Quantum computing and qubit decoherence

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 SiP. 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
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)

• Quantum search algorithm
• L. Grover (1995)

• Quantum Cryptography

Process optimization Industry Military
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Quantum computation
Bit Qubit

• Two states classical bit

• Two levels quantum system (qubit)

Polarization vector
S(Sf S? SRconst)
Density matrix

• Equalities

• Single qubit operations

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Quantum computation
Entanglement


Non-separable quantum states
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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.

O u t p u t
Unitary transformation
I n 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
e
e
E1

• Photon based QC

E0
P

• Spin qubit

• Superconducting qubit

Nuclear spin (liquid state NMR, solid state NMR)
Electron spin
Cooper pair box
SQUID
?
I
N pairs -
N1 pairs -
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Quantum computation
Decoherence. Interaction with macroscopic
environment.
Markov process T1 T2 concept
Non-exponential decay
t
t
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Quantum computation
Measure of Decoherence

• Basis independent.

• Additive for a few qubits.

• Applicable for any timescale and
• complicated system dynamics.

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Donor electron spin in SiP
Structure
Si atom (group-IV)
Diamond crystal structure
Natural Silicon 28Si 92 29Si 4.7
I1/2 30Si 3.1
5.43Å
31P electron spin (T4.2K) T1 min T2 msecs
P atom (group-V)
b 15 Å


a 25 Å
Natural Phosphorus 31P 100 I1/2
In the effective mass approximation electron wave
function is s-like
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Donor electron spin in SiP
Application for QC
Bohr Radius Si a 25 Å b 15 Å Ge a
64 Å b 24 Å
J - gate
A - gate
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 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
S2
HEx
I2
I1
Qubit 1
Qubit 2
Qubit 1
Qubit 2
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Donor electron spin in SiP
Sources of decoherence

• Interaction with phonons
• Gate errors
• Interaction with 29Si nuclear spins
• Theory
• Experiments

D. Mozyrsky, Sh. Kogan, V. N. Gorshkov, G. P.
BermanPhys. Rev. B 65, 245213 (2002)
X.Hu, S.Das Sarma, cond-mat/0207457
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)
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 340342, 895 (2003) E. Abe, K. M. Itoh, J.
Isoya S. Yamasaki, cond-mat/0402152 (2004)
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Donor electron spin in SiP
Spin Hamiltonian
28Si
H
31P
e-
Effect of external field
Electron- nuclei interaction
Nuclei- nuclei interaction
29Si
Electron spin Zeeman term
Effective Bohr radius 20-25 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
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Donor electron spin in SiP
Hyperfine interaction
Dipole-dipole interaction
Contact interaction
Hyperfine interaction
Approximations
Contact interaction High magnetic field
Contact interaction only
High magnetic field
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Donor electron spin in SiP
Energy level structure (high magnetic field)
H

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Donor electron spin in SiP
Effects of nuclear spin bath (low field)
S. Saikin, D. Mozyrsiky and V. Privman, Nano
Lett. 2, 651-655 (2002)
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Donor electron spin in SiP
Effects of nuclear spin bath (high field)
(a) S?
(b) S?

Electron spin system
Hz
Nuclear spin system
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Donor electron spin in SiP
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 161302(R), 1-4 (2003)
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Donor electron spin in SiP
Spin echo modulations. Experiment.
M. Fanciulli, P. Hofer, A. Ponti Physica B
340342, 895 (2003)
Si-nat
T 10 K H 0 0 1
E. Abe, K. M. Itoh, J. Isoya S. Yamasaki,
cond-mat/0402152
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Conclusions

• Effects of nuclear spin bath on decoherence of
  an electron spin qubit in a SiP 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

• Initial drop of spin coherence

M. Fanciulli, P. Hofer, A. Ponti Physica B
340342, 895 (2003)
A. M. Tyryshkin, S. A. Lyon, A. V. Astashkin,
and A. M. RaitsimringPhys. Rev. B 68, 193207
(2003)
Developing of error avoiding methods for spin
qubits in solids.

• Control for spin-spin coupling in solids

S. Barretts Group, Yale M. Fanciullis 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
Use perturbative Markovian schemes
New short-time approximations
(De)coherence in Transport
Deviation measures of decoherence and their
additivity
Measurement by charge carriers
Coherent spin transport
Measurement by charge carriers
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