnel barrier to reduce defect density (Delft, UCSB, NIST) *operate at qubit symmetry point

18. RC-shunts or C-shunts across each junction or Robertson, Plourde et al. PRB, 72, 024513 (2005) C-shunt across entire SQUID Chiorescu, Nature 431, 159 (2004) Readout improvements Narrow SQUID escape distribution by: • Lowering temperature • Adding damping across SQUID junctions • Increase effective mass of SQUID junctions

19. Alternative readout techniques Inherent dissipation in standard switching readout contributes to decoherence Inductive readout *Josephson inductance of SQUID LJ depends on flux, even for Ib < Ic *Detect change in LJ for two different states of qubit Lupascu et al. PRL, 93, 177006 (2004) Other promising non-dissipative readouts: *Josephson bifurcation amplifier [Siddiqi et al., PRL 94, 027005 (2005)] *circuit-QED [Wallraff et al., Nature 431, 162 (2004)]

20. *need flux offset following manipulation for measurable flux difference between ground and excited states *use pulse to shift and offset total qubit flux bias *adjust to give Protection at symmetry points (1) manipulation of qubit state at degeneracy point -- protection against flux noise (2) operation at symmetry point of readout SQUID -- protection against noise from readout circuitry and SQUID asymmetry Bertet, et al. cond-mat/0412485 \nu vs \Phi_Q plot

21. Roadmap • Introduction • Variety of superconducting qubits • Principle of flux qubit • Implementation • Fabrication and measurement techniques • Spectroscopy and quantum coherent manipulation • Decoherence • Relaxation and dephasing • Sources of decoherence • Optimization of readout to reduce decoherence • Improving coherence at symmetry points • Scalability • Coupling • Direct coupling with fixed interaction • SQUID-based controllable coupling scheme

22. Scalable biasing scheme • Operate at arbitrary to adjust SQUID sensitivity • Vary while maintaining fixed • Set separately for ith qubit • With multiple on-chip flux bias lines driven by independent current sources, possible to combine biases to address multiple qubits and SQUIDs • Add additional flux bias line and current source for each new element *Plourde et al., PRB 72, 060506(R) (2005)

23. Controllable coupling of qubits • Need qubit-qubit interaction to generate entanglement, for example, singlet state • For flux qubits, natural interaction is through the flux. For example, screening flux of qubit 1 changes the flux biasof qubit 2. Thus interaction has the form • The generalized Hamiltonian with this interaction is given by • For coupling via the mutual inductance Mqq of the two qubits, K is fixed at • Fixed coupling complicates the implementation of an entangling gate. • New proposed scheme enables one to vary both the magnitude and sign of K: in particular, K can be made zero.

24. Circulating current in dc SQUID vs. applied flux (T = 0)

25. Variable flux qubit coupling using dc SQUID • When Qubit 2 changes state, circulating current reverses direction, coupling flux to the SQUID. The change in circulating current J is • In turn, ΔJ couples a flux to Qubit 1 in addition to the directly coupled flux: • The net coupling strength K is thus • Thus, one can use the same SQUID to vary K and to read out the flux state of the qubits. *Plourde et al., PRB 70, 140501(R) (2004)

26. Progress and remaining challenges • Controlled manipulation of flux qubit state achieved • Promising techniques for improving coherence times • Demonstration of scalable biasing scheme • New proposed scheme for controllable coupling

27. Peak width measurement of dephasing A. Abragam, The Principles of Nuclear Magnetism. Strong-driving limit Weak-driving limit

28. Entangling operation with variable coupling For feasible parameters: • When combined with appropriate single qubit rotations with K = 0, a single pulse of current can generate the CNOT gate in 29 ns. • Qubit states can be determined immediately afterwards with a larger Ib pulse to measure SQUID critical current without changing the static flux. • Pulse parameters can be adjusted to compensate for both crosstalk terms and finite risetime of Ib pulse. CNOT