Semiconductor Qubits for Quantum Computation

7/27/2019 Semiconductor Qubits for Quantum Computation

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Semiconductor qubits forquantum computation

Matthias FehrTU Munich

JASS 2005

St.Petersburg/Russia

Is it possible to realize a quantum

computer with semiconductor technology?


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Contents

1. History of quantum computation

2. Basics of quantum computationi. Qubits

ii. Quantum gates

iii. Quantum algorithms

3. Requirements for realizing a quantum computer

4. Proposals for semiconductor implementationi. Kane concept: Si:31Pii. Si-Ge heterostructure

iii. Quantum Dots (2D-electron gas, self-assembly)

iv. NV-

center in diamond


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The history of quantum computation

1936 Alan Turing

1982 Feynman

1985 Deutsch

Church-Turing thesis:

There is a Universal Turing machine, that can efficiently

simulate any other algorithm on any physical device

Computer based on quantum mechanics mightavoid problems in simulating quantum mech.systems

Search for a computational device to simulate anarbitrary physical system

quantum mechanics -> quantum computer

Efficient solution of algorithms on a quantum computerwith no efficient solution on a Turing machine?


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1994 Peter Shor

1995 Lov Grover

In the 1990s

1995 Schumacher

1996 Calderbank,Shor, Steane

Efficient quantum algorithms

- prime factorization

- discrete logarithm problem->more power

Efficient quantum search algorithm

Efficient simulation of quantum mechanical systems

Quantum bit or qubit as physical resource

Quantum error correction codes

- protecting quantum states against noise

The history ofquantum computation


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The basics of quantum computation

Classical bit: 0 or 1

2 possible values

Qubit:

are complex -> infinite possiblevalues -> continuum of states

10 +=

,

Qubit measurement: result 0 with probability

result 1 with probability

Wave function collapses during measurement,

qubit will remain in the measured state

2

2

122

=+

0010 %1002

+=


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Qubits

Bloch sphere:

We can rewrite our state with

phase factors

Qubit realizations: 2 level systems

1) ground- and excited states of electron orbits

2) photon polarizations

3) alignment of nuclear spin in magnetic field4) electron spin

+= 1

2sin0

2cos

iiee

,,

Bloch sphere [from Nielson&Chuang]


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Single qubit gates

Qubits are a possibility to storeinformation quantum mechanically

Now we need operations toperform calculations with qubits

-> quantum gates:

NOT gate:classical NOT gate: 0 -> 1; 1 -> 0

quantum NOT gate:

Linear mapping -> matrixoperation

Equal to the Pauli spin-matrix

0110 ++

xX =

01

10

=

+

X

10

X

X


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Single qubit gates

Every single qubit operation can bewritten as a matrix U

Due to the normalization condition everygate operation U has to be unitary

-> Every unitary matrix specifies a valid

quantum gate

Only 1 classical gate on 1 bit, but

quantum gates on 1 qubit.

Z-Gate leaves unchanged, and flipsthe sign of

Hadamard gate = square root of NOT

IUU =*

=

=

11

11

2

1

10

01

H

Z

0

11


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Hadamard gate

Bloch sphere:

- Rotation about the y-axis by 90

- Reflection through the x-y-plane

Creating a superposition

2

10

2

1010

+

++ H

Bloch sphere [from Nielson&Chuang]

2101

2

100

+

H

H


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Decomposing single qubit operations

An arbitrary unitary matrix can bedecomposed as a product of rotations

1st

and 3rd

matrix: rotations about the z-axis

2nd matrix: normal rotation

Arbitrary single qubit operations with a finiteset of quantum gates

Universal gates

=

2/

2/

2/

2/

0

0

2/cos2/sin

2/sin2/cos

0

0

i

i

i

i

i

e

e

e

eeU


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Multiple qubits

For quantum computation multiple qubits are needed!

2 qubit system:

computational bases stats:superposition:

Measuring a subset of the qubits:

Measurement of the 1st qubit gives 0 with probability

leaving the state

11,10,01,00

11100100 11100100 +++=

1st qubit 2nd qubit

2

01

2

00 +

2

01

2

00

0100 0100`

+

+=


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Entanglement

Bell state or EPR pair:

prepare a state:

Measuring the 1st qubit gives

0 with prop. 50% leaving

1 with prop. 50% leaving

The measurement of the 2nd qubits alwaysgives the same result as the first qubit!

The measurement outcomes are correlated!

Non-locality of quantum mechanics

Entanglement means that state can not bewritten as a product state

11=

2

1100 +=

00=

2

1100

2

11011000

2

10

2

1021

+

+++

=

+

+==


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Multiple qubit gates, CNOT

Classical: AND, OR, XOR, NAND, NOR -> NAND is universal

Quantum gates: NOT, CNOT

CNOT gate:

- controlled NOT gate = classical XOR

- If the control qubit is set to 0, target qubit is the same

- If the control qubit is set to 1, target qubit is flipped

CNOT is universalfor quantum computation

Any multiple qubit logic gate may be composed from CNOT and single qubit gates

Unitary operations are reversible

(unitary matrices are invertible, unitary -> too )

Quantum gate are always reversible, classical gates are not reversible

2mod,,

1011;1110;0101;0000

ABABA

=

11

10

01

00

0100

10000010

0001

CNU

U 1U


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Qubit copying?

classical: CNOT copies bits

Quantum mech.: impossible

We try to copy an unknown state

Target qubit:

Full state:

Application of CNOT gate:

We have successful copied , but only in the case

General state

No-cloningtheorem: major difference between quantum andclassical information

10 ba +=

0

[ ] 1000010 baba +=+

=+ 1100 ba

10 or=

11100100

22

bababa+++=


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Quantum parallelism

Evaluation of a function:

Unitary map: black box

Resulting state:

Information on f(0) and f(1) witha single operation

Not immediately useful, becauseduring measurement thesuperposition will collapse

}1,0{}1,0{:)( xf

)(,,: xfyxyxUf

2)1(,1)0(,0 ff

+=

)1(,1

)0(,0

f

f

Quantum gate [from Nielson&Chuang (2)]


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Deutsch algorithm

Input state:

Application of :

010 =

( ) 2/10210

2

101 =

+

= x

( ) 2/10)1( )(2 = xxf

fU


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Deutsch algorithm

)1()0(1)1()0(

)1()0(0)1()0(

ffifff

ffifff

===

=

2

10)1()0(3 ff

global property determined withone evaluation of f(x)

classically: 2 evaluations needed

Faster than any classical device

Classically 2 alternatives excludeone another

In quantum mech.: interference


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Quantum algorithms

22 )(log nN =n

n

N

nNN

2

2)log(

=

=

Classical steps quantum logic steps

Fourier transform

e.g.:- Shors prime factorization

- discrete logarithm problem

- Deutsch Jozsa algorithm

- n qubits

- N numbers

- hidden information!

- Wave function collapseprevents us from directlyaccessing the information

Search algorithms

Quantum simulation cN bits kn qubits

N N


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The Five Commandments of DiVincenzo

1. A physical system containing qubits is needed

2. The ability to initialize the qubit state

3. Long decoherence times, longer than the gate

operation time Decoherence time: 104-105 x clock time

Then error-correction can be successful

4. A universal set of quantum gates (CNOT)

5. Qubit read-out measurement

...000


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Realization of a quantum computer

Systems have to be almostcompletely isolated from their

environment

The coherent quantum state hasto be preserved

Completely preventingdecoherence is impossible

Due to the discovery of quantumerror-correcting codes, slightdecoherence is tolerable

Decoherence times have to bevery long -> implementation

realizable

Performing operations on severalqubits in parallel

2- Level system as qubit:

Spin particles

Nuclear spins

Read-out:

Measuring the single spin states

Bulk spin resonance


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Si:31P, Kane concept from 1998

Logical operations on nuclear spinsof31P(I=1/2) donors in a Si host(I=0)

Weakly bound 31P valence electron atT=100mK

Spin degeneracy is broken by B-field Electrons will only occupy the lowest

energy state when

Spin polarization by a strong B-field

and low temperature

Long 31P spin relaxation time ,

due to low T

s1810

TkBB >>2


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Single spin rotations

Hyperfine interaction at the nucleus

: frequency separation of the nuclear levels

A-gate voltage pulls the electron wave functionenvelope away from the donors

Precession frequencyof nuclear spins iscontrollable

2nd magnetic field Bac in resonance to thechanged precession frequency

Selectively addressing qubits

Arbitrary spin rotations on each nuclear spin

2A

B

AABgh

ABgBH

B

nnA

nen

znn

e

zBen

2222 ++=

+=

A

h2/acBeff Bg

=


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Qubit coupling

J-gates influence the neighboringelectrons -> qubit coupling

Strength of exchange couplingdepends on the overlap of thewave function

Donor separation: 100-200

Electrons mediate nuclear spininteractions, and facilitatemeasurement of nuclear spins

o

A

+++=

BBB

eenen

a

r

a

r

a

erJ

JAABHH

2exp6.1)(4

)(2/5

2

1222211

e2


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Qubit measurement

J < BB/2: qubit operation

J > BB/2: qubit measurement

Orientation of nuclear spin 1 alonedetermines if the system evolvesinto singlet or triplet state

Both electrons bound to samedonor (D- state, singlet)

Charge motion between donors

Single-electron capacitancemeasurement

Particles are indistinguishable


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Many problems

Materials free of spin(isotopes)

Ordered 1D or 2D-donor array

Single atom doping methodes

Grow high-quality Si layers onarray surface

100-A-scale gate devices

Every transistor is individual ->large scale calibration

A-gate voltage increases theelectron-tunneling probability

Problems with low temperatureenvironment

Dissipation through gate biasing

Eddy currents by Bac

Spins not fully polarized

0I


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SRT with Si-Ge heterostructures

Spin resonance transistors, at a size of2000 A

Larger Bohr radius (larger )

Done by electron beam lithography

Electron spin as qubit

Isotropic purity not critical

No needed spin transfer betweennucleus and electrons

Different g-factors

Si: g=1.998 / Ge: g=1.563 Spin Zeeman energy changes

Gate bias pulls wave function awayfrom donor

*,m


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Confinement and spin rotations

Confinement through B-layer

RF-field in resonance with SRT -> arbitrary spin phase change


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2-qubit interaction

No J-gate needed

Both wave functions are pullednear the B-layer

Coulomb potential weakens

Larger Bohr radius

Overlap can be tuned

CNOT gate


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Detection of spin resonance

FET channel:

n-Si0.4Ge0.6 ground plane counter-electrode

Qubit between FET channel andgate electrode

Channel current is sensitive to

donor charge states: ionized / neutral /

doubly occupied (D- state)

D- state (D+ state) on neighbortransistors, change in channelcurrent -> Singlet state

Channel current constant -> tripletstate


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Electro-statically defined QD

GaAs/AlGaAsheterostructure -> 2DEG

address qubits with

high-g layer

gradient B-field

Qubit coupling by loweringthe tunnel barrier


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Single spin read-out in QD

Spin-to-charge conversion of electronconfined in QD (circle)

Magnetic field to split states

GaAs/AlGaAs heterostructure -> 2DEG

Dots defined by gates M, R, T

Potential minimum at the center

Electron will leave when spin-

Electron will stay when spin-

QPC as charge detector

Electron tunneling between reservoirand dot

Changes in QQPC detected bymeasuring IQPC


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Two-level pulse on P-gate


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Self-assembled QD-molecule

Coupled InAs quantum dots quantum molecule

Vertical electric field localizescarriers

Upper dot = index 0

Lower dot = index 1

Optical created exciton

Electric field off -> tunneling ->entangled state


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Self-assembled Quantum Dots array

Single QD layer

Optical resonant excitation of e-hpairs

Electric field forces the holes intothe GaAs buffer

Single electrons in the QD ground

state (remains for hours, at low T)

Vread: holes drift back andrecombine

Large B-field: Zeeman splitting ofexciton levels


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Self-assembled Quantum Dots array

Circularly polarized photons

Mixed states

Zeeman splitting yields either

Optical selection of pure spinstates

=

+=

eJ

eJ

ze

ze

h

h

2/1

2/1

,

,

=

+=

hJ

hJ

zh

zh

h

h

2/3

2/3

,

,

=+=

+

+

heandhe

JJJ zhzez h

h

h

1

1

1

,,

heorhe

h

h

1

1

+=

=

Jhe

Jhe


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NV- center in diamond

Nitrogen Vacancy center: defect indiamond, N-impurity

3A -> 3E transition: spin conserving

3E -> 1A transition: spin flip

Spin polarization of the ground state Axial symmetry -> ground state

splitting at zero field

B-field for Zeeman splitting of tripletground state

Low temperature spectroscopy


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NV- center in diamond

Fluorescence excitation with laser

Ground state energy splittinggreater than transition line with

Excitation line marks spinconfiguration of defect center

On resonant excitation:

Excitation-emission cycles 3A -> 3E

bright intervals, bursts

Crossing to 1A singlet small

No resonance

Dark intervals in fluorescence


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Thank you very much!


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References

1. Kane, B.E. A silicon-based nuclear spin quantum computer, Nature 393, 133-137 (1998)

2. Nielson & Chuang, Quantum Computation and Quantum Information, Cambridge University Press (2000)

3. Loss, D. et al. Spintronics and Quantum Dots, Fortschr. Phys. 48, 965-986, (2000)

4. Knouwenhoven, L.P. Single-shot read-out of an individual electron spin in a quantum dot, Nature 430, 431-435(2004)

5. Forchel, A. et al., Coupling and Entangling of Quantum States in Quantum Dot Molecules, Science 291, 451-453(2001)

6. Finley, J. J. et al., Optically programmable electron spin memory using semiconductor quantum dots, Nature432, 81-84 (2004)

7. Wrachtrup, J. et al., Single spin states in a defect center resolved by optical spectroscopy, Appl. Phys. Lett. 81(2002)

8. Doering, P. J. et al. Single-Qubit Operations with the Nitrogen-Vacancy Center in Diamond, phys. stat. sol. (b)233, No. 3, 416-426 (2002)

9. DiVincenzo, D. et al., Electron Spin Resonance Transistors for Quantum Computing in Silicon-GermaniumHeterostructures, arXiv:quant-ph/9905096 (1999)

10. DiVincenzo, D. P., The Physical Implementation of Quantum Computation, Fortschr. Phys. 48 (2000) 9-11, 771-783

Semiconductor Qubits for Quantum Computation

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