Principles of superconducting (Josephson)

electronics

Valeriy V. Ryazanov

Moscow Institute of Physics and Technology,

Dolgoprudny

Institute of Solid State Physics, Russian

Academy of Sciences, Chernogolovka

February 4, Skoltech

Review of materials and devices for nano- and optoelectronics

Term 3, 2020 Tuesday Skoltech (16.00, R3-C2-2009)

February 4, Valeriy Ryazanov (Institute of Solid State Physics, Russian Academy of

Scienses, Chernogolovka). Principles of superconducting (Josephson) electronics

February 11, Nikolai Klenov (Lomonosov Moscow State University, Dukhov All-Russia

Research Institute of Automatics) Physical foundations of macroscopic quantum

effects in superconducting electronics and spintronics

February 18, Alexey Ustinov, (Karsruhe Institute of Technology, Germany, Russian

Quantum Center, MISIS). Dynamics of magnetic flux quanta

February 25, Valery Koshelets, (Kotel'nikov Institute of Radio Engineering and Electronics

RAS, Moscow) Superconducting Low-noise Terahertz Receivers

March 3. Igor Soloviev. (Lomonosov Moscow State University, Dukhov All-Russia Research

Institute of Automatics). Superconducting digital electronics

March 10/11. Jukka Pekola (Aalto University, Helsinki, Finland) Josephson current

standards, calorimetry and bolometry. (?)

March 17, 24 – Seminars...

Superconductivity

Magnetic flux quantization

Josephson junctions

SQUIDs

B

n0 C

Magnetic flux quantization in superconductors

The superconducting flux quantum was predicted by London (1948) using a phenomenological semiclasical model.

Superconductivity is a macroscopic quantum phenomenon.

= e i is the single superconducting wave function described all

condensed collective of Cooper electron pairs (2e, 2m). 2= nS / 2

p=2mvs+2eA is the gauge-invariant momentum of Cooper pairs

Meissner effect ħ =2mvs+2eA in bulk superconductor C C

ħ dl =2e A dl = B dS

2 n = (2e/ ħ) = 2 /0

= n0

Magnetic flux quantum

0 = h/ (2e) =2.067833636×10−15 Wb

(2.067833636×10−7 G cm2)

The phase running due to magnetic flux : =2 /0

Magnetic flux quanta in type II superconductors and

Josephson junctions

Is () = Iс sin Josephson equation I

2eV= ħ= ħd/dt Josephson equations II

B B

L

d

J Meissner state destruction in

type II superconductors

+

1

2

Josephson vortex

J dm; dm =d+2

I

I

V ~ dn/dt ~ ~ d /dt

=2 -1

2eV= ħ= ħd/dt

V= [ħ /(2e)]d/dt = 0 /(2)d/dt

Is () = Iс sin

I>Ic B>Bc1

-

SC II Long Jos. junction

0 1 2 3

0.5

1

I _______

Imax

/0

2Ic

Superconducting quantum interferometer

(dc-SQUID) 3 2

13+42 =3- 1 + 2- 4 =(2e/ħ)Adl + (2e/ħ) Adl

1 4

2- 1 + 3- 4 = a- b = (2e/ħ) Adl

a- b =2/0 a= 2- 1; b= 4-

2

Ia = Ic sin a; Ib = Ic sin b; I = Ic(sin a+ sinb)

sin a+ sinb=2sin[(a+ b)/2] cos[(a- b)/2]; a- b =2/0

I = 2Ic cos(/0)sin(b+ /0) a= b+2/0

Maximum of the current I is

Imax = 2Ic |cos(/0)|

3

4

1

2

I

Ib Ia

a b V

SQUID is analog

of light

interference on

double slit

Resistive-Capacitive Shunted Junction (RCSJ-model)

IJ=Ic sin - “Josephson channel”

In=V/Rn= [0/(2Rn)]t - “resistive channel”

ID=CdV/dt =[0C/(2)]tt - “capacitive channel”

Ic sin + [0/(2Rn)]t + [0C/(2)]tt = Ie |*EJ/Iс

[ħ/(2e)]2C tt + [ħ /(2e)]2Rn-1t + EJ sin = EJ (Ie/Iс)

Tunnel Josephson junction B.D. Josephson, 1962

Ie

In IJ

ID

J Rn С

V= [ħ /(2e)]d/dt

= 0 /(2)d/dt

EJ=Ic0/2

Josephson weak links

• Is () = Iс sin ; I<Iс;

• 2eV= ħ= ħd/dt ; I>Iс;

V= [ħ /(2e)]d/dt = (0/2)d/dt

J

I

2

1

=2 - 1

S S N S S I

N S

~ x

~ 2 nm ~ 1 m

a) b) c)

~

~20 nm

e) d)

S x

S

Different types of

Josepson weak links

Josephson weak link is a region of

suppressed superconductivity

F, TI…

Sensitive SQUID-magnetometer

H

H

sample

Sensitivity with feedback

better than10-6 0/Hz

0=2x10-15 Wb (V c)

/0

Imax

2Ic

1

Sensitivity to magnetic field better than 10-11 G

/0 0.5 -0.5 1.0 -1.0

Magnetic flux – voltage

dependence (for I Ic )

V

0

0

Imax= 2Iccos( /0)

Scanning SQUID-microscopy

x-y scan

jooint 10m 100m Spatial resolution ~10 m Magnetic flux resolution

~10-6 0

Different magnetic structure visualization

Sensitive SQUID-picovoltmeter

SAMPLE (Rx)

Rst

Ix

Ist

R

Rst Ist = RxIx

Sensitivity better than 10-13 V

V, 10-13 V

100 200 I, 10-6 A

SNS (SFS) junction I-V characteristics

magnetic

flux

feedback

loop

output

amplifier current

source

current feedback loop

Josephson junctions with

ferromagnetic barrier

Josephson magnetic switches

Superconducting phase inverters

Problem of superconductivity/ferromagnetism

coexistence

B is Bohr magneton, is the energy gap of superconductor,

g is Lande giromagnetic factor

F

Orbital and magnetic depairing.

Paramagnetic limit Hp of superconductivity:

S

BHp=√2(/g)

Non-uniform superconductivity by

Larkin-Ovchinnikov-Fulde-Ferrel

in magnetic superconductor

LOFF-state

Paramagnetic limit

ErRh4B4,

HoMo6S8

etc

S

N

LOFF-state was not still observed

reliably in magnetic superconductors!

S and F have antagonistic

spin ordering ! Spin-triplet superconductivity

is possible too…

Proximity effect in SF-structures

(x)=0 exp(-x/xF1) cos (x/xF2)

~ ħ/Eex (Eex >> kBT)

xF1 =1/k1~(ħD/ Eex)1/2

Buzdin, Bulaevskii, Panjukov JETP Lett. 35, 178 (1982)

Buzdin, Vujicic, Kupriyanov Sov. Phys. JETP 74, 124 (1992)

h – exchange field

(x)=0exp(-kx)=0exp(-x/xN)

Pair-breaking time ~ħ/kBT

xN ~(D)1/2~(ħD/ kBT)1/2

E ~ħ

15

Spatial oscillations

of induced superconducting order parameter in

a ferromagnet in close proximity to a superconductor

Q ≠ 0 is the center of the pair

mass momentum

Clean limit

p'2/2m–p2/2m = pFQ/m = Eex ;

Q ~ Eex / vF

Ψ(x) = Ψ0 (ei2Qx+e-i2Qx)/2=

= Ψ 0Cos(2Qx)

Ψ(x)~exp(-k1x)cos(-k2x)

k=k1+ik2

Recent fundamental observations on

superconductor/ferromagnet (SF-) structures

-Observation of the supercurrent through a

ferromagnet in superconductor-

ferromagnet-superconductor (SFS-)

junctions (LS ISSP 1999).

-Observation of nonuniform (“sign-

reversal”) superconductivity close to a

superconductor-ferromagnet (FS-)

interface, realization of Josephson SFS -

junctions with inversion of

superconducting phase. (LS ISSP 2001)

- Prediction and realization of FS-spin-

valves and spin-switches.

- Observation of superconducting long-

range spin-triplet component.

- Ferromagnetically assisted Cooper pair

splitting.

S

F

S

Possible applications

- Josephson SFS (MJJ) junction like

a novel memory element for

superconducting and other digital

electronics

- Josephson SFS -junctions like

superconducting phase invertors

for superconducting digital and

quantum (qubit) logics

- FS-spin-valves and spin-switches

- Spin-polarized supercurrents for

spintronics

Magnetic Josephson

Junction (MJJ) S

F

S

F

F S

M

ds~xs

Distributed Josephson junctions

Magnetic field penetration

(x,y) const at the junction plane. H 0

H

L>4J

w<<J

C contour: 1-2-3-4 (out of dm region, supercurrent = 0), (2-4) = dx.

ħ =2mvs+2eA Phase winding over path 1-3 + 4-2 is only due to magnetic field:

2e Adl =2e d, where d =Bdm dx

3-1+2-4 = (x+dx) - (x) = d =2 d /0d/dx=(2 /0)d/dx Magnetic induction in the junction: B(x)=(1/dm)d/dx =[0 /(2 dm)]d/dx

dm =2L+tox– “magnetic depth”

dm

dx

J >>dm ,L

3- 4 1-2 i.e. 3-1+2-4=3- 4 -1-2

Critical current vs. magnetic flux

in the short junctions (L<4J)

Supercurrent Is through the junction can change only

due to the phase difference С: sin С1

B=[0 /(2 dm)]d/dx = const (uniformly at the short junctions)

i.e. (x) is linear d/dx=(2 dm/0)B =const

(x)=(2 dm/0)B x+С; where С is an integartion constant

Supercurrent distribution: js (x)= jс sin[(2 x /av)+ С], where av= 0/(B dm) Total current over the junction: Is= jс sin[(2 x /av)+ С] dx=-(av /2) jс cos [(2 x /av)+ С]|

cos (+) = cos cos - sin sin and cos(2 x /av)cosC | = 0

Is= jсL (av / L) sin( L/av) sin С = Iс[sin( L/av) /( L/av)] sin С

-L/2

L/2 L/2

-L/2

-L/2

L/2 x L/2 -L/2 0

I

Imax/Ic

/0

1.0

IсsinZ/Z,

where

Z=( /0)

Critical current Imax:

Imax = Iсsin( L/av) /(L/av)= Iсsin( /0) /( /0)

where Iс= jсL, L/av= /0 , = BLdm

per unit widths of the

junction Fraunhofer pattern

x

Magneto-hysteretic behavior of the critical current

Si-substrate

SiO bottom Nb

Weak ferromagnetic alloys

Cu1-xNix, Pd1-xFex top Nb

(10-20 nm)

x=0.53-0.59,

TCurie=30-150 K;

Cu1-xNix,

-25 -20 -15 -10 -5 0 5 10 15

0,05

0,06

0,07

0,08

0,09

0,10

0,11

0,12

0,13

I c, m

A

H, Oe

M

H

Pd0.99Fe0.01

Cri

tic

al c

urr

en

t, m

A

Magnetic field, Oe

Very weak and soft ferromagnet

0

0max

/

)/sin(

=

cII

Pd1-xFex

x=0.01,

TCurie=15 K non-volatile switchable Magnetic

Josephson junctions (MJJs)

programmable using small field

Magnetic Josephson Memory with Fast Reading

Josephson magnetometry

Nb - PdFe – Nb (10 x 10 m2, 30 nm)

Ic = Ic0 sin(πΦ/Φ0)/(πΦ/Φ0) ,

Φ=H(dF+2λL) +4πMdF ,

Φmin=Φ0n, Φmax=Φ0(n+1/2)

Msat~10-15 Am2

SFS without a tunnel layer

Vc~3nV

Switching

rate ~ 1MHz

f~Vc/Φ0

JETP Lett (2012)

Magnetic Josephson Memory with Fast Reading

Nb

SIsFS junction

with thin s-layer

ds = 15 nm; ξS < ds << λ

SIs tunnel junction

“feels” F-layer

remagnetization

junction

Nb

tunnel layer PdFe

Si-substrate

Al/AlOx

SiO insulator

Nb wiring Top Nb electrode

Ferromagnetic layer

Nb interlayer

Bottom Nb electrode

Theoretical model of SIsFS structures

S.V. Bakurskiy, N.V. Klenov, I.I. Soloviev, V.V. Bol'ginov, V.V. Ryazanov, I.V.

Vernik, O.A. Mukhanov, M.Yu. Kupriyanov, and A.A. Golubov.

Appl. Phys. Lett. 102, 192603 (2013).

Oscillating superconductivity and Josephson -junction

Nb-Cu0.47Ni0.53-Nb

-state

I=Icsin(+ )= - Icsin()

“0”-state

I=Icsin “0”-state

I=Icsin

h – exchange field

Qx=; Q ~ h/vF

-junction

Prediction: Buzdin, Bulaevskii, Panjukov JETP Lett. (1982)

Buzdin, Vujicic, Kupriyanov Sov. Phys. JETP (1992) Realization: V.R., Oboznov, Rusanov et al, Phys. Rev. Lett. (2001)

Oboznov, Bolginov, Feofanov, V.R., and Buzdin, Phys. Rev. Lett. (2006)

Checkerboard

frustrated

Fully

frustrated

Imaging spontaneous flux in arrays of -junctions Frolov, Stoutimore, Crane, Van Harlingen, Oboznov, V.R. et al, Nature Physics 4, 32 (2008)

= =

= 2 /0

=0 /2

L 2LIc>0/2

I

2. Demonstration of SFS π-shifters (Icπ>> IcJ)

-JJ

0-JJ

A.K. Feofanov, V.A. Oboznov, V.V. Bol’ginov, J. Lisenfeld, S. Poletto, V.R., V.P.

Koshelets, P.Dmitriev, A.V. Ustinov et al, Nature Physics 6, 593 (2010).

Realization of the complementary cell-II

In collaboration with M.I. Khabipov, D.V. Balashov,

A B Zorin, PTB, Germany (2010)

Normal

RSFQ-cell

Complementary

-SFQ-cell

50 m

Physical compatibility of SFS and SIS JJs was proven

Superconducting digital electronics

Based on storage of the magnetic flux quanta 0 = h/(2e) = 2.07×10−15 Вб

Started in Russia in 1985:

RSFQ –logic (Rapid Single Flux Quantum logic )

Fast digital electronics 20 GHz - 700 GHz (Nb)

Ultra-low power, can be used for reversible computing

- JJ-based memory for Random Access Memories (RAM)

exists, but it has low density, low capacity, cannot be easily

pipelined, not energy-efficient

- Large cell sizes needed to keep the flux quantum is the

second problem of the superconducting logic

RSFQ- and -SFQ –Toggle Flip-Flop

RSFQ –Toggle

Flip-Flop

- large cell !

0 0

current bias

current bias

0

LIc >0

-SFQ –Toggle

Flip-Flop

The circuits demonstrated correct

functionality with the operation

parameter ranges of ± 20%

L 0

self-biased

in collaboration with

PTB, Braunschweig

proposed by Ustinov

& Kaplunenko

J. Appl. Phys. (2003).

Integrated SFS+SFQ circuit of T-flip-flop

in collaboration with M. Khabipov, D. Balashov, A. B. Zorin,

Physikalisch-Technische Bundesanstalt, Braunschweig, Germany

-SFQ –circuit with SFS -junction

combined Nb-AlOx-Nb, Nb-CuNi-Nb technology

in collobaration with M. Khabipov, D. Balashov, A. B. Zorin

In this cell, a large inductance required for the

fluxon storage is replaced by a -junction operated

as a passive phase shifter.

Nb-AlOx-Nb: 4x4 m2;

jc=100 A/сm2

Nb-CuNi-Nb: 8x8 m 2;

jc=500 A/сm2

Vи

нт (

отн

. ед

)

Josephson junctions in “quantum

limit”

Superconducting qubits

Nanotechnology at ISSP

shadow deposition at two angles

base pressure 10 -10 mbar

AFM image of Al/Cu/Al bridge

- Different technique of thin film

sputtering

- Optical and electron lithography

- Focused ion beam technique

- Chemical nanostructuring

- Ion and Reactive ion etching

~200 × 300 nm2

Cu deposition

Al deposition

PMMA

MMA

PMMA

MMA

SiO

SiO

SiO

lift-off

shadow evaporation at two angles

base pressure 10 -10 mbar

Al1 evaporation

Al2

evaporation

Lift-off

PMMA

MMA

Shadow evaporation technique

0.1 m

spin-injection spin-injection S(Al) S(Al)

F(Fe) N(Cu)

spin-

injection

shadow evaporation at three angles

Sample fabrication by means of electron beam lithography and shadow evaporation

35

)cos1(Ed sin2e

Idt

dt

dsinI

2eVdtIE J

0

c

t

0

c

t

0

s

-====

coup

Josephson junction energy

Is () = Iс sin Josephson equation I

2eV= ħ= ħd/dt Josephson equations II

=2 -1 ; V= [ħ /(2e)]d/dt =(2/0)d/dt

where

-4 -2 0 2 4

-8

-4

0

4

8

I< Ic

Eb Ecoup J

π2

I

2e

IE c0c

J

==

JE

E

2EJ

I=Ic

I=0.6 Ic

I=0

I

2πdt

dt

dI

2eVdtIE 0

t

0

t

0

b

===

-4 -2 - 0 2 3

]I

I)cos1[(EEEE

c

Jb --=-= coup

“Tilted washboard” potentials with

minima at satisfying I = Iс sin

(stationary states)

-

1

2

36

“Equation of motion” for Josephson tunnel junction.

Analogy with a pendulum.

Resistive-Capacitive Shunted Junction (RCSJ-model)

IJ=Ic sin - “Josephson channel”

In=V/Rn= [0/(2Rn)]t - “resistive channel”

ID=CdV/dt =[0C/(2)]tt - “capacitive channel” Ic sin + [0/(2Rn)]t + [0C/(2)]tt = Ie |*EJ/Iс

[ħ/(2e)]2C tt + [ħ /(2e)]2Rn-1t + EJ sin = EJ (Ie/Iс)

The pendulum motion equation

J tt + t + m g l sin = M

Phase angle Phase difference Moment of inertia J=m l

2 [ħ /(2e)]2C

Viscosity coefficient [ħ /(2e)]2Rn-1

Restoring moment (m g sin ) l EJ sin Applied torque moment T EJ (I/Iс)

Ie

In IJ

ID

J Rn C

m

l

(mg sin)l

mg

T

Submicron-scale tunnel junction with small enough capacitance C.

Single electron Coulomb (charging) energy EC =e2/(2C) is large.

Q is charge and Ec=Q2/(2C)=CV2/2 is energy of this capacitor.

Discharge of the capacitor is not favorable for some value Q:

new energy E'c=(Q-|e|)2/(2C) becomes larger for 0<Q<|e|/2 !

Coulomb blockade of tunneling for V: (V=Q/C)

0<V<|e|/(2C)

The increase of the differential resistance

around zero bias is called the Coulomb blockade.

Limitation on T and R:

Ec>kT (thermal fluctuations), C<10-15 F,T>1 K

1/RT+1/Re<1/RQ (quantum fluctuations)

RQ=h/ (4e2)~6 k is the quantum resistance

37

Submicron tunnel junctions in normal state

N I N

E'c-Ec>0

Q/e=

VC/e

Region of

Coulomb

blockade

Tunneling is

prohibited

E Ec E'c

Q/e

1/2 -1/2

38

Josephson junction in quantum limit

“Quantum Josephson junction” is a submicron-scale tunnel junction with small

enough capacitance C0.

By analogy with “Quantum pendulum”: when m0 null oscillations arise,

because M ~ ħ. (=0 и M=0 without oscillations!)

M= J t is angular momentum.

Josephson junction angular momentum is

M= J t =[ħ /(2e)]2Ct = [ħ /(2e)]C V=Q[ħ /(2e)]

Q ~ 2e or n ~ 1

where Q is the junction (capacitor) charge, n= Q/(2e) is excess Cooper pairs. Quantum Josephson junctions have large Coulomb energy EC ~ EJ due to

small capacitance: EC =(2e)2/(2C)

(Tunnel junctions with sizes smaller than 0.3x0.3 m2

, C~10-15F)

Thus total energy of the quantum Josephson junction is

E= (2e)2/(2C) + EJ (1- cos ) - [0/(2)]I

39

Thermal and quantum fluctuations of the critical

current

The resistive state is observed at Iс* < Iс= (2e/ħ)EJ

due to thermal activation through the barrier U0(I) The rate of the thermo-activated decay is

where (I) = p[1- (I/Iс)

2]1/4; p=(LJC)-

1/2

and U0(I)=(42/3) EJ[1- (I/Iс)]

3/2

Quantum decay for “quantum” Josephson junctions

(from ground state)

kT ħ (I), the null oscillation energy

U0(I)

(I)

T

)kT

)I(Uexp(ω(I)ω 0

T -=

)ω(I)

)I(Uexp(ω(I)aω 0

qQ

-=

L -parameter, energy of superconducting ring

with Josephson junction

Dimensionless potential

(normalized to EJ):

U(φ) = 1−cosφ + li2/2 = 1−cosφ + (φ−φe)2/2l,

where dimensionless ring inductance:

0

2

== c

L

LIl

π2

IE c0

J

=

=2 Φ/Φ0 ; e =2 Φe/Φ0

LI2/2=(Φ-Φe) 2/2L

Φ = Φe − LI L I

Φ

Φe

L >1, barriers exist

L <1, no barriers

Φe =0

Φe = Φ0/2

, i=I/Ic

Superconducting flux

qubit

ток в

кольце

по

часовой

против

часовой

0

1

Qubit states on Bloch sphere

Microwave manipulations on qubit states

Classical bit

0

Qubit

Superposition

of these two states

10 =

wave function

x

y

z

0

Bloch

sphere 1AWG

к образцу

Microwave pulses

with controlled amplitude,

duration and phase

Qubit is an artificial atom

4

1

qubit spectrum

0.45 0.5 0.550

10

20

ext

/ 0

E

frequ

en

cy,

GH

z magnetic flux

01f

0

1

Two-level atom

Qubit controlled by magnetic field

is a superconducting ring with

several Josephson junctions.

An artificial 2-level atom with

and states 0 1ħq

Ф=Ф0/2

Phase qubit: Rabi flopping

Phase qubit with integrated SFS -junction

SIS – junction and readout

dc-SQUID fabricated by

VTT, Finland

– junction integrated

afterwards in

Chernogolovka

-JJ

0-JJ

Here the -junction

always remains at the

zero-voltage state.

in collobaration with Alexey Ustinov´s group, Karlsruhe Inst. of Technology,

Germany

Experimental data: Rabi oscillations

phase qubit with a -junction conventional phase qubit

0/ 2

Flux qubit with SFS -junction

Ip

/

4

A.V. Shcherbakova et al, Supercond. Sci. Tech. 28, 025009 (2015).

- the resonator dispersive

shift due to coupling to a

qubit

28

© Martinis lab, 2015

line

capacitance

shunted a

Josephson

junction

емкостью ~ 20

фФ

resonator

qubit 100 мкм

Csh~20 fF; =0.52;

jc=0.5 кА/сm2; Ic=0.4

А

T2=1.3 мкс

Проект «Лиман» - 1 этап Flux qubit shunted by large capacitance

3D-transmons

The coherence

time

T2=5 μs

3D-transmon

in cavity

resonator

Devoret &

Schoelkopf,

Science 339,

1169 (2013)

PLANAR QUBITS (XMONS)

The coherence

time

T1=11.5 μs

The coherence time T2=7 μs

delay of readout (μs) delay of readout (μs)

delay of readout

MULTIQUBIT SPECTRA (2-XMON-QUBIT SYSTEM)

Q1 freq “z” Q1“xy”

Q2 freq “z”

Q2

“xy”

Q3 freq

“z”

Q3 “xy”

ro

f = 6.85 GHz

Q = 6000

f = 7.15 GHz

Q = 6000

f = 7.00 GHz

Q = 6000

1-qubit spectrum

𝐻int = 𝐽𝜎 𝑥1𝜎 𝑥

2

J ~20-30 MHz

2𝐽

The coherence time for each qubit T2 ~ 3 мкс

THREE-QUBIT SYSTEM (2 XMONS AND ‘COUPLER’)

2-qubit spectrum

coupling

energy

РЕЗУЛЬТАТЫGrover search algorithm

initialization Oracle

РЕЗУЛЬТАТЫ РЕАЛИЗАЦИИ АЛГОРИТМА

ГРОВЕРА

1

3

The blue columns is statistics of the states at 10000 launches for sequential storage

in each of the "cells“:

|0>=|00>, |1>=|01>, |2>=|10> и |3>=|11>.

The red columns is a hypothetical result for an ideal (error-free) quantum computer.

Еhe black dotted line is a 50% probability (the limit for a classic computer with a

single access to the “Oracle”). Measured probabilities of correct state search: |00> –

58%, |01> – 57%, |10> – 53%, I / 11> - 57%.

Grover search algorithm realization

search of |0> state search of |1> state

search of |2> state search of |3> state