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					            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
                            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
                                           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
                               Superconducting qubits
•   Low intrinsic dissipation in superconductor
•   Josephson junctions provide nonlinearity

       Phase                                 Charge                                    Flux




  Ground and first excited         Different superpositions of even and   Different superpositions of CW and
state in Josephson potential      odd numbers of Cooper pairs on island   CCW screening currents around loop


                                                                               Leggett, Garg, 1985
                                                                               Friedman, et al., 2000
                                                                               van der Wal, et al., 2000

                                                                           Berkeley, Delft, Jena, Kansas,
Kansas, Maryland, NIST,                                                    MIT, NTT, Rome, Stony
                                   Chalmers, NEC, Saclay, Yale
UCSB                                                                       Brook, Syracuse
                               Flux qubit
       Consider superconducting loop interrupted by Josephson junction

•   Total flux
                   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
                  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
                                                         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:
    L = 150 pH, M ≈ 3 pH
     q                  qf




                         Q ui kTi e™ and a
                             c m
                       G r aphi s decom pr essor
                              c
                                          s c
                   ar e needed t o see t hi pi t ur e.
                                                                Qubit junctions                                Qubit junctions

                                                                     AuCu quasiparticle             SQUID junction
     Qubit 2                                                               traps


                                                                 Two independently-controlled
     Qubit 1                                                     flux lines for biasing SQUID
                                                                 and qubits

                                                                                                                     200 nm


         35 m                                                                                       SQUID junctions
                                                                                                       2
                                                                                          175 x 200 nm , I ≈ 0.23 A, C ≈ 6.4 fF
                                                                                                          0             j
*Microwaves applied with superconducting coax with  ~ 1 mm short at end, ~ 3 mm above chip
Measurement configuration

                 *Dilution
                 refrigerator

                 *Extensive
                 filtering and
                 shielding

                      SQUID/qubit chip
                                         87 mm




                  Pb plating, also in                  Microwave coax
                  chamber lid

               *Expect transverse cavity resonance around 6.7 GHz
Qubit excitation and state readout
Spectroscopy
             Coherent manipulation of qubit state
•   Need ability to generate any arbitrary superposition.


•   Visualize state of spin-1/2 particle in static magnetic
    field B on Bloch sphere:
           0
                                           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
                           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
Ramsey fringe measurement of dephasing
Spin echo sequence




          *Fit echo envelope for each

          *Extract echo fringe amplitude and plot
          against corresponding        pulse separation
          of echo peak
                Sources of decoherence in flux qubits
                                Source                    Remedy
                          •   Microwave circuit         *with careful thermalization of coax,
                                                        = not a problem (T1, T2 > 1 ms)
                          •   Flux bias                 *weaken coupling to qubit (need
                                                        larger critical currents for flux bias
                                                        traces)
                                                        *operate at qubit symmetry point

                          •   SQUID bias circuitry      *weaken coupling to qubit (need to
                                                        compensate with enhanced readout
                                                        sensitivity)
                                                        *alternative readout techniques
                                                        *operate at SQUID symmetry point
*For useful qubit, want                                 (Delft, Saclay, Yale)

                          •   Junction 1/f noise & defect*improvements in materials for
                              states                     junction tunnel barrier to reduce
                                                         defect density (Delft, UCSB, NIST)
                          •   Local flux noise, e.g.
                                                         *operate at qubit symmetry point
                              motion of vortices in
                              nearby traces
Readout improvements
      Narrow SQUID escape distribution by:
       • Lowering temperature
       • Adding damping across SQUID junctions
       • Increase effective mass of SQUID junctions
      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)
                    Alternative readout techniques
Inherent dissipation in standard switching readout contributes to decoherence
    Inductive readout
      *Josephson inductance of SQUID L J
      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)]
                          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/0412485
                                           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
                       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)
                     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 M of the two
                                            qq
    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.
Circulating current in dc SQUID vs. applied flux (T = 0)
                   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, J 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)
            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
Peak width measurement of dephasing




                         A. Abragam, The Principles of
                         Nuclear Magnetism.



                                    Strong-driving limit



                                    Weak-driving limit
              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 I pulse to measure SQUID critical current
                                                     b
                                      without changing the static flux.

                                    • Pulse parameters can be adjusted to compensate for both
                                      crosstalk terms and finite risetime of I pulse.
                                                                             b




                                                                                 CNOT

				
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