The Integration Algorithm

1
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.
2
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.
3
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).
4
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).
5
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.
6
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.
7
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.
8
Integration Gate Sequence
Sequence of conditional rotations - rotate work
bit by some angle if the function bit is 1.
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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 .
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Integration of
Actual integration yields
The integration algorithm taking the four data
points shown above yields
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Integrating
work bit
function bits
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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
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Integration of
Actual integration yields
The integration algorithm taking the four data
points shown above yields
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Integrating
work bit
function bits
Controlled-NOT gate
15
Integration Algorithm Using CNOT
CNOT31
Amplitude of state .5
Hadamard on function bits
Hadamard on function bits
16
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
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Internal Hamiltonian

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

JIS
HintwIIzwSSz2p JISIzSz
I
S
9.6 T
interaction with B field
spin-spin coupling
2-3 Dibromothiophene
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External Hamiltonian

• Experimentally Controlled Hamiltonian
• Total Hamiltonian

Hext(t) wRFx(t)(IxSx)wRFy(t)(IySy)
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
19
The Alanine Spin System
20
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.
21
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!
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Decoherence Free Subspace
23
Noiseless Subsystem Experiment
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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.
25
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
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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
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NMR CNOT Simulation
Pseudo-pure state
CNOT31
Hadamard on function bits
Hadamard on function bits
Simulator correlation -.99
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NMR Experiment
Pseudo-pure state
CNOT31
projection .98
correlation .97
Hadamard on function bits
Hadamard on function bits
correlation .92
correlation .91
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Integration Results
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.