the integration algorithm a quantum computer could integrate a function in less computational time...

The Integration AlgorithmThe Integration AlgorithmA quantum computer could integrate a function in less computational time then a classical computer...

nn dxdxdxxxxfI ...),...,(... 21

1

0

1

0

1

0

21

The integral of a one dimensional function, f(x), is the area between the f(x) and the x-axis.

y = f(x)

x

y

Integration via SummationIntegration via Summationy=f(x)

y

x

The integral, I, can be approximated by a sum, S. Taking more equally spaced points in the summation, leads to a better the approximation of the integral.

y=f(x)

y

x

SummationSummation

y=f(x)

y

x

We first evaluate the sum where M is the number of points used in the approximation. This sums the height of all the boxes. Multiplying this by the width of each box gives the area under the boxes.

M

M

xf

1

M

M

xf

MS

1

1

Defining , we see that S is equal to the average value of f(a).

M

xfaf )(

Quantum AveragingQuantum AveragingThe average of a function can be found on a quantum computer in the following way...

0...000Initial state of quantum computer

1 work qubit log2(M) function qubits - these qubits store the number for which we will evaluate the function, f(a).

The Hadamard TransformThe Hadamard Transform

aMM

M

a

1

0

01

1...11...1...000...0001

The Hadamard transform, H, takes a qubit from a ‘classical’ 0 or 1 state, to a superposition of 0 and 1.

102

10 H 10

2

11 H

Hence, Hadamards on all function qubits in the initial state of our quantum computer will give an equal superposition of all possible states, a, allowing us to evaluate f(a) for all input states.

Quantum AveragingQuantum AveragingWe now conditionally rotate the work qubit by an amount f(a) depending on the state of the function qubits. This puts our quantum computer into the state...

aafaafM

M

a

1

0

1)(0)(11

If we now perform another set of Hadamards on the function qubits the state will have an amplitude

of from which we can get S.

0...001

1

0

)(1 M

a

afM

Quantum Averaging via NMRQuantum Averaging via NMR

Measurement of a quantum system in a superposition state is probabilistic. Therefore, we can only extract the amplitude of a particular state by repeated experiments and measurements of the system. The more experiments the closer we can estimate the amplitude.

An NMR quantum information processor allows us to read out the entire state of our system exactly - allowing us to bypass methods necessary to amplify the amplitude.

Integration Gate SequenceIntegration Gate Sequence

H

H

H

H

H

H

H

H

0

wor

k bi

tfu

nctio

n bi

ts

000

0

evaluate f(a)

Extract amplitude

of0...001

state

Sequence of conditional rotations - rotate work bit by some angle if the function bit is 1.

Integrating Sinusoidal FunctionsIntegrating Sinusoidal Functions

H

H

H

H

H

H

H

H

0

wor

k bi

tfu

nctio

n bi

ts

000

0

22 n 12 n n2

Extract amplitude

of0...001

state

a is stored as a binary number . Thus the sequence to evaluate f(a) is a series of conditional gates that rotate the work bit by an amount .

01 ...... aaaaa lnn

To integrate a sinusoidal function between 0 and 1 would require each state, a, to conditionally rotate the work bit by , where a

1))((

Mxffreq

l2

Integration ofIntegration of xxf sin)(

Actual integration yields: 637.sin1

0

dxx

The integration algorithm taking the four data points shown above yields:

433.3

sin4

1 3

0

x

x

0

1

1

work bit

func

tion

bits

Integrating Integrating xsin

H

H

H

H

0

00

3

3

2

conditional rotations

Extract amplitude

of001

state

0

1

1

Integration Algorithm forIntegration Algorithm for xsin

Pseudo pure state

Hadamard on function bits

Conditional rotation from least significant function bit

Conditional rotation from most significant function bit


Bits 1 and 3 are function bits.

Amplitude of state = .433

010

Integration ofIntegration of

xxf

2

3sin)( 2

Actual integration yields:

5.2

3sin

1

0

2

dxx

The integration algorithm taking the four data points shown above yields:

5.2

sin4

1 3

0

2

x

x

0

1

1

0

1

1

IntegratingIntegrating

x2

3sin 2

1101110000

H

H

H

H

0work bit

func

tion

bits

00

Extract amplitude

of001

state

Controlled-NOT gate

Initial state 000

Integration Algorithm Using Integration Algorithm Using CNOTCNOT


CNOT31


Amplitude of state = .5

100

Quantum Information Processing Quantum Information Processing using NMRusing NMR

BB00 11

Nuclear Spins as qubitsNuclear Spins as qubits

High field magnetHigh field magnet

RF WaveRF Wave

sample sample test tubetest tube

SpectrometerSpectrometer

ADC for data acquisitionADC for data acquisitionRF synthesizer and amplifierRF synthesizer and amplifier

Gradient controlGradient control

wave guideswave guides I SJIS

2-3 Dibromothiophene

9.6 T

RF wave

InternalInternal Hamiltonian Hamiltonian

• The evolution of a spin system is generated by Hamiltonians– Internal Hamiltonian:

HHintint==IIIIzz++SSSSzz+2+2JJISISIIzzSSzz

spin-spin couplingspin-spin coupling

interaction with B fieldinteraction with B field

I SJIS


9.6 T

External HamiltonianExternal 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)

The Alanine Spin SystemThe Alanine Spin System

C1 C2C3

J12= 54.1

J13= -1.3

J23= 35.0

Hz8.71671

Hz5.22862

Hz4.48813

n

k kl

lz

kzkl

kz

n

kk IIJIH

11int 2

Radio Frequency PulsesRadio Frequency PulsesRF pulses are designed to implement a single unitary operator on any number of spins. A computer program designed for the specific spin system is used to search for such a pulse based on the parameters: duration of pulse, power, phase, and frequency offset.

time

RF

nut

atio

n ra

te (

radi

ans)

This pulse implements a Hadamard gate on the second and third spins.

Start with an initial state and some extra spins

Single bit errors become correlated errors

Encode

No Error

Flip Bit 1

Flip Bit 2

Flip Bit 3

Decode

Measure the extra bits to collapse to

one error and learn what error occurred.

Then correct it.

Never need to know the original state!Never need to know the original state!

Quantum Error CorrectionQuantum Error Correction

Decoherence Free SubspaceDecoherence Free Subspace

30 60 900.4

0.6

0.8

1

Info

rma

tio

n

Noise strength (Hz)

Encoded

Un-Encoded

EngineeredEngineered

NoiseNoise

Encode Decode

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

Noiseless Subsystem ExperimentNoiseless Subsystem Experiment

EncodeEncode

U1

U2

U3

DecodeDecodeU1

†U3†

U2†

11 00E

ngi

nee

red

E

ngi

nee

red

C

olle

ctiv

e C

olle

ctiv

e N

oise

Noi

se

EncodeEncode

U1

U2

U3

DecodeDecodeU1

†U3†

U2†

DecodeDecodeU1

†U1†U3

†U3†

U2†U2†

11 00E

ngi

nee

red

E

ngi

nee

red

C

olle

ctiv

e C

olle

ctiv

e N

oise

Noi

se

Strong Noise Limit

Z-X Noise 0.24Un-Encoded

0.70NS-Encoded

No Noise0.700.70

Z-X NoiseZ-Y Noise

Info

2x 32

zx

TomographyTomographyNot all elements of the density matrix are observable on an

NMR spectra.

To observe the other elements of the density matrix requires repeating the experiment 7 times with readout pulses appended to the pulse program.

This is done without changing any other parameters of the pulse program.

Creation of a Pseudo-Pure StateCreation of a Pseudo-Pure State

Pseudo-pure state

thermal state 72o spin 2 rotation and gradient

Control2 90o y on 1 & 3

gradient Fake ‘swap’ 1 &2

Add some identity

NMR SimulationNMR Simulation xsin

Pseudo-pure state


Conditional rotation from least significant function bit

Conditional rotation from most significant function bit


Simulator correlation -.92

NMR CNOT SimulationNMR CNOT Simulation

Pseudo-pure state



CNOT31

Simulator correlation -.99

NMR ExperimentNMR ExperimentPseudo-pure state

projection = .98



CNOT31

correlation = .97

correlation = .92 correlation = .91

Integration ResultsIntegration ResultsThe element gives the result of the integration.100

element100

Amplitude = .497

ConclusionsConclusions•Concrete mapping between integration algorithm and NMR QIP implementation.

•Sufficient control with current NMR quantum information processors to execute integration in small Hilbert spaces.

•NMR QIP version of algorithm does not require amplitude amplification.

•General approach for integrating sinusoidal functions.

the integration algorithm a quantum computer could integrate a function in less computational time...

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