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Quantum computing and qubit decoherence

Semion Saikin

NSF Center for Quantum Device Technology

Clarkson University

- 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.

L particles – nL equations!

Quantum computation

Modeling of quantum systems

R. Feynman, Inter. Jour.

Theor. Phys. 21, 467 (1982)

RSA Code:

Military,

Banking

Alice

Pharmaceutical industry

Nanoelectronics

Eve

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

0

1

0

Quantum computation

Bit & Qubit

- Two states classical bit

- Two levels quantum system (qubit)

Polarization vector:

S=(SφSθ SR=const)

Density matrix:

- Equalities

- Single qubit operations

≡

≡

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

E1

E0

S

i

Quantum computation

QC Roadmap

http://qist.lanl.gov/

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

Cooper pair box

Nuclear spin

(liquid state NMR,

solid state NMR)

Electron spin

SQUID

I

N pairs -

N+1 pairs -

Quantum computation

Decoherence. Interaction with macroscopic environment.

Markov process

T1 T2 concept

Non-exponential decay

t

t

Sreal

A. Fedorov, L. Fedichkin,

V. Privman, cond-mat/0401248

Quantum computation

Measure of Decoherence

- Basis independent.

- Additive for a few qubits.

- Applicable for any timescale and
- complicated system dynamics.

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:

SixGe1-x

Si1-xGex

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

A - gate

J - 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

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 340–342, 895 (2003)

E. Abe, K. M. Itoh, J. Isoya S. Yamasaki, cond-mat/0402152 (2004)

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:

29Si

Donor electron spin in Si:P

Hyperfine interaction

Contact interaction:

Dipole-dipole interaction:

Hyperfine interaction:

Approximations:

Contact interaction

High magnetic field

Contact interaction only:

High magnetic field

- 31P nuclear spin

- 29Si nuclear spin

Donor electron spin in Si:P

Energy level structure (high magnetic field)

H

…

Effects of nuclear spin bath (low field)

S. Saikin, D. Mozyrsiky and V. Privman,

Nano Lett.2, 651-655 (2002)

“ - pulse”

e-

Hz

Heff

Heff

Hz

Ik

Ik

29Si

29Si

H

28Si

H

31P

31P

Donor electron spin in Si:P

Effects of nuclear spin bath (high field)

(a)S=“”

(b)S=“”

+

Electron spin system

Hz

Nuclear spin system

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. B67, article 161302(R), 1-4 (2003)

A()

Hx

Mx

t

2

0

Donor electron spin in Si:P

Spin echo modulations. Experiment.

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/0402152

- 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.

- 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. RaitsimringPhys. Rev. B 68, 193207 (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

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

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