1. Quantum computing and qubit decoherence
Semion Saikin
NSF Center for Quantum Device Technology
Clarkson University

2. 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.

3. Quantum computation Modeling of quantum systems
1 particle – n equations:
L particles – nL equations!
R. Feynman, Inter. Jour.
Theor. Phys. 21, 467 (1982)

4. Quantum computation Applications Modeling of quantum systems
Factorization of large integer numbers
P. Shor (1994)
Pharmaceutical industry
Nanoelectronics
RSA Code:
Military,
Banking
Quantum search algorithm
L. Grover (1995)
Quantum Cryptography
Alice
Bob
Eve
Process optimization:
Industry
Military

5. 1 1 ≡ ≡ Quantum computation Bit & Qubit Two states classical bit
Two levels quantum system (qubit)
Polarization vector: 1
S=(Sφ Sθ SR=const)
Density matrix:
Equalities
Single qubit operations 1 ≡ ≡

6. Quantum computation Entanglement + = ≠ + ≠ + Non-separable
quantum states:

7. 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. O u t p I n p u t
Unitary
transformation
D. P. DiVincenzo, G. Burkard, D. Loss,
E. V. Sukhorukov, cond-mat/
Classical control

8. Physical realization of a qubit
Quantum computation
QC Roadmap
Physical realization of a qubit
Ion traps and neutral atoms
Semiconductor charge qubit
Single QD
Double QD E0 E1 E2 e e
Photon based QC E1 E0 P
Spin qubit
Superconducting qubit
Cooper pair box
Nuclear spin
(liquid state NMR,
solid state NMR)
Electron spin
SQUID  S i I
N pairs -
N+1 pairs -

9. Decoherence. Interaction with macroscopic environment.
Quantum computation
Decoherence. Interaction with macroscopic environment.
Markov process
T1 T2 concept
Non-exponential decay t t

10. Measure of Decoherence
Quantum computation
Measure of Decoherence
Basis independent.
Additive for a few qubits.
Applicable for any timescale and
complicated system dynamics.
S ideal
S real
A. Fedorov, L. Fedichkin,
V. Privman, cond-mat/

11. Donor electron spin in Si:P
Structure
Si atom
(group-IV)
Diamond crystal structure
Natural Silicon:
28Si – 92%
29Si – 4.7% I=1/2
30Si – 3.1%
5.43Å
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:

12. Donor electron spin in Si:P
Application for QC
Bohr Radius:
Si: a ≈ 25 Å
b ≈ 15 Å
Ge: a ≈ 64 Å
b ≈ 24 Å
Si1-xGex
SixGe1-x
A - gate
J - gate Si
B.E.Kane, Nature (1998)
R.Vrijen, E.Yablonovitch, K.Wang, H.W.Jiang,
A.Balandin, V.Roychowdhury, T.Mor, D.DiVincenzo,
Phys. Rev. A 62, (2000)
31P donor
Qubit – nuclear spin
Qubit-qubit inteaction – electron spin
31P donor
Qubit – electron spin
Qubit-qubit inteaction – electron spin
HEx
J - gate S1 S2
HHf
A - gate S1
HEx S2 I1 I2
Qubit 1
Qubit 2
Qubit 1
Qubit 2

13. Donor electron spin in Si:P
Sources of decoherence
Interaction with phonons
Gate errors
Interaction with 29Si nuclear spins
Theory
Experiments
D. Mozyrsky, Sh. Kogan, V. N. Gorshkov, G. P. Berman Phys. Rev. B 65, (2002)
X.Hu, S.Das Sarma, cond-mat/
I.A.Merkulov, Al.L.Efros, M.Rosen, Phys. Rev. B 65, (2002)
S.Saikin, D.Mozyrsky, V.Privman, Nano Letters 2, 651 (2002)
R. De Sousa, S.Das Sarma, Phys. Rev. B 68, (2003)
S.Saikin, L. Fedichkin, Phys. Rev. B 67, (R) (2003)
J.Schliemann, A.Khaetskii, D.Loss, J. Phys., Condens. Matter 15, R1809 (2003)
A. M. Tyryshkin, S. A. Lyon, A. V. Astashkin, and A. M. Raitsimring, Phys. Rev. B 68, (2003)
M. Fanciulli, P. Hofer, A. Ponti, Physica B 340–342, 895 (2003)
E. Abe, K. M. Itoh, J. Isoya S. Yamasaki, cond-mat/ (2004)

14. Donor electron spin in Si:P
Spin Hamiltonian
28Si H e-
31P
Effect of
external field
Electron-
nuclei
interaction
Nuclei-
nuclei
interaction
29Si
Electron spin Zeeman term:
Effective Bohr radius ~ Å
Lattice 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
Nuclear spin Zeeman term:
Hyperfine electron-nuclear spin interaction:
Dipole-dipole nuclear spin interaction:

15. Donor electron spin in Si:P
Hyperfine interaction
Contact interaction:
Dipole-dipole interaction: e-
29Si
Hyperfine interaction:
Approximations:
Contact interaction
High magnetic field
Contact interaction only:
High magnetic field

16. Donor electron spin in Si:P
Energy level structure (high magnetic field) H
- 31P electron spin
- 31P nuclear spin
- 29Si nuclear spin …

17. Donor electron spin in Si:P
Effects of nuclear spin bath (low field)
S. Saikin, D. Mozyrsiky and V. Privman,
Nano Lett. 2, (2002)

18. Donor electron spin in Si:P
Effects of nuclear spin bath (high field)
(a) S=“”
(b) S=“” e-
“ - pulse” e- + Hz
Electron spin system
31P Hz
Heff
29Si Ik H
Nuclear spin system
31P
28Si Hz H
Heff
29Si Ik

19. Donor electron spin in Si:P
Hyperfine modulations of an electron spin qubit
|||| t
Threshold value of the magnetic field for a fault tolerant 31P electron spin qubit:
S. Saikin and L. Fedichkin,
Phys. Rev. B 67, article (R), 1-4 (2003)

20. Donor electron spin in Si:P
Spin echo modulations. Experiment.
Spin echo: t Hx Mx  2
A()
M. Fanciulli, P. Hofer, A. Ponti
Physica B 340–342, 895 (2003)
Si-nat
T = 10 K
H || [0 0 1]
E. Abe, K. M. Itoh, J. Isoya
S. Yamasaki, cond-mat/

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. Developing of error avoiding methods for spin qubits in solids.
Prospects for future
Spin diffusion
Initial drop of spin coherence
M. Fanciulli, P. Hofer, A. Ponti
Physica B 340–342, 895 (2003)
A. M. Tyryshkin, S. A. Lyon,
A. V. Astashkin, and A. M. Raitsimring Phys. Rev. B 68, (2003)
Developing of error avoiding
methods for spin qubits in solids.
Control for spin-spin coupling in solids
S. Barrett’s Group, Yale
M. Fanciulli’s Group, MDM Laboratory, Italy

23. 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
Use perturbative Markovian schemes
New short-time approximations
(De)coherence in Transport
“Deviation” measures of decoherence and their additivity
Measurement by charge carriers
Measurement by charge carriers
Coherent spin transport
Coherent spin transport
Task: Identify measures of decoherence and establish their approximate “additivity” for several qubits
Relaxation time scales: T1, T2, and additivity of rates
How to measure spin and charge qubits
Spin polarization relaxation in devices / spintronics
Task: Apply to 2DEG and other QC designs; improve or discard QC designs and measurement schemes
QHE QC
P in Si QC
Q-dot QC
QHE QC
P in Si QC
Q-dot QC
Improve and finalize solid-state QC designs once the single-qubit measurement methodology is established