View
222
Download
2
Tags:
Embed Size (px)
Citation preview
Quantum computing and qubit decoherence
Semion Saikin
NSF Center for Quantum Device TechnologyClarkson University
2
Outline
• Quantum computation. Modeling of quantum systemsApplicationsBit & QubitEntanglementStability criteriaPhysical realization of a qubitDecoherenceMeasure of Decoherence
• Donor electron spin qubit in Si:P. Effect of nuclear spin bath.
Structure Application for Quantum computation Sources of decoherenceSpin HamiltonianHyperfine interactionEnergy 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
• Prospects for future.
• Conclusions.
3
)(2 2
22
xVxm
H
Ht
i
xE
xE
xE
nn
......11
00
Quantum computationModeling of quantum systems
R. Feynman, Inter. Jour.
Theor. Phys. 21, 467 (1982)
1 particle – n equations:
L particles – nL equations!
............
...
...
1
0
1110
0100
HH
HH
H
Quantum computation
• Modeling of quantum systems • Factorization of large integer numbers P. Shor (1994)
• Quantum search algorithm L. Grover (1995)
• Quantum Cryptography
RSA Code:Military,Banking
Process optimization:IndustryMilitary
AliceBob
Eve
Pharmaceutical industry Nanoelectronics
Applications
5
1 0
1
0
≡
≡
100
01
010
00
2221
1211
S=(Sφ Sθ SR=const)
Quantum computationBit & Qubit
• Two states classical bit
• Equalities
• Two levels quantum system (qubit)
• Single qubit operations
Polarization vector:
Density matrix:
// )0()( iHtiHt eet
6
+ =
+ ≠
≠+
Non-separablequantum states:
Quantum computationEntanglement
0000
0110
0110
0000
2
10110
2
1BABA
7
D. P. DiVincenzo, G. Burkard, D. Loss,
E. V. Sukhorukov, cond-mat/9911245
Input
Unitarytransformation
Classical control
Output
(~103 qubits)• The machine should have a collection of bits.
• 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.
• 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
(less 10-4 )
Quantum computationStability criteria
8
Physical realization of a qubit
• Ion traps and neutral atoms
E0
E1
E2
• Superconducting qubit
• Semiconductor charge qubit
• Spin qubit
Nuclear spin(liquid state NMR,solid state NMR)
I
Electron spin
S
SQUIDCooper pair box
QC Roadmaphttp://qist.lanl.gov/
Double QD
e
0 1
N pairs - 0 1N+1 pairs -
Single QD
i
E0
E1
e
Quantum computation
• Photon based QC
P
0
1
Decoherence. Interaction with macroscopic environment.
0
Quantum computation
Markov process T1 T2 concept
2~12Tte
1~2211Tte
t
Non-exponential decay
t
10
Measure of Decoherence
S ideal
S real
...,max 21
idealreal
idealreal SS ~
A. Fedorov, L. Fedichkin, V. Privman, cond-mat/0401248
2221
1211
Quantum computation
• Basis independent.
• Additive for a few qubits.
• Applicable for any timescale and complicated system dynamics.
11
Natural Silicon: 28Si – 92%29Si – 4.7% I=1/230Si – 3.1%
Si atom(group-IV)
P atom(group-V)
+ =
Natural Phosphorus: 31P – 100% I=1/2
a ≈ 25 Å
b ≈ 15 Å
22222 //)(1)( bzayxe
abF
rIn the effective mass approximation electron wave function is s-like:
Diamond crystal structure
31P electron spin (T=4.2K)T1~ min T2~ msecs
5.43Å
Donor electron spin in Si:PStructure
12
Donor electron spin in Si:P
R.Vrijen, E.Yablonovitch, K.Wang, H.W.Jiang, A.Balandin, V.Roychowdhury, T.Mor, D.DiVincenzo,Phys. Rev. A 62, 012306 (2000)
B.E.Kane, Nature 393 133 (1998)
31P donor Qubit – nuclear spinQubit-qubit inteaction – electron spin
Si1-xGex
SixGe1-x
Si
31P donor Qubit – electron spinQubit-qubit inteaction – electron spin
Bohr Radius:
Si: a ≈ 25 Å b ≈ 15 Å
Ge: a ≈ 64 Å
b ≈ 24 Å
I1I2
Qubit 1 Qubit 2
HHf
A - gate
S1 S2
HEx
J - gate
S1 S2HEx
Qubit 1 Qubit 2
Application for QC
J - gateA - gate
13
• Interaction with phonons
• Gate errors
• Interaction with 29Si nuclear spins Theory
Experiments
X.Hu, S.Das Sarma, cond-mat/0207457
D. Mozyrsky, Sh. Kogan, V. N. Gorshkov, G. P. BermanPhys. Rev. B 65, 245213 (2002)
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)
Sources of decoherenceDonor electron spin in Si:P
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)
14
31P
29Si
28Si
e-H
Effective Bohr radius ~ 20-25 ÅLattice constant = 5.43 Å
In a natural Si crystal the donor electron interacts with ~ 80 nuclei of 29Si
System of 29Si nuclear spins can be
considered as a spin bath
jiii
e jiHiHiHHH ),()()( DipHf nucl
ZZSpin
Electron spin Zeeman term:
Nuclear spin Zeeman term:
Hyperfine electron-nuclear spin interaction:
Dipole-dipole nuclear spin interaction:
HSgH e Z
HIiiH )(nulcZ
iiiH ISA)(Hf
Effect ofexternal field
Electron-nucleiinteraction
Nuclei-nucleiinteraction
jij
ijiH IDI),(Dip
Spin HamiltonianDonor electron spin in Si:P
15
e-
29Si
53Dip
))((3
rrH nene rμrμμμ
Dipole-dipole interaction:
z
y
x
zzzyzx
yzyyyx
xzxyxx
zyx
I
I
I
AAA
AAA
AAA
SSSHHf
Hyperfine interaction
Hyperfine interaction:
Contact interaction:
SIAH Cont
Donor electron spin in Si:P
Approximations:
Contact interaction only:Contact interaction High magnetic fieldHigh magnetic field
zz
yy
xx
A
A
A
00
00
00
A
zzA00
000
000
A
zzzyzx AAA
000
000
A
16
zze SgH HZ
PP
PZ H zz IH
PPPHf zzzz ISAH
1SiZH
1SiHfH …
- 31P electron spin
- 31P nuclear spin
- 29Si nuclear spin
H
Energy level structure (high magnetic field)Donor electron spin in Si:P
1715 20 25
0
5
10
15
20
1/T
[s-1
]
Magnetic field [Oe]
zeH H~Z
Oe42constHP
)(~ SiSi ISIS
PHg
Sig
Oe3constSi
S. Saikin, D. Mozyrsiky and V. Privman, Nano Lett. 2, 651-655 (2002)
Effects of nuclear spin bath (low field)
/1 t Te
zz
yy
xx
A
A
A
00
00
00
ADonor electron spin in Si:P
18
(a) S=“” (b) S=“”
31P
28Si
Hz
H
Heff
29SiIk
31P
Hz
Heff
29Si
Ik
H
Electron spin system
Nuclear spin system
Hz
e-
“ - pulse”
)(H~ tx
+
e-
zzzyzx AAA
000
000
AEffects of nuclear spin bath (high field)
Donor electron spin in Si:P
19
Hyperfine modulations of an electron spin qubit
0.0 0.5 1.0 1.5 2.00.000
0.002
0.004
0.006
0.008
t (sec)
0.0 0.5 1.0 1.5 2.00.0
5.0x10-6
1.0x10-5
1.5x10-5
2.0x10-5
2.5x10-5
3.0x10-5
t (sec)
0.0 0.5 1.0 1.5 2.00.0
1.0x10-5
2.0x10-5
3.0x10-5
4.0x10-5
t (sec)
||||
t
T9H
H~
th
2max
Threshold value of the magnetic field for a fault tolerant 31P electron spin qubit:
S. Saikin and L. Fedichkin, Phys. Rev. B 67, article 161302(R), 1-4 (2003)
Donor electron spin in Si:P
20
Spin echo modulations. Experiment.
Si-nat
M. Fanciulli, P. Hofer, A. PontiPhysica B 340–342, 895 (2003)
T = 10 KH || [0 0 1]
Spin echo:
t
Hx Mx
20
A()
Donor electron spin in Si:P
E. Abe, K. M. Itoh, J. IsoyaS. Yamasaki, cond-mat/0402152
21
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.
22
Prospects for future
• Spin diffusion • Initial drop of spin coherence
• Control for spin-spin coupling in solids
M. Fanciulli, P. Hofer, A. PontiPhysica B 340–342, 895 (2003)A. M. Tyryshkin, S. A. Lyon,
A. V. Astashkin, and A. M. RaitsimringPhys. Rev. B 68, 193207 (2003)
S. Barrett’s Group, Yale
M. Fanciulli’s Group, MDM Laboratory, Italy
Developing of error avoiding methods for spin qubits in solids.
23
NSF Center for Quantum Device Technology
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
QHE QC
New short-time approximations
(De)coherence in Transport
Relaxation time scales: T1, T2, and additivity of rates
“Deviation” measures of decoherence and their additivity
P in Si QC
Q-dot QC
Measurement by charge carriers
QHE QC
P in Si QC
Q-dot QC
How to measure spin and charge qubits
Spin polarization relaxation in devices / spintronics
Coherent spin transport
Measurement by charge carriers
Coherent spin transport
Improve and finalize solid-state QC designs once the single-qubit measurement methodology is established
PI V. Privman