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
Hadamard on function bits
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
Hadamard on function bits
CNOT31
Hadamard on function bits
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
2-3 Dibromothiophene
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
2-3 Dibromothiophene
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
Hadamard on function bits
Conditional rotation from least significant function bit
Conditional rotation from most significant function bit
Hadamard on function bits
Simulator correlation -.92
NMR CNOT SimulationNMR CNOT Simulation
Pseudo-pure state
Hadamard on function bits
Hadamard on function bits
CNOT31
Simulator correlation -.99
NMR ExperimentNMR ExperimentPseudo-pure state
projection = .98
Hadamard on function bits
Hadamard on function bits
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.