Transcript Slide 1

Quantum computing and qubit decoherence
Semion Saikin
NSF Center for Quantum Device Technology
Clarkson University
Outline
• Quantum computation.
Modeling of quantum systems
Applications
Bit & Qubit
Entanglement
Stability criteria
Physical realization of a qubit
Decoherence
Measure of Decoherence
• Donor electron spin qubit in Si:P. Effect of nuclear spin bath.
Structure
Application for Quantum computation
Sources of decoherence
Spin Hamiltonian
Hyperfine interaction
Energy level structure (high magnetic field)
Effects of nuclear spin bath (low field)
Effects of nuclear spin bath (high field)
Hyperfine modulations of an electron spin qubit
• Conclusions.
• Prospects for future.
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Quantum computation
Modeling of quantum systems
i

 H
t
2  2
H 
 V ( x)
2
2m x
E0
E1
...
En
0  x 
1  x 
...
n  x 
1 particle – n equations:
 H 00

H   H10
 ...

L particles – nL equations!
H 01 ... 0 
 
H11 ... 1 
... ... ... 
R. Feynman, Inter. Jour.
Theor. Phys. 21, 467 (1982)
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Quantum computation
Applications
• Modeling of quantum systems
Pharmaceutical industry
Nanoelectronics
• Quantum search algorithm
L. Grover (1995)
• Factorization of large integer numbers
P. Shor (1994)
RSA Code:
Military,
Banking
• Quantum Cryptography
Process optimization:
Industry
Military
Bob
Alice
Eve
Quantum computation
Bit & Qubit
• Two states classical bit
• Two levels quantum system (qubit)
Polarization vector:
1
0
S=(Sφ Sθ SR=const)
Density matrix:
 11
  
  21
• Equalities
≡
 1 0
  
  1
0
0


0 ≡
  
  0
 0 1
1
12 

 22 
• Single qubit operations
 0 0
 (t )  eiHt /   (0)eiHt / 
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Quantum computation
Entanglement
+
Non-separable
quantum states:
=
+
≠
+
≠
0
0 0

1
 0 A1B  1A0B   1  0 1  1
2 0 1 1
2

0
0 0
0

0
0

0
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Quantum computation
Stability criteria
• The machine should have a collection of bits.
(~103 qubits)
• It should be possible to set all the memory bits to 0 before the start of each computation.
• The error rate should be sufficiently low.
(less 10-4 )
• It must be possible to perform elementary logic operations between pairs of bits.
• Reliable output of the final result should be possible.
0
0
0
0
I
n
p
u
t
Unitary
transformation
O
u
t
p
u
t
D. P. DiVincenzo, G. Burkard, D. Loss,
E. V. Sukhorukov, cond-mat/9911245
Classical control
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QC Roadmap
http://qist.lanl.gov/
Quantum computation
Physical realization of a qubit
• Ion traps and neutral atoms
• Semiconductor charge qubit
Single QD
Double QD
E2
e
E1
E0
• Photon based QC
e
E1
0
E0
0
1
P
1
• Spin qubit
• Superconducting qubit
Cooper pair box
SQUID

Nuclear spin
(liquid state NMR,
solid state NMR)
I
Electron spin
S
i
N pairs -
0
N+1 pairs -
1
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Quantum computation
Decoherence. Interaction with macroscopic environment.
Markov process
T1 T2 concept
Non-exponential decay
11  22 ~ et T
1
12 ~ et T
2
t
0
t
Quantum computation
Measure of Decoherence
• Basis independent.
• Additive for a few qubits.
• Applicable for any timescale and
complicated system dynamics.
S ideal
   real   ideal
 max 1 , 2 ...
S real
 11
  
  21
 ~ Sreal  Sideal
12 

 22 
A. Fedorov, L. Fedichkin,
V. Privman, cond-mat/0401248
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Donor electron spin in Si:P
Structure
Si atom
(group-IV)
Diamond crystal structure
Natural Silicon:
– 92%
– 4.7%
30Si – 3.1%
28Si
29Si
5.43Å
I=1/2
31P electron
spin (T=4.2K)
T1~ min T2~ msecs
P atom
(group-V)
+
=
b ≈ 15 Å
a ≈ 25 Å
Natural Phosphorus:
31P –
100%
I=1/2
In the effective mass approximation electron wave function is s-like:
1
F (r ) 
e
ab
( x 2  y 2 ) / a 2  z 2 / b2
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Donor electron spin in Si:P
Application for QC
A - gate
Bohr Radius:
Si: a ≈ 25 Å
b ≈ 15 Å
J - gate
Ge: a ≈ 64 Å
b ≈ 24 Å
SixGe1x
Si
Si1xGex
B.E.Kane, Nature 393 133 (1998)
31P donor
Qubit – nuclear spin
Qubit-qubit inteaction – electron spin
S1
HEx
J - gate
S2
R.Vrijen, E.Yablonovitch, K.Wang, H.W.Jiang,
A.Balandin, V.Roychowdhury, T.Mor, D.DiVincenzo,
Phys. Rev. A 62, 012306 (2000)
31P donor
Qubit – electron spin
Qubit-qubit inteaction – electron spin
HHf
A - gate
S1
HEx
S2
I2
I1
Qubit 1
Qubit 2
Qubit 1
Qubit 2
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Donor electron spin in Si:P
Sources of decoherence
• Interaction with phonons
D. Mozyrsky, Sh. Kogan, V. N. Gorshkov, G. P. Berman
Phys. Rev. B 65, 245213 (2002)
• Gate errors
X.Hu, S.Das Sarma, cond-mat/0207457
• Interaction with 29Si nuclear spins
Theory
I.A.Merkulov, Al.L.Efros, M.Rosen, Phys. Rev. B 65, 205309 (2002)
S.Saikin, D.Mozyrsky, V.Privman, Nano Letters 2, 651 (2002)
R. De Sousa, S.Das Sarma, Phys. Rev. B 68, 115322 (2003)
S.Saikin, L. Fedichkin, Phys. Rev. B 67, 161302(R) (2003)
J.Schliemann, A.Khaetskii, D.Loss, J. Phys., Condens. Matter 15, R1809 (2003)
Experiments
A. M. Tyryshkin, S. A. Lyon, A. V. Astashkin, and A. M. Raitsimring, Phys. Rev. B 68, 193207 (2003)
M. Fanciulli, P. Hofer, A. Ponti, Physica B 340–342, 895 (2003)
E. Abe, K. M. Itoh, J. Isoya S. Yamasaki, cond-mat/0402152 (2004)
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Donor electron spin in Si:P
Spin Hamiltonian
28Si
HSpin  H Ze   H Znucl (i )   H Hf (i )   H Dip (i, j )
i
H
i j
i
31P
e29Si
Effect of
external field
Electronnuclei
interaction
Electron spin Zeeman term:
Effective Bohr radius ~ 20-25 Å
Lattice constant = 5.43 Å
Nuclear spin Zeeman term:
Nucleinuclei
interaction
H Ze  gHS
H Znulc (i )   i HI
In a natural Si crystal the donor electron
interacts with ~ 80 nuclei of 29Si
29Si
System of
nuclear spins can be
considered as a spin bath
Hyperfine electron-nuclear spin interaction:
Dipole-dipole nuclear spin interaction:
HHf (i)  SAi Ii
HDip (i, j)  Ii DijI j
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Donor electron spin in Si:P
Hyperfine interaction
Contact interaction:
Dipole-dipole interaction:
e-
HCont  ASI
H Dip 
29Si
Hyperfine interaction:
H Hf  S x
Sy
 Axx

S z  Ayx
A
 zx
μ eμ n 3(μ er )( μ nr )

r3
r5
Axy
Ayy
Azy
Axz  I x 
 
Ayz  I y 
Azz  I z 
Approximations:
Contact interaction only:
 Axx

A 0
 0

0
Ayy
0
0 

0 
Azz 
Contact interaction
High magnetic field
High magnetic field
 0

A 0
A
 zx
0
0
Azy
0 

0 
Azz 
0 0

A  0 0
0 0

0 

0 
Azz 
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Donor electron spin in Si:P
Energy level structure (high magnetic field)
H
H Ze  gH z Sz
H ZP   PH z I zP
- 31P electron spin
-
31P nuclear
spin
- 29Si nuclear spin
H HfP  AzzP Sz I zP
H ZSi1
Si1
H Hf
…
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 Axx

A 0
 0

Donor electron spin in Si:P
Effects of nuclear spin bath (low field)
0
Ayy
0
0 

0 
Azz 
gSi
~ ( S I Si  S I Si )
gH P
H Ze ~ H z
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HP  const  42Oe
  1  et / T
-1
1/T [s ]
15
Si  const  3Oe
10
5
S. Saikin, D. Mozyrsiky and V. Privman,
Nano Lett. 2, 651-655 (2002)
0
15
20
Magnetic field [Oe]
25
17
 0

A 0
A
 zx
Donor electron spin in Si:P
Effects of nuclear spin bath (high field)
(a) S=“”
(b) S=“”
0
0
Azy
0 

0 
Azz 
e-
“ - pulse”
+
~ H x (t )
e-
Electron spin system
Hz
Hz
Heff
Nuclear spin system
Heff
Ik
28Si
H
31P
Hz
Ik
29Si
29Si
H
31P
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Donor electron spin in Si:P
Hyperfine modulations of an electron spin qubit
0.008

0.006
||||
0.004
0.002
0.000
0.0
0.5
1.0
t (sec)
1.5
2.0
0.5
1.0
t (sec)
1.5
2.0
0.5
1.0
t (sec)
1.5
2.0
-5
3.0x10
-5
2.5x10
-5
2.0x10
-5
1.5x10
-5
1.0x10
t
Threshold value of the magnetic field for a fault tolerant
31P electron spin qubit:
 max ~ H
2
H th  9T
S. Saikin and L. Fedichkin,
Phys. Rev. B 67, article 161302(R), 1-4 (2003)
-6
5.0x10
0.0
0.0
4.0x10
3.0x10
2.0x10
1.0x10
-5
-5
-5
-5
0.0
0.0
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Donor electron spin in Si:P
Spin echo modulations. Experiment.
Spin echo:
A()
Hx
Mx
M. Fanciulli, P. Hofer, A. Ponti
Physica B 340–342, 895 (2003)
Si-nat
0

2
t
E. Abe, K. M. Itoh, J. Isoya
S. Yamasaki, cond-mat/0402152
T = 10 K
H || [0 0 1]
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Conclusions
• Effects of nuclear spin bath on decoherence of an electron spin qubit in a Si:P
system has been studied.
• A new measure of decoherence processes has been applied.
• At low field regime coherence of a qubit exponentially decay with a characteristic
time T ~ 0.1 sec.
• At high magnetic field regime quantum operations with a qubit produce deviations
of a qubit state from ideal one. The characteristic time of these processes is T ~ 0.1
sec.
• The threshold value of an external magnetic field required for fault-tolerant
quantum computation is Hext ~ 9 Tesla.
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Prospects for future
• Spin diffusion
A. M. Tyryshkin, S. A. Lyon,
A. V. Astashkin, and A. M. Raitsimring
Phys. Rev. B 68, 193207 (2003)
• Control for spin-spin coupling in solids
• Initial drop of spin coherence
M. Fanciulli, P. Hofer, A. Ponti
Physica B 340–342, 895 (2003)
Developing of error avoiding
methods for spin qubits in solids.
S. Barrett’s Group, Yale
M. Fanciulli’s Group, MDM Laboratory, Italy
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NSF Center for Quantum Device Technology
PI V. Privman
Modeling of Quantum Coherence for Evaluation of QC Designs and Measurement Schemes
Task: Model the
environmental effects and
approximate the density
matrix
Task: Identify measures of
decoherence and establish
their approximate “additivity”
for several qubits
Task: Apply to 2DEG and
other QC designs; improve or
discard QC designs and
measurement schemes
Use perturbative
Markovian schemes
Relaxation time
scales: T1, T2, and
additivity of rates
QHE
QC
P in Si
QC
Q-dot
QC
New short-time
approximations
(De)coherence in
Transport
“Deviation”
measures of
decoherence and
their additivity
QHE
QC
P in Si
QC
Q-dot
QC
Measurement
by charge
carriers
How to
measure
spin and
charge
qubits
Coherent
spin
transport
Spin
polarization
relaxation
in devices /
spintronics
Improve and finalize solid-state QC designs once the
single-qubit measurement methodology is established
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