Transcript Integration via Summation

The Integration Algorithm
A quantum computer could integrate a function in less
computational time then a classical computer...
1 1
1
0 0
0
I    ... f ( x1 , x2 ,...xn )dx1dx2 ...dxn
y = f(x)
y
The integral of a one dimensional
function, f(x), is the area between the
f(x) and the x-axis.
x
Integration via Summation
y=f(x)
y
y=f(x)
y
x
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.
Summation
M
 x 
f 
M 
We first evaluate the sum 
where M is the number of
1
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.
y=f(x)
 1 M  x 
S    f  
M  1 M 
 x 
Defining f (a)  f  M  , we see
 
y
that S is equal to the average value
of f(a).
x
Quantum Averaging
The average of a function can be found on a quantum computer
in the following way...
Initial state of quantum computer
0 00...0
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, H, takes a qubit from a ‘classical’ 0
or 1 state, to a superposition of 0 and 1.
1
0  1 
H0 
2
1
0  1 
H1 
2
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.
1
1
0  00...0  00...1  ...  11...1  
M
M
M 1
0
a 0
a
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...
1
M
M 1

1  f (a) 0 a  f (a) 1 a
a 0
If we now perform another set of Hadamards on the
function qubits the state 1 00...0 will have an amplitude
of
1
M
M 1
 f (a) from which we can get S.
a 0
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.
function bits
work bit
Integration Gate Sequence
0
0
0
0
H
H
0
H
H
evaluate
f(a)
H
H
H
H
Sequence of conditional rotations - rotate work bit
by some angle if the function bit is 1.
Extract
amplitude
of
1 00...0
state
Integrating Sinusoidal Functions
function bits
work bit
To integrate a sinusoidal function between 0 and 1 would require each
state, a, to conditionally rotate the work bit by a , where   freq( f ( x)) M  1

0
0
0
0
H
H
H
2n  2 
2 n1
2n 
H
H
H
Extract
amplitude
of
1 00...0
state
0
H
H
a is stored as a binary number a  an an1...al ...a0 . Thus the
sequence to evaluate f(a) is a series of conditional gates that
rotate the work bit by an amount 2 l  .
Integration of f ( x)  sinx
1
Actual integration yields:
1
 sin x dx  .637
0
0
1
The integration algorithm taking the four data points
shown above yields:
1 3
 x 
sin   .433

4 x 0  3 
Integrating sin x 
1
0
work bit

3
function bits
0
0
0
1
2
3
H
H
H
H
conditional rotations
Extract
amplitude
of
1 00
state
Integration Algorithm for sin x 
Amplitude of 010
state = .433
Pseudo pure state
Hadamard on
function bits
Hadamard on function bits
Bits 1 and 3 are
function bits.
Conditional rotation
from most significant
function bit
Conditional rotation
from least significant
function bit
Integration of
1
 3 
f ( x)  sin 
x
 2 
2
Actual integration yields:
 3
0 sin  2
1
2
0

x dx  .5

1
The integration algorithm taking the four data points
shown above yields:
1 3
2
sin 

4 x 0
2

x   .5

Integrating
 3 
sin 
x
 2 
2
1
work bit
0
1
function bits
0
0
0
H
H
Controlled-NOT gate
H
H
Extract
amplitude
of
1 00
state
0  00  10  1  01  11 
Integration Algorithm Using
CNOT
Initial 000 state
CNOT31
Amplitude of 100
state = .5
Hadamard on
function bits
Hadamard on
function bits
Quantum Information Processing
using NMR
Spectrometer
Nuclear Spins as qubits
ADC for data acquisition
RF synthesizer and amplifier
Gradient control
wave guides
sample
test tube
0
1B
I
JIS
S
9.6 T
RF Wave
RF wave
High field magnet
2-3 Dibromothiophene
Internal Hamiltonian
• The evolution of a spin system is generated
by Hamiltonians
– Internal Hamiltonian:
Hint=wIIz+wSSz+2 JISIzSz
I
JIS
S
9.6 T
interaction with B field
spin-spin coupling
2-3 Dibromothiophene
External Hamiltonian
– Experimentally Controlled Hamiltonian:
Hext(t) =wRFx(t)·(Ix+Sx)+wRFy(t)·(Iy+Sy)
spins couple to RF field
– Total Hamiltonian:
I
JIS
S
9.6 T
Htotal (t) = Hint + Hext(t)
Htotal(t)
controlled via
Hext(t)
RF wave
2-3 Dibromothiophene
The Alanine Spin System
w2  2286.5Hz
J12= 54.1
w1  7167.8Hz
C1
J23= 35.0
C3
C2
J13= -1.3
n
n
w3  4881.4Hz
H int  w k I zk   2J kl I zk I zl
k 1
k 1 l  k
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.
RF nutation
rate (radians)
This pulse
implements
a Hadamard
gate on the
second and
third spins.
time
Quantum Error Correction
Start with an initial state and some extra spins


Encode


Single bit errors become correlated errors
Decode
No Error


Flip Bit 1


Flip Bit 2


Flip Bit 3


Measure the extra
bits to collapse to
one error and learn
what error occurred.
Then correct it.
Never need to know the original state!
Decoherence Free Subspace

Engineered
Noise
Decode
1
Information

Encode
Encoded
0.8
Un-Encoded
0.6
0.4
30
60
Noise strength (Hz)
90


Noiseless Subsystem Experiment
Weak Noise
Strong Noise Limit
Info
No Encoding, Y Noise
0.8
0.6
0.4
0
Encoded, Y, Z Noise
10
20
Noise Strength (Hz)
Encode
U1


U3
U2
Un-Encoded
Z-X Noise
NS-Encoded
No Noise
Z-X Noise
Z-Y Noise
0.24
0.70
0.70
0.70
30
Engineered
Collective
Noise
Information
1
Decode
U†3
U†1
U†2

1

0
Tomography
Not all elements of the density matrix are observable on an
NMR spectra.

2
x
 
2
x
3
z
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
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 sin x  Simulation
Pseudo-pure state
Hadamard on function bits
Hadamard on
function bits
Conditional rotation
from most significant
function bit
Conditional rotation
from least significant
function bit
Simulator correlation -.92
NMR CNOT Simulation
Pseudo-pure state
CNOT31
Hadamard on
function bits
Hadamard on
function bits
Simulator correlation -.99
NMR Experiment
Pseudo-pure state
projection = .98
Hadamard on function bits
correlation = .92
CNOT31
correlation = .97
Hadamard on function bits
correlation = .91
Integration Results
The 100 element gives the result of the integration.
100 element
Amplitude = .497
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.