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y = f(x). y. x. The Integration Algorithm. A quantum computer could integrate a function in less computational time then a classical computer... . The integral of a one dimensional function, f(x), is the area between the f(x) and the x-axis. y=f(x). y=f(x). y. y. x. x.

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the integration algorithm

y = f(x)

y

x

The Integration Algorithm

A quantum computer could integrate a function in less computational time then a classical computer...

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

integration via summation

y=f(x)

y=f(x)

y

y

x

x

Integration via Summation

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.

summation

y=f(x)

y

x

Summation

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.

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

quantum averaging
Quantum Averaging

The average of a function can be found on a quantum computer in the following way...

Initial 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 transform
The Hadamard Transform

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

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 averaging6
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...

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.

quantum averaging via nmr
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 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 sequence

work bit

Extract amplitude of

evaluate f(a)

state

function bits

H

H

H

H

H

H

H

H

Integration Gate Sequence

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

integrating sinusoidal functions

Extract amplitude of

state

H

H

H

H

H

H

H

H

Integrating Sinusoidal Functions

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

work bit

function bits

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 .

slide10

1

0

1

Integration of

Actual integration yields:

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

integrating

1

Extract amplitude of

0

1

state

H

H

H

H

conditional rotations

Integrating

work bit

function bits

integration algorithm for
Integration Algorithm for

Amplitude of state = .433

Pseudo pure state

Hadamard on function bits

Bits 1 and 3 are function bits.

Hadamard on function bits

Conditional rotation from most significant function bit

Conditional rotation from least significant function bit

slide13

1

0

1

Integration of

Actual integration yields:

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

integrating14

1

Extract amplitude of

1

0

state

H

H

H

H

Integrating

work bit

function bits

Controlled-NOT gate

integration algorithm using cnot

Initial state

Integration Algorithm Using CNOT

CNOT31

Amplitude of state = .5

Hadamard on function bits

Hadamard on function bits

slide16

JIS

I

S

9.6 T

RF wave

2-3 Dibromothiophene

Quantum Information Processing using NMR

Spectrometer

Nuclear Spins as qubits

ADC for data acquisition

RF synthesizer and amplifier

Gradient control

0

1

B

sample

test tube

wave guides

RF Wave

High field magnet

slide17

Internal Hamiltonian

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

JIS

Hint=wIIz+wSSz+2pJISIzSz

I

S

9.6 T

interaction with B field

spin-spin coupling

2-3 Dibromothiophene

slide18

External Hamiltonian

  • Experimentally Controlled Hamiltonian:
  • Total Hamiltonian:

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

spins couple to RF field

JIS

I

S

9.6 T

Htotal (t)= Hint + Hext(t)

Htotal(t)

controlled via

Hext(t)

RF wave

2-3 Dibromothiophene

the alanine spin system

J12= 54.1

J23= 35.0

C1

C3

C2

J13= -1.3

The Alanine Spin System
radio frequency pulses

RF nutation rate (radians)

time

Radio Frequency Pulses

RF 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.

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

slide21

Encode

Decode

No Error

Flip Bit 1

Flip Bit 2

Flip Bit 3

Quantum Error Correction

Start with an initial state and some extra spins

Single bit errors become correlated errors

Measure the extra bits to collapse to one error and learn what error occurred.

Then correct it.

Never need to know the original state!

slide22

Encode

Decode

Engineered

Noise

Information

Encoded

Un-Encoded

Noise strength (Hz)

Decoherence Free Subspace

slide23

Weak Noise

Strong Noise Limit

Info

No Encoding, Y Noise

Un-Encoded

0.24

Z-X Noise

Information

NS-Encoded

0.70

No Noise

Encoded, Y, Z Noise

Z-X Noise

0.70

Z-Y Noise

0.70

Noise Strength (Hz)

Noiseless Subsystem Experiment

tomography
Tomography

Not 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 state
Creation of a Pseudo-Pure State

thermal state

72o spin 2 rotation and gradient

Control2 90o y on 1 & 3

Add some identity

gradient

Fake ‘swap’ 1 &2

Pseudo-pure state

nmr simulation
NMR Simulation

Hadamard on function bits

Pseudo-pure state

Hadamard on function bits

Conditional rotation from most significant function bit

Conditional rotation from least significant function bit

Simulator correlation -.92

nmr cnot simulation
NMR CNOT Simulation

Pseudo-pure state

CNOT31

Hadamard on function bits

Hadamard on function bits

Simulator correlation -.99

nmr experiment
NMR Experiment

Pseudo-pure state

CNOT31

projection = .98

correlation = .97

Hadamard on function bits

Hadamard on function bits

correlation = .92

correlation = .91

conclusions
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.