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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
2-3 Dibromothiophene
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
2-3 Dibromothiophene
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.