# are the incoherent by MikeJenny

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• pg 1
```									Adiabatic Quantum Computation
with Noisy Qubits

M.H.S. Amin
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)
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•

• 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
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 •
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 •
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 •
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

• 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 •

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 •

In terms of spectral density

Ohmic environment:

 increases with T !

The larger the T, the less the

• 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

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 •

• 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 •
J. Roland and N.J. Cerf, PRA 65, 042308 (2001)
Gap size:     
N

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

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

• 52 •
M.H.S. Amin and D.V. Averin, arXiv:0708.0384

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

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

• 53 •
M.H.S. Amin and D.V. Averin, arXiv:0708.0384

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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