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