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 Computation and Quantum Information, Nielsen and Chuang, 2000
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 Implementation Decoherence Scalability Coupling
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
Low intrinsic dissipation in superconductor
Josephson junctions provide nonlinearity
Phase
Charge
Flux
Berkeley, Delft, Jena, Kansas, MIT, NTT, Rome, Stony Brook, Syracuse
Kansas, Maryland, NIST, UCSB
Chalmers, NEC, Saclay, Yale

5. Consider superconducting loop interrupted by Josephson junction
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
E-beam lithography, Al-AlOx-Al double-angle evaporation
Delft design
Berkeley design
2 qubits with on-chip flux lines: Lq = 150 pH, Mqf ≈ 3 pH
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
Pb plating, also in chamber lid
Microwave coax
*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 Implementation Decoherence Scalability Coupling
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
Remedy
Microwave circuit
Flux bias
SQUID bias circuitry
Junction 1/f noise & defect states
Local flux noise, e.g. motion of vortices in nearby traces
*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
*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. Readout improvements Narrow SQUID escape distribution by: or
Lowering temperature
Adding damping across SQUID junctions
Increase effective mass of SQUID junctions
RC-shunts or C-shunts across each junction
Robertson, Plourde et al. PRB, 72, (2005) or
C-shunt across entire SQUID
Chiorescu, Nature 431, 159 (2004)

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, (2004)
Other promising non-dissipative readouts:
*Josephson bifurcation amplifier
[Siddiqi et al., PRL 94, (2005)]
*circuit-QED
[Wallraff et al., Nature 431, 162 (2004)]

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

21. Roadmap Introduction Implementation Decoherence Scalability Coupling
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, (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 bias of 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, (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