are the incoherent by MikeJenny

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									Adiabatic Quantum Computation
        with Noisy Qubits


                M.H.S. Amin
    D-Wave Systems Inc., Vancouver, Canada
      Collaborators:
                                       Experiment:
      Theory:                          Andrew Berkley (D-Wave)
                                       Paul Bunyk (D-Wave)
      Dmitri Averin (Stony Brook)      Sergei Govorkov (D-Wave)
      Peter Love (D-Wave, Haverford)   Siyuan Han (Kansas)
      Colin Truncik (D-Wave)           Richard Harris (D-Wave)
      Andy Wan (D-Wave)                Mark Johnson (D-Wave)
      Shannon Wang (D-Wave)            Jan Johansson (D-Wave)
                                       Eric Ladizinsky (D-Wave)
                                       Sergey Uchaikin (D-Wave)

                                       Many Designers, Engineers,
                                       Technicians, etc. (D-Wave)

                                       Fabrication team (JPL)


•2•
      Quantum requirements for
      gate model quantum computation:

      • Coherence

      • Superposition

      • Entanglement




•3•
      For Adiabatic quantum computation:

      • Phase coherence is not required

      • Ground state superposition is required

      • Ground state entanglement is required

       Superposition and entanglement
      can be protected by the Hamiltonian
•4•
  Single Qubit Example
                                                      Measurement basis
      Hamiltonian:           H   1  x
                                   2

                                                          0 1       Superposition
         Energy eigenstates:                         
                                                            2           states
                                                       
         Energy eigenvalues:                    E  
                                                       2
                                                                    
      Initializing in state “0”:               (t  0)  0 
                                                                     2
      Time evolution:
                         eit / 2   eit / 2          t           t
               (t )                                cos    0  i sin    1
                                     2                    2            2

•5•
  Single Qubit Example

      Hamiltonian:        H   1  x
                                2



      Probability of finding the qubit in state “0”:

                                          2       1
                     P0 (t )  0  (t )            (1  cos t )
                                                  2



                            Coherent
                           Oscillations

•6•
  Density Matrix
                                                                       1         1
                                                                                      eit 
      Closed system density matrix:                               1 2it
                                                                       e
                                                                                  2
                                                                                      1
                                                                                           
                                                                                           
      (in the energy basis)                                           2              2    
      Open system density matrix:
      weak coupling limit                                    Dephasing (T2)
      Relaxation (T1)
         process                                                process

     Peq  ( 1  Peq )e t / T1               1
                                                     eit e t / T2   t   Peq       0 
  
    
               2
           1 it t / T2
                                                 2                     
                                                             t / T1 
                                                                                           
                                                                                          eq 
           2e    e                         P  ( 2  P )e
                                              eq   1     eq
                                                                            0         P 

                         e  E / T                        Equilibrium
             Peq   E / T                               (Boltzmann)
                   e          e  E / T                  Distribution


•7•
  Coherent Tunneling
      Probability of finding the qubit in state “0”:

      Decoherence rate             1 / T2  

      P0 (t )  0  (t ) 0  1 (1  e t cos t ), Coherent oscillations
                             2




                             0            1
                                                    = broadening
 Energy gap = 


                         well-defined gap
•8•
  Incoherent Tunneling
      Probability of finding the qubit in state “0”:

      Decoherence rate                    1 / T2  

                                            2
      P0 (t )  1 (1  e  t ),               Incoherent tunneling rate
                2
                                            

                                   0              1

 Energy gap =                                            = broadening


                            No well-defined gap
•9•
     Density Matrix
                                                        Peq    0 
         Density matrix in energy basis:             
                                                        0
                                                                     
                                                                  eq 
                                                               P 

         Density matrix in computation basis (“0” , “1”):
                                                                Signature
                           Superposition
                           1 1  P P         eq    eq
                                                                of coherent
                             eq
                        (coherent           
                                                              mixture
                            
                           2  P  P
                                 eq          mixture)
                                                  
                                               1          
                           can persist in
                           is diagonal only if
                            equilibrium
                        Peq  Peq    1
                                        2   i.e.,   T  


• 10 •
     Two-Qubit Example

         Hamiltonian:   H   1  ( 1   x )  1 J z  z ,
                              2      x
                                           2
                                                 2
                                                      1 2
                                                                 J  


                           Ferromagnetic coupling


         Lowest two                       00  11           Entangled
         energy eigenstates:         
                                               2             states
                                          2
         Energy eigenvalues:       E  
                                          2J




• 11 •
     Two-Qubit Entanglement

         Equilibrium density matrix (J  T , ):

                       P   P  
                               
                                eq
                                                  
                                                   eq



         Concurrence Entanglement (  )  P eq  P eq
                     (entanglement measure):
                                             C
         W.K. Wootters, PRL 80, 2245 (1998)                         
                              can persist in
                               equilibrium
                   C()  0, (i.e., unentangled) only if
                   P P 
                    
                     eq
                              
                               eq     1
                                      2       i.e., T   / J
                                                          2




• 12 •
   Summary:
         1. Classical limit is large T (compared to
            energy spacings) and not long t
            (compared to decoherence time)

         1. Without a Hamiltonian, the system will
            be classical after the decoherence time

         1. With a well-defined Hamiltonian (stronger
            than noise) system may stay quantum
            mechanical at small T


• 13 •
     Adiabatic Quantum Computation (AQC)
     E. Farhi et al., Science 292, 472 (2001)
                                                Energy Spectrum

     System Hamiltonian:


     H = (1 s) Hi + s Hf


     Linear interpolation: s = t/tf


     • Ground state of Hi is easily accessible.
     • Ground state of Hf encodes the solution
       to a hard computational problem.

• 14 •
     Adiabatic Quantum Computation (AQC)
     E. Farhi et al., Science 292, 472 (2001)
                                                         Energy Spectrum
                                           Effective
     System Hamiltonian:
                                           two-state
                                            system
     H = (1 s) Hi + s Hf
                                                       Gap = 

     Linear interpolation: s = t/tf


     • Ground state of Hi is easily accessible.
     • Ground state of Hf encodes the solution
       to a hard computational problem.

• 15 •
     Adiabatic Theorem
                                                Landau  Zener
                             Error              probabilit y :
                                                         2 / 2
                 E                              PLZ  e
                                                 ~ s ~ 1/ t f
                                                    
                                 


                           Success
                                            s


         To have small error probability:         tf >> 1/2


• 16 •
     System Plus Environment



         Smeared out              
         anticrossing
                                       Environment’s
                                       energy levels
                                       e.g., Harmonic oscillator



               Adiabatic theorem does not apply!



• 17 •
     Environment at Zero Temperature



                                                   




          At T=0 the excitation (Landau-Zener) probability
          is exactly the same as that for a closed system

         For spin environment: A.T.S. Wan, M.H.S. Amin, S.X. Wang, cond-mat/0703085

         For harmonic oscillator model: M. Wubs et al., PRL 97, 200404 (2006)


• 18 •
     Environment at Finite Temperature



                                                 Probabilit y of success:
                                                                          t f /t c
                                                  P0 (t f )  1 (1  e
                                                              2                       ),
         Energy level
                                                  1     tf
         Broadening = W                                dt Γ(t )
                                                  tc    0


                                                        1    1  ds
                                                            0 s Γ(s)
                                                                 
         If W > , transition will be via
                                                        tf

         incoherent tunneling process
                                                Incoherent tunneling rate




• 19 •
     Directional Tunneling Rates

                “0”            “1”   “0”         “1”

                       01                 10




         01 and 10 can
         be extracted from
         the initial slopes:



• 20 •
     Macroscopic Resonant Tunneling (MRT)

           “0”         “1”                  “0”         “1”

                 01                              10


                             1st resonant
                                  peak


                                                               2nd resonant
                                                                   peak

                                   10            01         wp




• 21 •
     Calculating Incoherent Tunneling Rate

         Two-State Model:           E
                                                       


                                                                 e
                                                   0

                                                 1
         System Hamiltonian:            H S   (e z   x )
                                                 2
         Interaction Hamiltonian:       H int  Q z

                                    Heat bath operator

• 22 •
     Non-Markovian Environment
         M.H.S. Amin and D.V. Averin, preprint


         1. For a Gaussian environment up to second order in :




             Noise spectral density:


         2. If S(w) is peaked at low frequencies, one can expand eiwt




• 23 •
     Non-Markovian Environment
         M.H.S. Amin and D.V. Averin, preprint


         Gaussian line-shape

                                                      1
         Width                                      ~ *
                                                     T2
         Shift


                                           t 2 / 2T2*2
          Decoherence ~ e
           1                sin 2 (wt / 2)
                cos   dw
                    2
                                           S (w ) ~ cos2   dwS (w ),     arctan( / e )
          T2*2                (wt / 2) 2

• 24 •
     Non-Markovian Environment
         M.H.S. Amin and D.V. Averin, preprint


         Gaussian line-shape


         Width                                   depends on symmetric part of S(w)


         Shift                            depends on anti-symmetric part of S(w)


         For a   classical noise, S(w)            is symmetric, hence


                                      Symmetric 01
• 25 •
     Quantum Noise
         Let e  0
                         01(e)                01(e)



          Absorption     Tunneling              Tunneling   Emission



            Bose-Einstein distribution:



           01(e)  01(e)              The peak is shifted
                                           toward positive e

• 26 •
     Equilibrium Environment
                         S(w) = Ss(w)  Sa(w)


                            depends on symmetric part Ss(w)


                            depends on anti-symmetric part Sa(w)

                    Can be tested                w      S (w )
                                   S (w )
                    experimentally  S (w ) coth 2T  2T w
                                                               a
  Fluctuation-Dissipation Theorem:      s         a




                                                  W2
           W 2  2Te p                 Teff               Effective
                                                  2e p   Temperature



• 27 •
     Experiment
                                                             Magnetic flux

         Tunable rf-SQUID qubit:
                      Josephson junctions:
                                                        F2        F1

         Double-well potential:


         Low             F2  F0 / 2         High                 F2  0
         barrier:                            barrier:

              ~ wp                                          wp
                                                                             ~0

                            wp = plasma frequency
• 28 •
     Pulse Sequence
                                High barrier



  Initializing



  Tunneling



                  Low barrier
                                      Measurement


• 29 •
     Transition Rate Measurements
         R. Harris et al., preprint available

                                    Tunneling to the 2nd level

                                            Tunneling to
                                            the 1st level



                                                            W
                                  10                           01




                                                    (mF0)

            Asymmetric tunneling  Quantum Noise
• 30 •
     Fit to Theory
         R. Harris et al., preprint available

                                    Tunneling to the 2nd level

                                            Tunneling to
                                            the 1st level



                                                            W




                                                    (mF0)

                   Good agreement with theory
• 31 •
     Width and Shift Measurements


                                 ep




                       W



              W (1 / T2* ) weakly depends on T
              e p decreases with T as ~ 1/T
• 32 •
     Effective Temperature




                                          Equilibrium
          Saturation                      distribution
         Temperature




                       Mixing chamber thermometer
• 33 •
     Tunneling Amplitude

         One can extract  from data using




                   Why T-dependent?
Why increase with T?

   is renormalized by high frequency environmental modes

• 34 •
     Adiabatic Renormalization of 

         Consider the Hamiltonian      H   1  0 x  Q z
                                             2

                                         1                                  
         At 0  0, the two states        0   g ,0  1   g ,1
                                                                            
                                                                             
                                          2                               
         are degenerate
                                                        Environment states


         Degeneracy is lifted by H, with a splitting:

                      H    H   0  g ,0 g ,1
                                             HF



                  Renormalization can only decrease 

• 35 •
     Adiabatic Renormalization of 

         In terms of spectral density




          Ohmic environment:


                                         increases with T !

                      The larger the T, the less the
                  environment cares about the system

• 36 •
     Sub-Conclusions:
          • The noise is coming from a quantum source,
           e.g. two-state fluctuators; J. Martinis et al. PRL. 95, 210503 (2005)

          • Low frequency part of the noise spectrum is
            peaked at zero frequency (e.g. 1/f noise).

          • High frequency noise is likely to be ohmic.
                                                                         1/f
         In agreement with:
         Astafiev et al.
         PRL 93, 267007 (2004)
                                                                            ohmic


• 37 •
     Back to AQC
         M.H.S. Amin and D.V. Averin, arXiv:0708.0384

                                                                                          t f /t c
         Probability of success:                  P0 (t f )  (1  e    1
                                                                        2                             )
                                                                                                                   1
                                                       t f dt Γ(t )    2                         de        
                                                                               1                     
                                                                         
         Characteristic time scale:              tc   
                                                       0
                                                                     
                                                                        t                        e Γ(e ) 
                                                                                                               
                                                                          f                                   
         For a non-Markovian environment:

               (e )  01 (e )  10 (e ) 
                                                2
                                               8 W
                                                   (e    ( e e p ) 2 / 2W
                                                                              e
                                                                                    ( e e p ) 2 / 2W
                                                                                                          )
                                                                                       2E
         Linear interpolation (global adiabatic evolution):                   e
                                                                                          const.
                                                                                       tf
                                                    E
                                               tc  2
                                                   
• 38 •
     Computation Time Scale
         M.H.S. Amin and D.V. Averin, arXiv:0708.0384

                                          2E
     Open system:                     tf ~ 2
                                                      Normalized
     Closed system:                      2E
                                     tf ~ 2
     (Landau-Zener probability)           0           Not normalized

             Broadening (low frequency noise)
             does not affect the computation time

  Incoherent tunneling rate ~ 1/ W
                                                        Cancel each other
  Width of transition region ~ W

• 39 •
     Gap Renormalization Effect

         High frequency modes:       T  w  wc

         Low frequency modes:        w W

         If W  T , the two regions overlap


                 Gap renormalization may not happen
                 for large n, or may happen differently
                  from the simple one-qubit example



• 40 •
     AQC vs. Classical Annealing

         Optimization problem:                               H  (1 s) Hi  s Hf
                  n                 n
         H f   hi       z
                              j
                                  J         jk    zj zk    Diagonal
                  j 1            j , k 1

                1 n
         H i     j xj                                   Non-diagonal
                2 j 1
                                                                               sH f       Hf
                                                                                     
         j  0                         Boltzmann factor:
                                                                       e      e T /s
                                                                                T

          Effective temperature Teff = T/s                     changes from  to T


                         Classical annealing
• 41 •
     AQC vs. Classical Annealing

         Optimization problem:                          H  (1 s) Hi  s Hf
                 n             n
         H f   hi  z
                         j
                             J         jk    zj zk    Diagonal
                j 1         j , k 1

                1 n
         H i     j xj                              Non-diagonal
                2 j 1


                       j  T                      Quantum regime
                       j  T                     Thermal regime
                       j ~ T                      Mixed regime!

• 42 •
     Noise Requirements for AQC

         1. Away from classical limit:     j >T
                     For Gaussian distribution
                      of the energy levels the
         2. Stable final ground state:    hj , Jjk > T
                     number of the levels near
                      the ground state will be
         3. Well defined tunnel splitting: j > Wj          Single
                          polynomial (in n)                 qubit
         4. Accurate final Hamiltonian:     hj , Jjk > Wj   noise

         5. Polynomial number of energy levels
            within energy T from the ground state
            during the evolution


• 43 •
     Our Multi-Qubit System
     Tunable rf-SQUID qubit + coupler:
         R. Harris et al., PRL 98, 177001 (2007)


                                            Qubits

                                       Tunable coupler


                  Fa
                   2
                           F1
                            a
                                               Fc        F1
                                                          b
                                                               Fb
                                                                2




   Controls j                           Controls Jjk         Controls j

                                         Controls hj

• 44 •
     Annealing Process

         Double-well potential:

                F2  F 0 / 2                           F2  0

                               Annealing
   j ~ wp                                        wp
                                                                j ~ 0
                          wp = plasma frequency

          Initial Hamiltonian                 Final Hamiltonian



• 45 •
     Annealing Regime
                                                wp ~ 400 mK   2nd resonant peak
                          1st   resonant peak


                                                              wp

    Effective
    Temperature:
    Teff ~ 20 mK                                  W ~ 80mK


                      Unlikely to be
         j ~ wp   classical annealingsplittings
                   >W      Well-defined energy
         j ~ wp >> T                   Quantum regime
         hj , Jjk > W,T                 Well-defined final Hamiltonian
• 46 •
     Conclusions
         1. Our environment model correctly describes
            resonant tunneling in superconducting devices

         2. Theoretical + experimental MRT investigations:
            dominant low frequency noise has quantum origin
         3. Phase coherence (energy basis) is not required
            (global AQC not for local adiabatic evolution)
         4. Ground state superposition and entanglement
            (computation basis) are required for AQC;
            protected by the Hamiltonian
         5. Our multi-qubit system is in the right regime for
            AQC/Quantum Annealing
• 47 •
         Additional Slides




• 48 •
     Classical and Quantum Annealing

         Annealing         Disorder                      Order
                                        Slow transition

    Classical annealing:
          Disorder = Thermal mixing (introduced by entropy)


    Quantum annealing:
          Disorder = Superposition (introduced by a Hamiltonian)



• 49 •
     AQC vs. Quantum Annealing (QA)
         Optimization problem:                                           H  (1 s) Hi  s Hf
                 n                      n
         H f   hi       z
                              j
                                      J         jk    zj zk        Problem Hamiltonian (Diagonal)
                j 1                  j , k 1

                1 n
         H i     j xj                                        Disordering Hamiltonian (Non-Diagonal)
                2 j 1

                           Quantum phase transition
                       Large                                                          Small
                    superposition                                                  superposition


                         z
                1
     ( 0)                                                        Quantum                (t f )  z  solution
                N      z{ 0,1}   n                              Critical point
• 50 •
     AQC vs. Quantum Annealing (QA)

         1. For optimization problems:
            AQC = QA

         2. Superposition and entanglement
            are important for AQC/QA
                                                    Single
         3. Quantum critical point happens          qubit
            when j(s) ~ hj(s) , Jjk(s)             noise


           Condition: j(s) , hj(s) , Jjk(s) > Wj
• 51 •
     Global vs. Local Adiabatic Evolution
         Adiabatic Grover search:                                           E
     J. Roland and N.J. Cerf, PRA 65, 042308 (2001)
                                                           Gap size:     
                                                                            N

         Closed system:                               N  2 n , n  number of qubits
                    Quantum
         Global adiabatic evolution:
                 advantage only
                   possible viat f ~ 2  O(N )
            1                        E                              The same as
         s   const.
         
            tf   local evolution                                    classical

         Local adiabatic evolution:

         s  gap(s )
                        2
                                             1
                                         tf ~ O N
                                             
                                                      ( )          The same as
                                                                   Grover search


• 52 •
     Global vs. Local Adiabatic Evolution
         M.H.S. Amin and D.V. Averin, arXiv:0708.0384

                                                                               1
         Open system:                                   2      de        
                                                   tc        e Γ(e ) 
                                                        t                 
                                                         f                
               No Quantum
         Global adiabatic evolution:
               advantage!
         s  only prefactor t f ~ 2  O(N )
             1                       E                              The same as
               const.
            t f speedup                                             classical

         Local adiabatic evolution:
                                                                    The same as
                                         t f ~ 2  O( N )
                                              W
          s  gap(s )
                          2
          
                                                                    classical


• 53 •
     Global vs. Local Adiabatic Evolution
         M.H.S. Amin and D.V. Averin, arXiv:0708.0384


         Quantum advantage is only possible
                                       The same as
                      if W            gate model
                                        decoh.
           on the      W ~ 1/d          quantum
                                         time           tf < d
           other :                     computation
           hand
                       tf ~ 1/

            Global coherence is required for local AQC


• 54 •

								
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