quantum studies in iisc

8/11/2019 quantum studies in IISC

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Quantum Computation and Quantum InformationProcessing by NMR:

Introduction and Recent Developments.

Anil KumarCentre for Quantum Information and Quantum Computing (CQIQC)

Department of Physics and NMR Research Centre Indian Institute of Science, Bangalore

IIT-Madras 7-Nov-2012


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Introduction to QC/QIP


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What is Special about Quantum Systems?

Classicalbit

0 1

Qubit

c1|0 +c2|1

Coherent Superposition

New Possibilities

EPR States: 2(-1/2) ( 00 + 11 ) NOTresolvable into tensor product of

individual particles 0 1+ 1 1)0 2+ 1 2)

Entanglement

Teleportation Quantum

Computation

|0 |1

All present day computers (classical computers) use binary (0,1)logic, and all computations follow this Yes/No answer.

Feynman (1982) suggested that it might be possible to simulate theevolution of quantum systems efficiently, using a quantum simulator


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Quantum Algorithms 1. PRIME FACTORIZATION

10 10years (Age of the Universe)

Classically :exp [2(ln c) 1/3(ln ln c) 2/3 ]

400 digit

Shor s algorithm : (1994)(ln c) 3

3 years

2. SEARCHING UNSORTED DATA -BASE Classically : N/2 operationsGrover s Search Algorithm : (1997) N operations

3.DISTINGUISH CONSTANT AND BALANCED FUNCTIONS:

Classically : ( 2 N-1 + 1) stepsDeutsch-Jozsa(DJ) Algorithm : (1992) . 1 step

4. Quantum Algorithm for Linear System of Equation:Harrow, Hassidim and Seth Lloyd; Phys. Rev. Letters, 103, 150502 (2009).

Exponential speed-up


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5. Simulating a Molecule: Using Aspuru-Guzik AdiabaticAlgorithm

(i) J.Du, et. al, Phys. Rev. letters 104, 030502 (2010).

Used a 2-qubit NMR System ( 13CHCl 3 ) to calculated theground state energy of Hydrogen Molecule up to 45 bit accuracy.

(ii) Lanyon et. al, Nature Chemistry 2, 106 (2010).

Used Photonic system to calculate the energies of the groundand a few excited states up to 20 bit precision.

Recent Developments


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Experimental Techniques for Quantum Computation:

1. Trapped Ions

4. Quantum Dots

3. Cavity QuantumElectrodynamics (QED)

6. NMR

Ion Trap:Ion Trap:Linear Paul-Trap

~ 16 MHz

1kV

1kV

: . . . _ _ .

Laser Cooled

IonsT ~ mK

7. Josephson junction qubits

8. Fullerence based ESR quantum computer

5. Cold Atoms

2. Polarized Photons


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Ion Trap:Ion Trap:Linear Paul-Trap

~ 16 MHz

1kV

1kV

: - . . . _ _ .

Laser

CooledIons

T ~ mK

Nobel Prize in Physics 2012

Trapped Ions Cavity Quantum

Electrodynamics (QED)

Haroche has been one of the pioneersin the field of cavity quantum

electrodynamics, He observed asingle atom interacting with a few

photons inside a reflective cavity andcan keep a photon bouncing back andforth in a centimeter-sized cavity

billions of times before it escapes

Wineland performed quantum-probingexperiments with trapped ions. The tightconfinement of ions in these electric fieldtraps causes ion motion to be restricted todistinct quantum states, each of whichrepresents a different frequency of

bouncing back-and-forth between the

electric field walls. Michael Schirber

David Wineland Serge Haroche


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0

1. Nuclear spins have small magneticmoments (I) and behave as tinyquantum magnets.

2. When placed in a large magneticfield B 0 , they oriented either along

the field (|0 state) or opposite to thefield (|1 state) .

4. Spins are coupled to other spins by indirect spin-spin (J) coupling, andcontrolled (C-NOT) operations can be performed using J-coupling.

Multi-qubit gates

Nuclear Magnetic Resonance (NMR)

3. A transverse radiofrequency field (B 1) tuned at the Larmor frequency ofspins can cause transition from |0 to |1 (NOT Gate by a 180 0 pulse).Or put them in coherent superposition (Hadamard Gate by a 90 0 pulse).

Single qubit gates.

SPINS ARE QUBITS

B1


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DSX300

AMX400

AV500

AV700

DRX500

NMR Research Centre, IISc1 PPB

Field/Frequencystability

= 1:109


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Why NMR?

> A major requirement of a quantum computer is that thecoherence should last long.

> Nuclear spins in liquids retain coherence ~ 100s millisecand their longitudinal state for several seconds .

> A system of N coupled spins (each spin 1/2) form an Nqubit Quantum Computer.

> Unitary Transform can be applied using R.F. Pulses and

J-evolution and various logical operations and quantumalgorithms can be implemented.


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1. Preparation ofPseudo-Pure States

2. Quantum Logic Gates

3. Deutsch-Jozsa Algorithm

4. Grover s Algorithm

5. Hogg s algorithm 6. Berstein-Vazirani parity algorithm

7. Quantum Games

8. Creation of EPR and GHZ states

9. Entanglement transfer

Achievements of NMR - QIP

10 . Quantum State Tomography

11. Geometric Phase in QC

12. Adiabatic Algorithms

13. Bell-State discrimination

14. Error correction

15. Teleportation

16. Quantum Simulation

17. Quantum Cloning

18. Shor s Algorithm

19. No-Hiding Theorem

Maximum number of qubits achieved in our lab: 8

Also performed in our Lab.

In other labs.: 12 qubits;Negrevergne, Mahesh, Cory, Laflamme et al., Phys. Rev. Letters, 96, 170501 (2006).


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NMR sample has ~ 10 18 spins.

Do we have 10 18 qubits?

No - because, all the spins can t be individually addressed.

Spins having different Larmor frequencies can beindividually addressed in Frequency space as many qubits

Progress so far

One needs resolved couplings between the spinsin order to encode information as qubits.


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NMR Hamiltonian

= Zeeman + J-coupling

= w i Izi + J i j I i I j i i < j

Weak coupling Approximationw i - w j >> J ij

Two Spin System (AM)

A2 A1 M2 M1

wA

wM

M 1= |0A

M 2= |1A

A1= |0M

A2= |1M

|aa = |00

|bb = |11

|ab = |01 |ba = |10

= w i Izi + Ji j Izi Izj

Under this approximation all spinshaving same Larmor Frequency

can be treated as one Qubit

i i < j

Spin States are eigenstates


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13CHFBr 2

An example of a three qubit system.

A molecule having three different nuclear spins (each spin ) having different Larmor frequencies, coupled to each

other, forms a nice 3-qubit (F-C-H) system

13C

Br (spin 3/2) is a quadrupolar nucleus, is decoupled from therest of the spin system and can be ignored in isotropic solution


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1 Qubit

00

0110

11

0

1

CHCl 3

000

001010

011

100

101110

111

2 Qubits 3 Qubits

Homo-nuclear spins having different Chemical shifts (Larmorfrequencies) and J-Coupled also form multi-qubit systems


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Two methods of Addressing

Qubit selective pulsesAnd Coupling (J) Evolution

Transition-selectivePulses and no evolution

I 1

I 2

y x

1/4J 1/4J

00

0110

11p

XOR/C-NOT

I 1

I 2

NOT1

00

0110

11

p

p

I1z

+I2z

I 1z+I 2x

I 1z+2I 1zI2y

I 1z+2I 1zI2z

y

(1/2J)

x


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I 1

I2

y x -y

y x -y

00

0110

11

p1SWAP

I 1

I 2

y y -x -x

I 3

ToffoliC 2-NOT

000

001010

011

100

101110

111

I 1

I 2

y y -x x

I 3

OR/NOR

000

001010

011

100

101110111non-selective p pulse+ a p on 000 001

p2p

3

p

p


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Pseudo-Pure States

Pure States:

Tr( ) = Tr ( 2

) = 1For a diagonal density matrix, this condition requiresthat all energy levels except one have zero populations.

Such a state is difficult to create in NMR

We create a state in which all levels except one have

EQUAL populations. Such a state mimics a pure state.

= 1/N ( 1 + )

Under High Temperature Approximation

Here = 10 5 and U 1 U -1 = 1


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Pseudo-Pure State

In a two-qubit Homo-nuclear system:(Under High Field Approximation)

(i) Equilibrium:

= 10 5 + = {2, 1, 1, 0}

~ I z1+I z2 = { 1, 0, 0, -1}

(ii) Pseudo-Pure

= {4, 0, 0, 0}

0 | 11

1 | 10 2

| 00

1 | 01

0

| 11 0

| 10 4

| 00

0 | 01 ~ I z1+I z2 + 2 I z1I z2

= { 3/2, -1/2, -1/2, -1/2}


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Spatial Averaging

Logical Labeling

Temporal Averaging

Pairs of Pure States (POPS)

Spatially Averaged Logical Labeling Technique (SALLT)

Cory, Price, Havel, PNAS, 94, 1634 (1997)

E. Knill et al., Phy. Rev. A57 , 3348 (1998)

N. Gershenfeld et al, Science, 275, 350 (1997)Kavita, Arvind, Anil Kumar, Phy. Rev. A 61 , 042306 (2000)

B.M. Fung, Phys. Rev. A 63 , 022304 (2001)

T. S. Mahesh and Anil Kumar, Phys. Rev. A 64 , 012307 (2001)

Preparation of Pseudo-Pure States

Using long lived Singlet StatesS.S. Roy and T.S. Mahesh, Phys. Rev. A 82 , 052302 (2010).


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1 Spatial Averaging: Cory, Price, Havel, PNAS, 94, 1634 (1997)

p/3)X(2) p/4)X

(1) p

1/2J

2 4 5 6 1 3

G x

p/4)Y(1)

I1z + I 2z + 2I 1zI 2z = 1/23 0 0 00 -1 0 00 0 -1 00 0 0 -1

Pseudo-purestate

I 1z = 1/2

1 0 0 00 1 0 00 0 -1 00 0 0 -1

I 2z = 1/2

1 0 0 00 -1 0 00 0 1 00 0 0 -1

2I 1z I 2z = 1/2

1 0 0 00 -1 0 00 0 -1 00 0 0 1

Eq.= I 1z+I 2zI1z + I 2z + 2I 1zI 2z


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2. Logical Labeling

N. Gershenfeld et al, Science,

1997, 275, 350

Kavita, Arvind,and Anil Kumar

Phys. Rev. A

61 , 042306 (2000). DRX-500SIF


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3. Temporal Averaging

1 | 111

| 102

| 00

0 | 01

0 | 111

| 102

| 00

1 | 01

1 | 110

| 102

| 00

1 | 01

2 | 11

2 | 106

| 00

2 | 01

+ +

=Pseudo-purestate

p p

E. Knill et al., Phys. Rev. A 57 , 3348 (1998)


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SubsystemPseudo-purestates of2 qubits

T. S. Maheshand Anil Kumar,Phys. Rev. A 64 ,012307 (2001)

4. Pseudo Pure State by SALLT:(Spatially Averaged Logical Labeling Technique)

This method does not scale with number of qubits


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Logic Gates

By 1D NMRUsing Transition Selective Pulses


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Logic GatesUsing 1D NMR

NOT(I 1)

C-NOT-2XOR2

C-NOT-1XOR1

Kavita Dorai, PhD Thesis, IISc, 2000.


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Logical SWAP

Kavita, Arvind, and Anil KumarPhys. Rev. A 61 , 042306 (2000).

0 0

0 1

1 0

1 1

0 0

1 0

0 1

1 1

INPUT OUTPUT

p

XOR+SWAP

0

| 11

1

| 102

| 00

1

| 01p2 p

1 p3

p

e 1 , e2 e 2 , e1 e1 e2


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Kavita Dorai, PhD Thesis, IISc, 2000.

Toffoli Gate = C 2-NOT

e 1 , e2 , e3

e 1 , e2 , e3 (e1 e2)

Input Output

000 000001 001010 010011 111

100 100101 101110 110111 011

AND

NAND p

^

e1

e3 e2

Eqlbm.

Toffoli

e2 e3

e1


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Logic Gates By 1D NMRUsing J-Evolution Method


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CNOT GATE

1

2 y

2p

x

2p p

p

z

x

y

22

11 z z eq I I r +

22

112 x z I I r +=

212

113 2 y z z I I I -+= r

212

114 2 z z z I I I r +=

IN OUT

|00> |00>

|01> |01>

|10> |11>

|11> |10>

( )2121

+

( )2121

-

( )2121

--

( )2121

+-

( )2121

+

( )2121

--

( )2121

-

( )2121

+-

eq r 4 r


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1

2 y

2

p

x

2

p p

p

x

2

p

p

p

y-

2

p

y-

2

p

y

2

p

1 2 3 4 5

11

01

00

10

( )2121

+

( )2121 - ( )212

1 --

( )2121

+-

2

2

1

1 z z eq I I r +

1

SWAP GATE

2

22

111 x x I I r +=

212

2112 22 z y z y I I I I -+-= r

212

2113 22 y z y z I I I I -+= r

12

214 x x I I r +=

12215 z z I I r +=

3

4

2,1

2 x

p

J

2,1

2 x-

p

|00>

|01>

|10>

|11>

( )212

1 +

( )212

1 -

( )212

1 --

( )212

1 +-

( )212

1 +

( )212

1 --

( )212

1 -

( )212

1 +-

5


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Logic Gates

By 2D NMR

Two-Dimensional Gates


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T. S. Mahesh,Kavita Dorai,Arvind andAnil Kumar,J. Magn Res.,148 , 95 (2001)

A completeset of 24Reversible,One-to-one,

2-qubitGates

Two Dimensional Gates


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T. S. Mahesh,et al, J. Magn.Reson, 148 ,

95, 2001

Three-qubit 2D-Gates:

000 001 010 011 100 101 110 111

0 0 0

0 0 1

0 1 0

0 1 1

1 0 0

1 0 1

1 1 0

1 1 1

NOT1

TOFFOLI OR/NOR I N P U T

w1

OUTPUT2

NOP

0 0 0

0 0 1

0 1 0

0 1 1

1 0 0

1 0 1

1 1 0

1 1 1

000 001 010 011

100 101 110 111

I 0

I 1

I 2

I 3


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Quantum Algorithms

(a) DJ

(b) Grovers Search

l h b h k b


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DJ algorithm on ONE qubit with one work bit:Constant Balanced

x f 1 (x) f 2 (x) f 3 (x) f 4 (x) 0 0 1 0 11 0 1 1 0

Uf |x |y |x |y f(x)

|x is input qubit and |y is work qubitO/PI/P

Constant Balancedx y f 1 (x) y f 1 (x) f 2 (x) y f 2 (x) f 3 (x) y f 3 (x) f 4 (x) y f 4 (x) 0 0 0 0 1 1 0 0 1 10 1 0 1 1 0 0 1 1 01 0 0 0 1 1 1 1 0 01 1 0 1 1 0 1 0 0 1

O p e r a

t o r

Uf

1 0 0 00 1 0 00 0 1 00 0 0 1

0 1 0 01 0 0 00 0 0 10 0 1 0

1 0 0 00 1 0 00 0 0 10 0 1 0

0 1 0 01 0 0 00 0 1 00 0 0 1

Cleve Version

U f 1 NOT-2 C-NOT-1,2 C-NOT-2,1

h l h


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Kavita, Arvind, Anil Kumar, Phys. Rev. A 61 , 042306 (2000)

Deutsch-Jozsa Algorithm

Experiment One qubit DJ Two qubit DJ

Eq

Grover s Search Algorithm


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Grover s Search Algorithm Steps:

Pure State

Superposition

Avg

Inversion aboutAverage

Measure

SelectivePhase Inversion

|00..0

Groveriteration

N times

Grover search algorithm using 2D NMR


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Grover search algorithm using 2D NMR

Ranabir Das and Anil Kumar, J. Chem. Phys. 121 , 7603(2004)


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Other works

(i) Non-Adiabatic Geometric Phase in NMR QC

(ii) Adiabatic Quantum Algorithms by NMR

(iii) Quantum Games by NMR


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Geometric Phase ?

When a vector is parallel transported on a curved surface, it acquires a phase. A part of the acquired phase depends on the geometry of the path.

A state in a two-level quantum system is a vector which can betransported on a Bloch sphere. The geometrical part of the acquired

phase depends on the solid angle subtended by the path at the centre ofthe Bloch sphere and not on the details of the path.


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Adiabatic Geometric PhaseM. V. Berry, Quantum phase factors accompanying adiabatic changes,

Proc. R. Soc. Lond. A, 392, 45 (1984)

When a quantum system, prepared in one of the eigenstates ofthe Hamiltonian, is subjected to a change adiabatically, thesystem remains in the eigenstate of the instantaneous

Hamiltonian and acquires a phase at the end of the cycle.The phase has two parts. The dynamical and the geometric part.

)/exp( iEt D -= (Dynamical Phase)

)exp( i g = (Geometric Phase)

g D += (Total Phase)

Pancharatnam, S.

Proc. I nd. Acad. Sci. A 44 , pp. 247-262 (1956).


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Non-adiabatic Geometric phases inNMR QIP using transition selective pulses

1. Geometric phases using slice & triangular circuits.- Use of above in DJ & Grover algorithms.

Ranabir Das et al, J. Magn. Reson., 177 , 318 (2005).

2. Experimental measurement of mixed state geometric phase byquantum interferometry using NMR.

Ghosh et al, Physics Letters A 349 , 27 (2005).

3. Geometric phases in strongly dipolar coupled spins .- Fictitious spin- subspaces.- Geometric phase gates in dipolar coupled 13CH 3CN.- Collins version of 2-qubit DJ algorithm.- Qubit-Qutrit parity algorithm.

Gopinath et al, Phys. Rev A 73 , 022326 (2006).

Geometric phase acquired by a slice circuit


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Geometric phase acquired by a slice circuit

The state vector of the two level sub space cuts a slice on the Blochsphere.

The slice circuit can be achieved by two transition selective pulses

The resulting path encloses a solidangle W= 2 .

A spin echo sequence t-p-t is appliedto refocus the evolution under internalHamiltonian (the dynamic phase).

Second ( p) pulse is applied to restore thestate of the first qubit altered by the ( p) pulse.

(p)x (p)-x

(p)q 10 11 (p)

q+p+ 10 11.A.B =

z

Unitary operator associated with the slice circuit:


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Unitary operator associated with the slice circuit:

A.B =

( ) ( )010010 . i B A

e+ +Solid angle W = 2.

13C Spectra of 13CHCl3



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Grover search algorithm using geometric phases (by slice circuit):


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Using Geometric phase gates

H

H

H

H

H C i j C 00

|0

|0

| i

| j

Superposition

Pseudo purestate

SelectiveInversion

Inversion aboutAverage

Measure

H

Inversion about Average C 00 = H U 00 H

U00 = C 00(p)

Selective Inversion:

= p


00= x

01= x

10= x

11= x

Experimental Measurement of Mixed State Geometric Phase by NMR


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Creation of |00 PPS

Experimental Measurement of Mixed State Geometric Phase by NMR

Preparation of Mixed state by the a degree pulse and a gradient.

Implementation of Slice circuit tointroduce Geometric Phase .

Measurement

WResults:


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The shift of the interference pattern as a function of W and r.

W-= -

2tantan 1 r

The shift directly gives the geometric phase acquired by the spin qubit.

Results:Geometric Phase (Shift)

Ghosh et al, Physics Letters A 349 , 27 (2005).


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Adiabatic QuantumAlgorithms

Adi b i Q C i


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Adiabatic Quantum Computing

Based on Adiabatic Theorem of Quantum Mechanics: A quantum system inits ground state will remain in its ground state provided that the HamiltonianH under which it is evolved is varied slowly enough.

U

i

f

s

( s

) E

e21s0 g

s0dt

sdHs1

min

;)(

;max

The system evolvesfrom H i to H f with aprobability (1- e2)provided the evolutionrate satisfies thecondition

Where e


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Adiabatic Algorithms

by NMR(i) NMR Implementation of Locally Adiabatic

Algorithms

(a) Grover s search Algorithm (b) D-J Algorithm

(ii) Adiabatic Satisfibility problem using StronglyModulated Pulses

Avik Mitra


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Step 3. Implementation of Adiabatic Evolution

3x

2x

1xB IIIH z

3z

2z

1zF IIIH

p2180

M

m

1i

180

M

m

i2

180

M

m

1im

xzx eeeU

FB HMm

HMm

1mH

pulse 3,2,1

x

m

2 pulse 3,2,1zm pulse

3,2,1

x

m

2

m th step of the interpolatingHamiltonian .

m th step ofevolution operator

using Trottersformula

Pulse sequence for adiabatic evolution Total number of iteration is 31 time needed = 62 ms

(400 s x 5 pulses x 31 repetitions)

I H l i


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In a Homonuclear spin systems

spins are close(~ kHz) in frequency

space

Pulses are oflonger duration

Decoherenceeffects cannot beignored

Strongly Modulated Pulses circumvents the above problems

Strongly Modulated Pulses.


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Strongly Modulated Pulses.

lllrf

leff iHl1z

lllSMP eUU tt

w r

f

2SMPT

NUUTrF =

Nedler-MeadSimplex Algorithm(fminsearch)

Using Concatenated SMPs


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Using Concatenated SMPs

Avik Mitra et al, JCP, 128 , 124110 (2008)

Duration: Max 5.8 ms, Min. 4.7 ms

Results for all Boolean Formulae


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Avik Mitra et al,JCP, 128 , 124110 (2008)


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Quantum Games

Game Theory


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Von Neumann and MorgensternTheory of games and Economic behaviour

(Princeton University Press, 1947)

John F. Nash (Princeton University) Nobel Prize in Economics (1994)

Robert Aumann (Hebrew University) Nobel Prize in Economics (2005)

Game Theory

Evolutionary GamesNash Equilibrium


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Quantum Games by NMR

Avik Mitra

(i) Ulam s Quantum Game Guessing a number

(i) Three player Dilamma Game- Optimum strategy for

going to a Bar with two stools

(i) Battle of Sexes Game Should Bob and Alice togetherwatch TV or go to a football game

UlamsGame


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Classical Algorithm

1 11

Boblice

Total number of queries = log 22n = n

Quantum Version of UlamsGame


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Bob performs a unitary transform and stores the result in register B.

Alice makes a superposition of all the qubits .

Total number of queries = 1

Alice again interferes the qubits and then performs a measurement .

The quantum system consists of two registers .

Register A consists of n qubits

Register B consists of 1 qubit

--S. Mancini and L. Maccone, quant-ph/0508156

Quantum Version of Ulam sGame

Protocol


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Creationof

PPS

Application

ofHadamardgate

Unitary

TransformofBOB

Application

ofHadamardgate

MEASURE

xperimental implementation


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Preparation of Pseudo-pure state(PPS).

000

110101

100

011

001 010

111

p p

C CIF

F F

a

b c

Equilibrium Spectrum

001 PPS Spectrum

Results


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000

110101

100

011

001 010

111 population

coherence

Avik Thesis I.I.Sc. 2007


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Recent Developments in our Laboratory

1. Experimental Test of Quantum of No-Hiding theorem.Jharana Rani Samal, Arun K. Pati and Anil Kumar,Phys. Rev. Letters, 106, 080401 (25 Feb., 2011)

2. Use of Nearest Neighbour Heisenberg XY interaction for

creation of entanglement on end qubits in a linear chain of 3-qubit system.K. Rama Koteswara Rao and Anil Kumar (IJQI, 10, 1250039 (2012) .

Use of Genetic Algorithm for QIP.

V.S. Manu and Anil Kumar, Phys. Rev. A 86, 022324 (2012)


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No-Hiding Theorem

S.L. Braunstein & A.K. Pati, Phys.Rev.Lett. 98, 080502 (2007).

Any physical process that bleaches out the original informationis called Hiding. If we start with a pure state, this bleachingprocess will yield a mixed state and hence the bleachingprocess is Non -Unitary. However, in an enlarged Hilbertspace , this process can be represented as a unitary.

The No-Hiding Theorem demonstrates that the initial purestate, after the bleaching process, resides in the ancilla qubitsfrom which, under local unitary operations , is completelytransformed to one of the ancilla qubits.


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Quantum Circuit for Test of No-Hiding Theorem using StateRandomization (operator U).

H represents Hadamard Gate and dot and circle representCNOT gates.

After randomization the state | > is transferred to the secondAncilla qubit proving the No-Hiding Theorem.

(S.L. Braunstein, A.K. Pati , PRL 98, 080502 (2007).


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|0>) 1 ()1

= Cos ( /2) |0>) 1+ e [i( - /2)] Sin ( /2) |1>) 1

|00> 2,3 (/2) 2,3 |A2,3> = [|(0 + 1)>] 2 O [(0 + 1)>] 3X

Creation of and Hadamard Gates after preparation of |000> PPS


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The randomization operator is given by,

Are Pauli Matrices

With this randomization operator it can be shown that anypure state is reduced to completely mixed state if the ancillaqubits are traced out.

Using Eqs (1) and (2), the U is given by,

Eq. (1)

Eq. (2)

where

The Randomization Operator is obtained as


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|000> |001> |010> |011> |100> |101> |110> 111>

|000> 1

|001> 1

|010> 1

|011> 1

|100> 1

|101> 1

|110> -1|111> -1

U =

p

Blanks = 0

Local unitary for transforming information to one of the ancilla qubits

This shows that the first two qubitsare in Bell State and the > has

been transferred to the 3 rd qubit.


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Conversion of the U-matrix into an NMR Pulse sequence has been achieved here by a Novel Algorithmic Technique,

developed in our laboratory by Ajoy et. al (PRA 85, 030303(2012)). This method uses Graphs of a complete set of Basisoperators and develops an algorithmic technique for efficientdecomposition of a given Unitary into Basis Operators and theirequivalent Pulse sequences.

The equivalent pulse sequence for the U-Matrix is obtained as

NMR Pulse sequence for the Proof of No-Hiding Theorem


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The initial State isprepared for differentvalues of and

Jharana et al


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Three qubit

Energy LevelDiagram

EquilibriumSpectra ofthree qubits

Spectracorrespondingto |000> PPS

Experimental Result for the No-Hiding Theorem.


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The state is completely transferred from first qubit to the third qubit

325 experiments have been performed by varying and in steps of 15 o

All Experiments were carried out by Jharana (Dedicated to her memory)

Input State

Output State

s

s

S = Integral of real part of the signal for each spin

Tomography of first two qubits showing


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that they are in Bell-States.

Jharana Rani Samal, Arun K. Pati and Anil Kumar,

Phys. Rev. Letters, 106, 080401 (25 Feb., 2011)

Bell States are Maximally Entangled 2-qubit states.

Bell states play an important role in teleportation protocols


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Use of nearest neighbourHeisenberg-XY interaction.

Until recently we had been looking for qubit systems in which


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Solution

Use Nearest Neighbour Interactions

Until recently we had been looking for qubit systems, in whichall qubits are coupled to each other with unequal couplings, sothat all transitions are resolved and we have a complete access

to the full Hilbert space.

However it is clear that such systems are not scalable,

since remote spins will not be coupled.


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Creation of Bell states between end qubits and a W-stateusing nearest neighbour Heisenberg-XY interactions

in a 3-spin NMR quantum computer

Rama K. Koteswara Rao et al, IJQI, 10 (4) 1250039 (2012)

Heisenberg XY interaction is normally not present inliquid state NMR: We have only ZZ interaction available.

We create the XY interaction by transforming the ZZinteraction into XY interaction by the use of 90 0 RF pulses.

Heisenberg-interaction


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g

-

=+++

-

=+ ++==

1

1 111

1

1 1 )()(

N

i

z

i

z

i

y

i

y

i

x

i

x

ii

N

i iii J J H s s s s s s s s

-

= ++ +=

1

1 112

1 )( N

i

y

i

y

i

x

i

x

ii XY J H s s s s

= ji

jiij J H )( s s Where are Pauli spin matrices

Linear Chain: Nearest Neighbour Interaction

( )-=

++

--+

+ +=1

111

N

iiiiii J s s s s

Nearest neighbour Heisenberg XY Interaction

Jingfu Zhang et al., Physical Review A, 72, 012331(2005)

J JConsider a linear Chain of 3 spins with equal couplings


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3232212121

y y x x y y x x XY J H s s s s s s s s +++=

t iH XY

et U -

=)(

( )32212)( y y x xi Jt

A et U s s s s +-=

( )32212)( x x y yi Jt

B et U s s s s +-=

( ) ( )3221232212)()()( x x y yi

y y x xi Jt Jt

B A eet U t U t U s s s s s s s s +-+-==

0, 32213221 =++ x x y y y y x x s s s s s s s s

jx/y are the Pauli spin matrices and J is the coupling constant between two spins.

21 3J J

Divide the H XY into two commuting parts


( )3221i Jt ssss +- I y y x x = + 23221 s s s s


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)()sin()cos( 3221222 y y x x

Jt i Jt I s s s s +-=

( )22

3221 y y x x Jt i

e

s s s s +-

=

( )2)( y y x x Jt A et U s s s s +=

)()sin()cos( 3221222

x x y y Jt i Jt I s s s s +-=

( )22

3221 x x y y Jt i

es s s s +-

=

( )32212)(

x x y yi Jt

B et U s s s s +-

=

)(sincos)(sincos 32212

3221

2 x x y yi

y y x xi I I s s s s s s s s +-+-=

)()()( t U t U t U B A=

2/ Jt =

I 2

-=

=

- Ai I e

then I Athat suchmatrixais A

iA )sin()cos(

,2

q q q

The operator U in Matrix form for the 3-spin system


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--

--

--

--

--

--

=

10000000

0cos)2sin(0sin000

0)2sin()2cos(0)2sin(000

000cos0)2sin(sin0

0sin)2sin(0cos000

000)2sin(0)2cos()2sin(0

000sin0)2sin(cos0

00000001

)(

2

2

2

22

2

2

2

2

2

2

22

2

2

2

i

ii

i

i

ii

i

t U

= Jt/2 where


Quantum State Transfer


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-

-

-

-

-

-

=

10000000

00001000

00100000

00000010

01000000

00000100

00010000

00000001

2 J U

p

100001 -=U

110011 -=U

001100 -=U

011110 -=U

000000 =U

010010 -=U

101101 -=U

111111 =U

/2 ,2/When p p == J t

Interchanging the states of 1 st and 3 rd qubit

[Ignore the phase (minus sign)]

Two qubit Entangling Operator


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---

--

--

--

--

--

=

10000000

021

20

21

000

02

002

000

00021

022

10

021

20

21

000

0002

002

0

00021

022

10

00000001

22

i

ii

i

i

ii

i

J U

p

( ) 213211001101 += - iU

( ) 213200110010 += - iU

,

22

J

t When p = 4/p =

Bell States

Three qubit Entangling Operator


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--

--

--

--

--

--

=

10000000

079.03

021.0000

033

10

3000

00079.003

21.00

021.03

079.0000

000303

1

30

00021.003

79.00

00000001

)(

i

ii

i

i

ii

i

t U

,2

)2(tan

1

J t When

-

=2

)2(tan 1-=

110011101101)(333

1 iit U --=

1100111013

13

13

1 ++

Phase Gate on 2 nd spin

1100 i+

W - State

Experiments using nearest neighbour interactionsin a 3 spin system


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in a 3-spin system

1. Pseudo-Pure States.

2. Bell states on end qubits.

3. W-state .

13 CHFBr2

J HC = 224.5 Hz, J CF = -310.9 Hz and J HF = 49.7 Hz.

Energy Level Diagram Equilibrium spectra

1H

13 C

19 F

Pseudo-Pure States using only nearest neighbour interactions


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[45] x [45] -y [45] x

[45] x

[45] -y

[45] -y

[90] -y

[90] -y

[90] x

[90] x

[90] y

[90] y

[57.9]

1H

13 C

19 F

G z 1/2J HC 1/J CF 1/2J CF 1/2J HC

321313221321000 4222 z z z z z z z z z z z z I I I I I I I I I I I I ++++++= r

C z C

F z F

H z H eq I I I r ++= )25.094.0( C z

F z

H z H I I I ++=


Pseudo-Pure States:


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000

001

010

011

1H 13 C 19 F


Bell state on end qubits:


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,

22

J

t When p = 4/22 p == Jt

Starting from 010 pps the U yields Bellstates in which the Middle qubit in state 0

Starting from 101 pps the U yields Bellstates in which the Middle qubit in state 1( ) 2132 11001101 += - iU

( ) 213200110010 += - iU


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Bell state on end qubits:( )

213211001101 += - iU


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ExperimentSimulated

Real Part

Imaginary Part

( ) 2132


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ExperimentSimulated

1100111013

13

13

1 ++W-State:


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ExperimentSimulated

Real Part

Imaginary Part


The Genetic Algorithm


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Directed search algorithms based on themechanics of biological evolution

Developed by John Holland, University of

Michigan (1970s)

John HollandCharles Darwin 1866

1809-1882

Genetic Algorithm


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Genetic Algorithms are good at taking large,

potentially huge, search spaces and navigating them,looking for optimal combinations of things, solutionsyou might not otherwise find in a lifetime

Here we apply Genetic Algorithm to Quantum Information Processing

We have used GA for

Quantum Logic Gates ( Operator optimization)and

Quantum State preparation (state-to-state optimization)

V.S. Manu et al. Phys. Rev. A 86 , 022324 (2012)


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The Individual, which represents a valid solution can berepresented as a matrix of size (n+1)x2m. Here m is thenumber of genes in each individual and n is the numberof channels (or spins/qubits).

So the problem is to find an optimized matrix, in which theoptimality condition is imposed by a Fitness Function

Fitness function


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Fitness function

In operator optimizationGA tries to reach a preferred target Unitary Operator (U tar ) from an

initial random guess pulse sequence operator (U pul ).

Maximizing the Fitness function

F pul = Trace (U pul U tar )

In State-to-State optimization

F pul = Trace { U pul ( in ) U pul (-1) tar }

Two-qubit Homonuclear case


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H = 2 (I 1z 1 2z ) + 2 J 12 (I 1z I 2z )

Single qubit rotation

= 500 Hz, J= 3.56 Hz

1 = 2 , 2 = , = /2 , = /2

1 = , 2 = 0 = /2 , = /2

/2 /2

Simulated using J = 0

Hamiltonian used

Non-Selective ( Hard) Pulsesapplied in the centre


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Fidelity for finite J/

Controlled- NOT:


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Equilibrium

00

01 1011

1

-1

00

00

01 1011

1

-10

0

00

01 1011

1

-1 0

0

00

01 10111

-1

0

000

01 1011

1

-1

0

0

Pseudo Pure State (PPS) creation


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Pseudo Pure State (PPS) creation

All unfilled rectangles represent 900 pulseThe filled rectangle is 180 0 pulse.

Phases are given on the top of each pulse.

Fidelity w.r.t. to J/

0001 10 11


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F t reDirections:


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F uture Directions:

(i) Use Collective M odes of l inear chains.

(ii) Backbone of a C-13, N-15 labeled protein forming a linearchain:

(i i i) Use of Genetic Algor i thm f or designing pulsesequences for NM R QI P.

Acknowledgements


112/133

Other IISc CollaboratorsProf. Apoorva Patel

Prof. K.V. RamanathanProf. N. Suryaprakash

Prof. Malcolm H. Levitt - UKProf. P.Panigrahi IISER KolkataDr. Arun K. Pati HRI-AllahabadMr. Ashok Ajoy BITS-Goa-MIT

Dr. ArvindDr. Kavita DoraiDr. T.S. MaheshDr. Neeraj SinhaDr. K.V.R.M.MuraliDr. Ranabir Das

Dr. Rangeet Bhattacharyya

- IISER Mohali- IISER Mohali- IISER Pune- CBMR Lucknow- IBM, Bangalore- NCIF/NIH USA

- IISER KolkataDr.Arindam Ghosh - NISER BhubaneswarDr. Avik Mitra - Varian PuneDr. T. Gopinath - Univ. MinnesotaDr. Pranaw Rungta - IISER Mohali

Dr. Tathagat Tulsi IIT Bombay

Other Collaborators

Funding: DST/DAE/DBT

Former QC- IISc-Associates/Students

This lecture is dedicatedto the memory of

Ms. Jharana Rani Samal*

(*Deceased, Nov., 12, 2009)

Current QC IISc - Students

Mr. Rama K. Koteswara RaoMr. V.S. Manu

Thanks: NMR Research Centre IISc, for spectrometer time


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Thank You

BookQ t C t ti d Q t I f ti P i g


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Quantum Computation and Quantum Information Processing by

Michael A. Nielsen and Isaac L. Chuang,Cambridge University Press (2000)

Indian Edition available for Rs. 317/-

Review Article onQuantum Computing by NMR

ByJonathan A Jones

University of Oxford, UK

arXiv:1011.1382v1 [quant-ph] 5 Nov 2010

2D NMR Quantum Computing Scheme

l D i


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Preparation Evolution Mixing Detection

y -y y

t 1 t 2 I1 , I 2 etc.

G z

Creation of Initial States

Labeling of the initial states

Computation Reading output states

Computation

I0

Ernst & co-workers, J. Chem. Phys., 109 , 10603 (1998).

Three-spin system:


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Transitions of I0 arelabeled by the

states of I 1 and I 2

00 01 10 11

I 0 I 1 I 2 H H

Br C C COOH

H Br

I 1 I 0

I 2

I 0

2 qubits Typical systems used for NMR-QIP using J-Couplings

Chuang et al,quant ph/0007017


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3 qubits

4 qubits

5 qubits

Marx et al, Phys. Rev. A62, 012310 (2000)

Knill et al, Nature,404, 368 (2000)

7 qubits

7 qubits

quant-ph/0007017

67


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How to increase thenumber of qubits?

Use Molecules Partially Oriented (~ 10-3

)in Liquid Crystal Matrices

(i) Quadrupolar Nuclei (spin >1/2). Reduced

Quadrupolar Couplings(ii) Spin =1/2 Nuclei. Reduced Intramolecular

Dipolar Couplings


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QuadrupolarSystems

Using spin-3/2 ( 7Li) orientedsystem as 2-qubit system

Pseudo-pure states


121/133

Neeraj Sinha,T. S. Mahesh, K. V. Ramanathan, and

Anil Kumar, J. Ch em. Phys., 114 , 4415 (2001).

y q y

2-qubit Gates


122/133

using 7Li

oriented system

Neeraj Sinha,T. S. Mahesh,

K. V. Ramanathan,and Anil Kumar,

JCP, 114, 4415 (2001).

133 Cs system spin 7/2 system [Cs pentadeca-fluoro-octonate + D 2O]

Half Adder SubtractorEquilibrium


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Optimal Labeling

Half-Adder p(5,6,5,7,6)

5-pulses

Subtractor p(3,2,3,5,4,3,2,5,7)

9-pulses

765

4321

111

110

101

100

011

010

001

000

-7/2

-5/2

-3/2

-1/2

1/2

3/2

5/2

7/2

765

4321

000

010

011

001

101

110

111

100

-7/2

-5/2

-3/2

-1/2

1/2

3/2

5/2

7/2

Half-Adder p(1,3,2)3-pulses

Subtractor

p(6,4,2)3-pulses

Half-Adder SubtractorEquilibrium

Conventional Labeling

Murali et.al.Phys. Rev. A66 .022313 (2002)

R. Das et al, Phys. Rev.

A 70, 012314 (2004)

Collins version of 3-qubit DJ implemented on the 7/2 spin of Cs-133.


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There are 2 constantand 70 Balancedfunctions. Half differin phase of theUnitary transform.

12 are shown here.

1-Constant and 11-

Balanced

C B B

BB B

B

B

B

B B B

Gopinath and Anil Kumar, J.Magn. Reson., 193, 163 (2008)


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DipolarSystems

Advantages of Oriented Molecules


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Large Dipolar coupling - ease of selectivity - smaller Gate time Long-range coupling - more qubits

DisadvantageFor Homo-nuclear spin system

Spins become Strongly coupledA spin can not be identified as a qubit

2 N

energy levels are collectively treated as an N-qubit system

Solution

Weak coupling Approximationwi - w j >> D ij

3-Qubit Strongly Dipolar Coupled Spin SystemBromo-di-chloro-benzene


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Z-COSY (90-t 1-10- m -10-t 2) wasused to label the various transitions

GHZ state (|000 + |111 )

C 2-NOT gate

POPS(|000 000| -

|001 001| )

populations coherences

(|110 |111 )

T.S. Mahesh et.al., Current Science 85 , 932 (2003).

5-qubit systemHET-Z-COSY spectrum for labeling


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(in liquid crystal)

Eqlbm

p g

E-level diagram

Proton transitions Fluorine transitions


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8-qubit system


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Equilibrium spectrum 1H spectrum

19F spectrum

F Molecule: 1-floro naphthalene

(in liquid crystal ZLI1132)

(1-512)

(513-630)

H

H H

H H

H H

HET-Z-COSY spectrum


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R. Das, RBhattacharyyaand Anil

Kumar,QuantumComputing Back Action,AIP Proc.,864, 313 (2006)


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C 7-NOT [ p1]


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POPS(1) [Eq- p1]

POPS(40) [Eq- p40]

quantum studies in iisc

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