Transcript Document

Adiabatic Quantum Computation
with Noisy Qubits
M.H.S. Amin
D-Wave Systems Inc., Vancouver, Canada
Collaborators:
Theory:
Dmitri Averin (Stony Brook)
Peter Love (D-Wave, Haverford)
Colin Truncik (D-Wave)
Andy Wan (D-Wave)
Shannon Wang (D-Wave)
Experiment:
Andrew Berkley (D-Wave)
Paul Bunyk (D-Wave)
Sergei Govorkov (D-Wave)
Siyuan Han (Kansas)
Richard Harris (D-Wave)
Mark Johnson (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
H   12  x
Hamiltonian:
0 1
Energy eigenstates:
 
Energy eigenvalues:

E  
2
2
 (t  0)  0 
Initializing in state “0”:
Superposition
states
  
2
Time evolution:
 (t ) 
•5•
eit / 2   e it / 2 
2
t
t
 cos
0  i sin
1
2
2
Single Qubit Example
Hamiltonian:
H   12  x
Probability of finding the qubit in state “0”:
P0 (t )  0  (t )
2

1
(1  cos t )
2
Coherent
Oscillations
•6•
Density Matrix
Closed system density matrix:
(in the energy basis)
 12
      1 it
2e
1
2
e it 


1
2

Open system density matrix:
weak coupling limit
Relaxation
(T1)
process
 Peq  ( 12  Peq )e t / T1
  
1 it t / T2
e
2e

 E / T
e
Peq   E / T
e
 e  E / T
•7•
Dephasing (T2)
process
 t   Peq
 




t
/
T
eq
eq
 0
P  ( 12  P )e 1 

1
2
eit e  t / T2
Equilibrium
(Boltzmann)
Distribution
0 

eq 
P 
Coherent Tunneling
Probability of finding the qubit in state “0”:
  1/ T2  
Decoherence rate
P0 (t )  0  (t ) 0  12 (1  e t cos t ),
0
Energy gap = 
Coherent oscillations
1
 = broadening
well-defined gap
•8•
Incoherent Tunneling
Probability of finding the qubit in state “0”:
  1/ T2  
Decoherence rate
P0 (t )  12 (1  e  t ),

0
Energy gap = 
2

Incoherent tunneling rate
1
 = broadening
No well-defined gap
•9•
Density Matrix
 Peq
  
 0
Density matrix in energy basis:
Density matrix in computation basis (“0” , “1”):
Superposition
1
P P 
1
 eq
 (coherent
eq

2  P  P
eq

eq


mixture)

1

can persist in
equilibrium
 is
diagonal only if
Peq  Peq 
• 10 •
1
2
i.e., T  
0 

eq 
P 
Signature
of coherent
mixture
Two-Qubit Example
Hamiltonian:
H   12 ( 1x   x2 )  12 J z1 z2 ,
J  
Ferromagnetic coupling
• 11 •
00  11
Lowest two
energy eigenstates:
 
Energy eigenvalues:
2
E  
2J
2
Entangled
states
Two-Qubit Entanglement
Equilibrium density matrix (J  T , ):
 P   P  
eq

eq

Concurrence Entanglement
(entanglement measure):
W.K. Wootters, PRL 80, 2245 (1998)
C( )  P  P
eq

can persist in
equilibrium
C()  0, (i.e., unentangled) only if
P P 
eq

• 12 •
eq

1
2
i.e., T   / J
2
eq

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
•
•
• 14 •
Ground state of Hi is easily accessible.
Ground state of Hf encodes the solution
to a hard computational problem.
Adiabatic Quantum Computation (AQC)
E. Farhi et al., Science 292, 472 (2001)
Energy Spectrum
System Hamiltonian:
H = (1 s) Hi + s Hf
Effective
two-state
system
Gap = 
Linear interpolation: s = t/tf
•
•
• 15 •
Ground state of Hi is easily accessible.
Ground state of Hf encodes the solution
to a hard computational problem.
Adiabatic Theorem
Landau  Zener
probabilit y :
Error
2 / 2
PLZ  e
 ~ s ~ 1 / t f
E

Success
s
To have small error probability:
• 16 •
tf >> 1/2
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

Energy level
Broadening = W
If W > , transition will be via
incoherent tunneling process
Probabilit y of success :
P0 (t f )  12 (1  e
 t f /tc
),
tf
1
  dt Γ(t )
0
tc
1

tf
ds
0 s Γ( s)
1
Incoherent tunneling rate
• 19 •
Directional Tunneling Rates
“0”
“1”
01
01 and 10 can
be extracted from
the initial slopes:
• 20 •
“0”
“1”
10
Macroscopic Resonant Tunneling (MRT)
“0”
“0”
“1”
01
“1”
10
1st resonant
peak
2nd resonant
peak
10
• 21 •
01
wp
Calculating Incoherent Tunneling Rate
Two-State Model:
E

0
System Hamiltonian:
Interaction Hamiltonian:
1
H S   (e z   x )
2
H int  Q z
Heat bath operator
• 22 •
e
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
Width
1
~ *
T2
Shift
Decoherence ~ e
 t 2 / 2T2*2
1
sin 2 (wt / 2)
2
2

cos

d
w
S
(
w
)
~
cos
  dwS (w ),
*2
2

T2
(wt / 2)
• 24 •
  arctan(  / e )
Non-Markovian Environment
M.H.S. Amin and D.V. Averin, preprint
Gaussian line-shape
Width
Shift
For a
depends on symmetric part of S(w)
depends on anti-symmetric part of S(w)
classical noise, S(w)
is symmetric, hence
Symmetric 01
• 25 •
Quantum Noise
Let e  0
01(e)
Absorption
Tunneling
01(e)
Tunneling
Emission
Bose-Einstein distribution:
01(e)  01(e)
• 26 •
The peak is shifted
toward positive e
Equilibrium Environment
S(w) = Ss(w)  Sa(w)
depends on symmetric part Ss(w)
depends on anti-symmetric part Sa(w)
Can be tested
S (w )
w
 2T
Fluctuation-Dissipation Theorem: S (w )  S (w ) coth
experimentally
2T
w
a
s
W 2  2Te p
• 27 •
Teff
a
W2

2e p
Effective
Temperature
Experiment
Magnetic flux
Tunable rf-SQUID qubit:
Josephson junctions:
F1
F2
Double-well potential:
Low
barrier:
F2  F0 / 2
wp
 ~ wp
wp = plasma frequency
• 28 •
F2  0
High
barrier:
~0
Pulse Sequence
High barrier
Initializing
Tunneling
Low barrier
• 29 •
Measurement
Transition Rate Measurements
R. Harris et al., preprint available
Tunneling to the 2nd level
Tunneling to
the 1st level
W
10
(mF0)
01
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
Saturation
Temperature
Equilibrium
distribution
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   12  0 x  Q z
At 0  0, the two states  
are degenerate
1 
 0   g ,0  1   g ,1
2


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)
• 37 •
ohmic
Back to AQC
M.H.S. Amin and D.V. Averin, arXiv:0708.0384
P0 (t f )  (1  e
1
2
Probability of success:
2
tf


tc    dt Γ(t )   
t
 0

 f
1
Characteristic time scale:
 t f /tc
)

de
 e Γ(e ) 


For a non-Markovian environment:
(e )  01 (e )  10 (e ) 
(
e
8W
 2
(e e p ) 2 / 2W
Linear interpolation (global adiabatic evolution):
E
tc  2

• 38 •
e
(e e p ) 2 / 2W
)
2E
e 
 const.
tf
1
Computation Time Scale
M.H.S. Amin and D.V. Averin, arXiv:0708.0384
Open system:
Closed system:
(Landau-Zener probability)
2E
tf ~ 2

Normalized
2E
tf ~ 2
 0
Not normalized
Broadening (low frequency noise)
does not affect the computation time
Incoherent tunneling rate ~ 1 / W
Width of transition region ~ W
• 39 •
Cancel each other
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:
n
H f   hi 
j
z
j 1
1 n
H i     j xj
2 j 1
j  0

n
J
j , k 1
H  (1 s) Hi  s Hf
j k

jk z  z
Diagonal
Non-diagonal
sH f
e
T
Classical annealing
• 41 •

Hf
 e T /s
changes from  to T
Boltzmann factor:
Effective temperature Teff = T/s

AQC vs. Classical Annealing
Optimization problem:
n
H f   hi 
j 1
j
z
n
J
j , k 1
j k

jk z  z
1 n
H i     j xj
2 j 1
• 42 •
H  (1 s) Hi  s Hf
Diagonal
Non-diagonal
j  T
 Quantum regime
j  T
 Thermal regime
j ~ T
 Mixed regime!
Noise Requirements for AQC
1. Away from classical limit:
 >T
j
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
polynomial (in n)
4. Accurate final Hamiltonian:
hj , Jjk > Wj
5. Polynomial number of energy levels
within energy T from the ground state
during the evolution
• 43 •
Single
qubit
noise
Our Multi-Qubit System
Tunable rf-SQUID qubit + coupler:
R. Harris et al., PRL 98, 177001 (2007)
Qubits
Tunable coupler
F a2
Controls j
F1a
Fc
Controls Jjk
Controls hj
• 44 •
F 1b
F b2
Controls j
Annealing Process
Double-well potential:
F2  F0 / 2
F2  0
Annealing
wp
j ~ wp
wp = plasma frequency
Initial Hamiltonian
• 45 •
j ~ 0
Final Hamiltonian
Annealing Regime
1st
resonant peak
wp ~ 400 mK
2nd resonant peak
wp
Effective
Temperature:
Teff ~ 20 mK
j ~ wp
• 46 •
W ~ 80mK
Unlikely to be
Well-defined
energy splittings
>W
classical
annealing
j ~ wp >> T
Quantum regime
hj , Jjk > W,T
Well-defined final Hamiltonian
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:
n
H f   hi 
j 1
j
z
n
J
j , k 1
j k

jk z  z
1 n
H i     j xj
2 j 1
H  (1 s) Hi  s Hf
Problem Hamiltonian (Diagonal)
Disordering Hamiltonian (Non-Diagonal)
Quantum phase transition
Small
superposition
Large
superposition
 ( 0) 
1
N
• 50 •
z
z{ 0,1}n
Quantum
Critical point
 ( t f )  z  solution
AQC vs. Quantum Annealing (QA)
1. For optimization problems:
AQC = QA
2. Superposition and entanglement
are important for AQC/QA
3. Quantum critical point happens
when j(s) ~ hj(s) , Jjk(s)
Condition: j(s) , hj(s) , Jjk(s) > Wj
• 51 •
Single
qubit
noise
Global vs. Local Adiabatic Evolution
Adiabatic Grover search:
Gap size:
J. Roland and N.J. Cerf, PRA 65, 042308 (2001)
E

N
N  2n , n  number of qubits
Closed system:
Quantum
Global adiabatic evolution:
advantage only
E
1
possible viat f ~ 2  O(N )
s   const.
tf
local evolution 
The same as
classical
Local adiabatic evolution:
s  gap(s)
• 52 •
2
1
tf ~ O

( N)
The same as
Grover search
Global vs. Local Adiabatic Evolution
M.H.S. Amin and D.V. Averin, arXiv:0708.0384
2
tc  
t
 f
Open system:
No Quantum
Global adiabatic evolution:
advantage!
E
1only prefactor
t f ~ 2  O( N )
s   const.

t f speedup

de
 e Γ(e ) 

1

The same as
classical
Local adiabatic evolution:
s  gap(s)
• 53 •
2
W
t f ~ 2  O( N )

The same as
classical
Global vs. Local Adiabatic Evolution
M.H.S. Amin and D.V. Averin, arXiv:0708.0384
Quantum advantage is only possible
if W  
on the
other :
hand
W ~ 1/td
The same as
gate model
decoh.
quantum
time
computation
tf < td
tf ~ 1/
Global coherence is required for local AQC
• 54 •