David G. CoryDepartment of Nuclear

EngineeringMassachusetts Institute of

Technology

Using Nuclear Spins forQuantum

Information Processing

and Quantum

Computing

Quantum Information Processing

• The precise control of a set of coupled

2-level systems.

HHintint==IIIIzz

interaction with B fieldinteraction with B field

E

|0 >

|1 >

E

| >

| >

• Qubit can be in a continuum of states

• Most states are superpositions of 0 and 1

0

1

“0 and 1”qubit spin

Addressable Qubits• Chemically distinct spins

HHintint==IIIIzz++SSSSzz+2πJIzSz

interaction with B fieldinteraction with B field

I S

2-3 Dibromothiophene

coupling between spinscoupling between spins

JIS

External Hamiltonian

– Experimentally Controlled Hamiltonian:

– Total Hamiltonian:

HHextext(t)(t) ==RFxRFx(t)(t)··(I(Ixx+S+Sxx)+)+RFyRFy(t)(t)··(I(Iyy+S+Syy))

HHtotaltotal(t)(t)

controlled viacontrolled via

HHextext(t)(t)

I SJIS

9.6 T

RF wave

spins couple to RF fieldspins couple to RF field

HHtotal total (t)(t) = H= Hintint + H + Hextext(t)(t)

Single Qubit Gates

I S

qubit selectiveinversion pulse

Conditional Qubit Gates

I S

selectiveπ/2

coupling

selectiveπ/2

Quantum & Classical ChannelQuantum & Classical Channel

0

0.5

1

0

0.5

1

0

0.5

1

0

0.5

1

0

0.5

1

0

0.5

1

Partial Trace

System

Environment

0.95 0.99

0.98 1.000.96

0.99

Decoherence Free Subspacephase noise:

φjeφje−

Collective phase noise:φje

φje−

φjeφjeφje−

φje−

φjeφje−

φ2je

φ2je−

EncodeEncode

Logical Logical QubitQubit

DFS for Memory

30 60 900.4

0.6

0.8

1

Info

rma

tio

n

Noise strength (Hz)

Encoded

Un-Encoded

EngineeredEngineered

NoiseNoise

Encode Decode

Samples and Hamiltonians

Alanine (3 Qubits)Alanine (3 Qubits) Crotonic Acid (4 Qubits)Crotonic Acid (4 Qubits)

C4C1

C2

C3

J12

J23

J34

∑∑∑= <=

+=n

k kl

lz

kzkl

kz

n

kk IIJIH

11int 2πω

C3C1 C2

J12 J23

J13

CN

N

C

C

CC

C

O–

ND3+

D H

H

H H O

H

Control of EntanglementControl of Entanglement

( )1110002

1+=GHZ

X Measurement C=0.92

Z Measurement C=0.89

GHZ StateC=0.88

Traced state C=0.71 Traced state

( )1101010113

1++=W

Z Measurement C=0.80

X Measurement C=0.77

W StateC=0.73

Pseudopure state - Product of two Singlet states

(real part of the density matrix)

Strong Measurement

Entangle bits 1 & 2, and bits 3 & 4

Map bits 2 & 3 onto the Bell basis

H

Measurement of bits 2 & 3

|01> + |10>

|01> + |10>

Final Results – After Selective Strong Measurement in the Bell basis

n (number of H-CNot pairs)

Fin

al C

orre

lati

on

C

C

H

0

0

0 H

H( )n

H

|000 GHZ|000 for n = 0, 8, 16, …|100 for n = 4, 12, 20, ...

Output for n = 128

Correlation: 96.65%

Correlation: 90.89%

Output for n = 64

16

Output state

Quantum Fourier Transform

• Shor’s algorithm

• Quantum simulations

• Quantum chaos

Input state

QFT SuperoperatorFidelity = .99 Fidelity = .80

QFT Superoperator

Theoretical QFT Superoperator

Experimental QFT Superoperator

Statistical Verification of Control

Q =2−2nq

Tr ρi2

[ ]i=1

nq

∑ = number of qubits

reduced density matrix for qubit i

qn

iρ

m = 16 (+) m = 24 (.) m = 32 (x) m = 40 (o)

Inset: average Q approaches CUE average exponentially.

Random Circuit on nq = 8 qubits

= 4, 6, 8, 10.

P(Q)

0.999

0.99

0.9

Gat

e F

idel

ity

0.9999

RF Power (Hz)

Strongly Modulated Pulses

modulation frequency

add

ress

abil

ity

Why is quantum noise so bad?Why is quantum noise so bad?Consider an entangled state:

If any are disturbed, they all collapse

How do we protect our information?How do we protect our information?

To make matters worse:•Can’t Copy•Can’t Look No Majority Coding

Quantum Error CorrectionIf the noise is sufficiently weak compared with your control rates then you correct a subset of errors with only a finite accuracy requirements.

Encoder

Memory

Extra Bits

RegularlyCorrect Error

Decoder

Discarded Bits

Arbitrary Collective NoiseExpand your state space in the joint eigenstates of J2 and Jz

€

J =J32( ) ⊕ J

12( ) ⊗ J

12( )⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

J can’t distinguish between the 2 paths to the (1/2) state.

(1/2)1 (1/2)2

(0)12 (1/2)3

(1)12 (1/2)3

(1/2)123

(1/2)123

(3/2)123

Encode 22

22

==expexp((--i2i2ππ/3)/3)

Encode

λλ

λλ

zz

zz

0 10 20 300.4

0.6

0.8

1

Noise Strength (Hz)Noise Strength (Hz)

Encoded, Y, Z Noise

No Encoding, No Encoding, YY Noise Noise

Info

rma

tion

Info

rma

tion

Weak NoiseWeak Noise

Experimental Results

Strong Noise Limit

Z-X Noise 0.24Un-Encoded

0.70NS-Encoded

No Noise0.700.70

Z-X NoiseZ-Y Noise

Info

Nuclear Spins in the Solid StateM = -N/2

N/2

N/2 -1

N/2 -2

-N/2 +1

-N/2 +2

0Liquid state is a good test-bed for QIP,

not a scalable approach to QC.

Solid State appears to be scalable.

Spin Hamiltonian

Zeeman Hamiltonian + +

= B0

HZ = hωIz

i

i∑

HD = hωD

ij 3Izi Iz

j −I i ⋅ I j( )

i<j∑

Secular Dipolar Hamiltonian

Htot = HZ + HD

>> Dij)

• use chemistry locally for error correction

• use spatial addressing to define qubits (magnetic field gradients)

++

• dipolar to nearest neighbor coupling

• single spin detection

++

The selective decoupling problem• Consider a system consisting of pairs of spins.

• nearest neighbor coupling is well defined (d), • there is a quasi-continuous broadening that arises from coupling

to distant spins (D ) • Typically d >> D.

increasing D decreasing Dd is fixed

dij =

γ2h

rij3 3cos2θij −1( )

Dipolar -> nn

€

˜ H D(t)=h2

dij σziσz

j +σyiσy

j −2σxi σx

j( )

i<j∑ +

3h2

dij

σziσz

j −σyiσy

j( )cos

2ω1

ωmsinωmt

⎛

⎝ ⎜

⎞

⎠ ⎟

−σziσy

j +σyiσz

j( )sin

2ω1

ωmsinωmt

⎛

⎝ ⎜

⎞

⎠ ⎟

⎧

⎨

⎪ ⎪

⎩

⎪ ⎪

⎫

⎬

⎪ ⎪

⎭

⎪ ⎪

i<j∑

The RF Hamiltonian is

€

HRF

( t ) = h ω1

cos ωm

t σx

and the corresponding RF propagator is

€

URF

( t ) = exp − i

ω1

ωm

sin ωm

t σx

⎛

⎝

⎜

⎞

⎠

⎟

The interaction frame dipolar Hamiltonian is time-dependent

Status of NMR QIP

The functions J0(x) (blue) and H0(x) (red).

€

˜ H D(t)=h2

dij σziσz

j +σyiσy

j −2σxi σx

j( )

i<j∑ +

3h2

dij

J 02ω1

ωm

⎛

⎝ ⎜

⎞

⎠ ⎟ σz

iσzj −σy

iσyj

( )

−H02ω1

ωm

⎛

⎝ ⎜

⎞

⎠ ⎟ σz

iσyj +σy

iσzj

( )

⎧

⎨

⎪ ⎪

⎩

⎪ ⎪

⎫

⎬

⎪ ⎪

⎭

⎪ ⎪

i<j∑

The zero-order average Hamiltonian is,

where J0(x) and H0(x) are the zeroth order Bessel and Struve functions

Starting from IxA

without control (left) and with control (right)

BLUE: IxA; BROWN: Ix

B; BLACK: IxC

0 0.2 0.4 0.6 0.8 1.02

1.5

1

0.5

0

0.5

1

1.5

2

fraction of cycle time for weaker coupling0 0.2 0.4 0.6 0.8 1.0

0.5

0

0.5

1

1.5

2

fraction of cycle time for weaker coupling

BC coupling = 1/8 AB coupling

Starting from IxA

without control (left) and with control (right)

Quantum Simulation, Spin Diffusion

€

Hdiff = bi,j(σ i+σ j

−+σ i−σ j

+)i,j∑

spin-spin correlation time ~ 6 µs

diffusion time 10 -> 100 s

steps ~ 10^8

mean displacement ~ 1 µm

# spins involved 10^11

To connect with theorists combine this

with nn coupling scheme.

Diffusion measurements

k2s/cm2)

k2scm2)

(~10-12 cm2/s)D=D|| [001] [111]

Zhang et al. (T1 = 114-157 s) 7.14 ±0.52 5.31±0.34

Boutisetal.(T1=256-288s) 6.4±0.9 4.4±0.5

29±3 33±4

k2s/cm2)

k2s/cm2)

1 single spin:Result of a quantum computation

Set of N spins:Collective measurement

Transfer of polarization Transfer of polarization single spin single spin transducer spins. transducer spins.

The final state of transducer spins is determined by the The final state of transducer spins is determined by the

state of the controlling (single) spinstate of the controlling (single) spin

Spin TransducerSpin Transducer

1H19F

Global CNOT

• Ideal behavior:

|0>0|00000…0> |0>0 |00000…0>1-N

|1>0|00000…0>1-N |1>0 |11111…1>1-N

(initial state) (final state)

|0>

|2>|1>

|N>

…… Tra

ce

Mea

sure

men

t

Series of Cnots

Requisites on Control

Addressable spins.

Interactions w/ single spin

Control operator

Entanglement none

Gates or basic steps # for max

Contrast

Gate: CNOT

n

Final Contrast 2

−

=

+ +∏ 01

0 EE xi

n

i

σ −

=

+ +∏ 01

0 EE xi

n

i

σ( )

−−

=

+

=

−

=

+

=

+

+⎭⎬⎫

⎥⎦

⎤⎢⎣

⎡+−

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡+−

∏∏

∏∏

022

1

2210

2/2

12/2

Ei

EEE

i

n

ii

n

i

z

i

n

ii

n

i

x

σσσ

σ−

=

+ +∑∏ 0}{

2/

10 EE

i

xi

n

i

σ

Maximum Entanglement Scheme

ρ0|0 00..0> |>(|1 00..>+|0 00..>)/ 2 |>(|1 10..>+|0 00..>)/ 2

|> (|1 11..>+|0 00..>)/ 2 |1>(|0 11..>+|1 00..>)/ 2 |1>(|0 11..>+|1 01..>)/ 2

|1> (|0 11..>+|1 11..>)/ 2 |1>(|0 11..>+ |1 11..>- |1 11..>+|0 11..>)/2 =

ρfin |1> |0 111111...1> vs. |0> |0 00000..0>

U1=UHAD U2=UCNOT12*… UCNOT1N U3= UCNOT01 U4=U2-1 U5=U1

-1=UHAD

|0>

|2>|1>

|N>

H ………… H

Mea

sure

men

t

UU11

UU22

UU33

UU44

UU55

Series of CnotsEntanglement

w/ CnotsEntanglement w/

MQCPerturbative

approach

Requisites on Control

Addressable spins.

Interactions w/ single spin

Addressable spins.

Only one interaction w/ single spin

• Collective control

• Refocusing of the control operator

• Collective control

Control operator

Entanglement none Cat State Cat StateGround State of

Heisenberg Hamiltonian

Gates or basic steps # for max

Contrast

Gate: CNOT

n

Gate:CNOT

2n-1Sequence: DQ &

Dipolar Ham:

2n +1CNOT

Sequence: DQ1 & Dipolar

Hamiltonian: ~n

Final Contrast 2 2 ~1 ~1

−

=

+ +∏ 01

0 EE xi

n

i

σ −

=

+ +∏ 01

0 EE xi

n

i

σ( )

−−

=

+

=

−

=

+

=

+

+⎭⎬⎫

⎥⎦

⎤⎢⎣

⎡+−

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡+−

∏∏

∏∏

022

1

2210

2/2

12/2

Ei

EEE

i

n

ii

n

i

z

i

n

ii

n

i

x

σσσ

σ−

=

+ +∑∏ 0}{

2/

10 EE

i

xi

n

i

σ

Creation of Cat State

With the N-quantum Grade Raising operator:

the pure state is transformed into the cat state:

( )∏∏ −+ +=k

k

k

kNGR bH σσ)(

2

111000000

)(4 KK

K+

=− N

GRHbie

π

Series of CnotsEntanglement

w/ CnotsEntanglement w/

MQCPerturbative

approach

Requisites on Control

Addressable spins.

Interactions w/ single spin

Addressable spins.

Only one interaction w/ single spin

Collective control

Refocusing of the control operator

• Collective control

Control operator

Entanglement none Cat State Cat StateGround State of

Heisenberg Hamiltonian

Gates or basic steps # for max

Contrast

Gate: CNOT

n

Gate:CNOT

2n-1Sequence: DQ &

Dipolar Ham:

2n +1CNOT

Sequence: DQ1 & Dipolar

Hamiltonian: ~n

Final Contrast 2 2 ~1 ~1

−

=

+ +∏ 01

0 EE xi

n

i

σ −

=

+ +∏ 01

0 EE xi

n

i

σ( )

−−

=

+

=

−

=

+

=

+

+⎭⎬⎫

⎥⎦

⎤⎢⎣

⎡+−

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡+−

∏∏

∏∏

022

1

2210

2/2

12/2

Ei

EEE

i

n

ii

n

i

z

i

n

ii

n

i

x

σσσ

σ−

=

+ +∑∏ 0}{

2/

10 EE

i

xi

n

i

σ

• Limited number of spins.

• Simulated all the pulse sequences needed for

the wanted propagators,

• varying the # of repetitions of the cycle.

Quantum Transducer• Model:

– linear chain of spins.

– all the couplings are taken into account.

Results: Entanglement Scheme

0 5 10 15 20 25 30 35 40 45 500

0.2

0.4

0.6

0.8

1

# of repetitions

Entanglement of the 1st SpinGlobal Entanglement

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

# of repetitions

Contrast

• Simulations (8 spins)

Global EntanglementEntanglement of 1st

spin

1

Conclusions• Liquid state NMR is a useful QIP test-bed ~ 8 qubits.• Systems of dipole coupled spins are universal for QIP.• Solid state, nuclear spin approaches appear to be scalable.• For selected problems we are already beyond the

capabilities of classical computing.• Introduced a quantum transducer that uses entanglement to

make a classical measurement that could not otherwise be realized.

Dr. Timothy HavelProfessor Seth LloydProfessor Raymond LaflammeDr. Chandrasekhar RamanathanDr. Chandrasekhar RamanathanDr. Joseph EmersonDr. Joseph EmersonDr. Grum TeklemariamDr. Marco Pravia Dr. Evan Fortunato Dr. Greg Boutis Dr. Yaakov WeinsteinNicolas BoulantPaola Cappellaro Zhiying (Debra) ChenHyung Joon ChoDaniel GreenbaumJonathan HodgesSuddhasattwa Sinha