Integration by Summation

7/30/2019 Integration by Summation

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The 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


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Integration 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


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Summation

y=f(x)

y

x

We first evaluate the sum where Mis 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 seethat Sis equal to the average value

off(a).

M

xfaf )(


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Quantum 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).


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The 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.

10210 H 102

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 inputstates.


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Quantum Averaging

We 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

af

M


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Quantum 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 estimatethe 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.


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Integration Gate Sequence

HH

H

H

HH

H

H

0workbit

functionbits

0

0

0

0

evaluatef(a)

Extract

amplitudeof

0...001state

Sequence of conditional rotations - rotate work bit

by some angle if the function bit is 1.


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Integrating Sinusoidal Functions

HH

H

H

HH

H

H

0

workbit

functionbits

00

0

0

22 n 12 n n2Extract

amplitude

of

0...001state

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 , wherea1

))((

M

xffreq

l

2


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Integration of xxf sin)(

Actual integration yields:

637.sin1

0

dxx

The integration algorithm taking the four data pointsshown above yields:

433.3

sin4

1 3

0

x

x

0

1

1


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workbit

functio

nbits

Integrating xsin

H

H

H

H

0

0

0

3

3

2

conditional rotations

Extract

amplitude

of

001state

0

1

1


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Integration Algorithm for xsin

Pseudo pure state

Hadamard on

function bits

Conditional rotation

from least significant

function bit

Conditional rotationfrom most significant

function bit

Hadamard on function bits

Bits 1 and 3 are

function bits.

Amplitude of

state = .433

010


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Integration of

xxf

2

3sin)(

2

Actual integration yields:

5.2

3

sin

1

0

2

dxx

The integration algorithm taking the four data pointsshown above yields:

5.2

sin4

1 3

0

2

x

x

0

1

1


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0

1

1

Integrating

x

2

3sin

2

1101110000

H

H

H

H

0workbit

func

tionbits

0

0

Extract

amplitude

of

001

state

Controlled-NOT gate


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Initial state000

Integration Algorithm Using

CNOT

Hadamard on

function bits

CNOT31

Hadamard on

function bits

Amplitude of

state = .5

100


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Quantum Information Processing

using NMR

B0 1

Nuclear Spins as qubits

High field magnet

RF Wave

sample

test tube

Spectrometer

ADC for data acquisition

RF synthesizer and amplifier

Gradient control

wave guidesI SJIS

2-3 Dibromothiophene

9.6 T

RF wave


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Internal Hamiltonian

The evolution of a spin system is generated

by Hamiltonians

Internal Hamiltonian:

Hint=wIIz+wSSz+2JISIzSz

spin-spin coupling

interaction with B field

I SJIS


9.6 T


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External Hamiltonian

Experimentally Controlled Hamiltonian:

Total Hamiltonian:

Hext(t)=wRFx(t)(Ix+Sx)+wRFy(t)(Iy+Sy)

Htotal(t)

controlled via

Hext(t)

I SJIS


9.6 T

RF wave

spins couple to RF field

Htotal(t)= Hint+ Hext(t)


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The Alanine Spin System

C1 C2C3

J12= 54.1

J13= -1.3

J23= 35.0

Hz8.71671 w

Hz5.22862 w

Hz4.48813 w

n

k kl

l

z

k

zkl

k

z

n

k

k IIJIH

11

int 2w


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Radio 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

RFnutation

rate(ra

dians)

This pulse

implements

a Hadamardgate on the

second and

third spins.


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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 learnwhat error occurred.

Then correct it.

Never need to know the original state!

Quantum Error Correction


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Decoherence Free Subspace

30 60 900.4

0.6

0.8

1

Infor

mation

Noise strength (Hz)

Encoded

Un-Encoded

Engineered

Noise

Encode Decode


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0 10 20 300.4

0.6

0.8

1

Noise Strength (Hz)

Encoded, Y, Z Noise

No Encoding, Y Noise

Info

rmation

Weak Noise

Noiseless Subsystem Experiment

EncodeEncode

U1

U2

U3

DecodeDecode

U1

U3

U2

11 00Engineered

Engineered

Collective

Collective

Noise

Noise

EncodeEncode

U1

U2

U3

DecodeDecode

U1

U3

U2

DecodeDecode

U1

U

1

U3

U3

U2

U2

11 00Engineered

Engineered

Collective

Collective

Noise

Noise

Strong Noise Limit

Z-X Noise 0.24Un-Encoded

0.70NS-Encoded

No Noise0.70

0.70

Z-X NoiseZ-Y Noise

Info


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2

x32

zx

TomographyNot all elements of the density matrix are observable on an

NMR spectra.

To observe the other elements of the density matrixrequires 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.


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Creation of a Pseudo-Pure State

Pseudo-pure state

thermal state 72o

spin 2 rotation andgradient

Control2 90o y on1 & 3

gradient Fake swap 1 &2

Add some

identity


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NMR Simulation xsin

Pseudo-pure state

Hadamard on

function bits

Conditional rotation

from least significant

function bit

Conditional rotationfrom most significant

function bit

Hadamard on function bits

Simulator correlation -.92


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NMR CNOT Simulation

Pseudo-pure state

Hadamard on

function bits

Hadamard on

function bits

CNOT31

Simulator correlation -.99


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NMR ExperimentPseudo-pure state

projection = .98

Hadamard on function bits Hadamard on function bits

CNOT31

correlation = .97

correlation = .92 correlation = .91


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Integration Results

The element gives the result of the integration.100

element100

Amplitude = .497


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Conclusions

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.

Integration by Summation

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