Download Integration via a Quantum Information Processor

Integration via a Quantum Information
Processor
Quantum computers are devices made up of two level quantum systems or qubits
that can process information in a way that preserves quantum coherence. Unlike a
classical bit, a qubit can be in a superposition of states 0 and 1 at the same time. In
addition, quantum bits may become entangled, that is, there may arise correlations
between quantum bits that are not allowed classically. These features allow quantum
computers to solve certain problems much faster then their classical counterparts. These
problems include the factoring of large numbers and database search.
Quantum computers may also be used to integrate in a manner more efficient then
current classical algorithms. This review summarizes a basic implementation of the
quantum integration algorithm on a system of a few (three) qubits – a quantum
information processor.
 1 M  x 
An integrals may be approximated by a sum S    f   . Where
M  1 M 
y=f(x)
M is the number of points used in the
approximation. The more points used in the
summation (the higher the value of M) the
better the approximation.
y
 x 
If we define f (a)  f   the summation S
M 
is simply the average of value of f (a ) . This
average can be calculated on a quantum
computer.
x
Integration (shaded area) approximated
by a sum (area under the M boxes).
We start our quantum algorithm with one work qubit and log2M function qubits,
where M is the number of points used in our summation, all in the state 0 . Hadamard
gates are applied to each function qubit. The Hadamard gate performs the following
operation
1
1
0  1 
 0  1 .
H0 
H1 
2
2
Hence, Hadamards on all function qubits puts them into an equal superposition of
all possible states, a. This allows for the evaluation of all possible f(a).
1
1 M 1
0  00...0  00...1  ...  11...1  
0 a
M
M a 0
State of the quantum computer after application of Hadamard gates on all function qubits.
The ‘evaulation’ of f(a) is now performed by conditionally rotating the work qubit
dependent on the state of the function qubits. A final set of Hadamards on the function
1 M 1
qubits will return the 1 00...0 state with an amplitude of
 f (a) which is the
M a 0
value of S.
The complete gate sequence of the algorithm is as follows:
State of system after function Hadamards
function bits
work
bit
1
1
0  00...0  00...1  ...  11...1  
M
M
State of system after evaluating f(a)
M 1

0 a
a 0
1
M
M 1

1  f (a) 0 a  f (a) 1 a
a 0
0
0
0
0
H
H
H
0
H
Extract
amplitude
of
H
H
H
evaluate
f(a)
1 00...0
state
H
Sequence of conditional rotations - rotate work bit
by some angle if the function bit is 1.
function bits
work bit
For quantum systems it is impossible to accurately measure the amplitude of any
one state with a single measurement. Instead, the experiment must be repeated a number
of time and a series of measurements made. The number of measurements made can be
reduced through a quantum process called amplitude amplification that allows the
quantum algorithm to be faster then the classical one.
The gate sequence to evaluate f(a) depends on the function to be integrated.
However, a general sequence for sinusoidal functions is as follows. Where phi is the

0
0
0
0
H
H
H
2n2
2n1
2n
H
H
H
Extract
amplitude
of
1 00...0
state
0
H
frequency of the function f(x).
H
Our implementation utilizes a liquid-state nuclear magnetic resonance (NMR)
quantum information processor. A quantum information processor is a system of a few
qubits over which quantum coherence can be controlled. The NMR sample contains
many copies of a molecule with spin ½ atoms. The inermolecular interactions are
cancelled out due to movement of the moecules. Therefore, detection of a spin is an
ensemble measurement which does not induce ‘collapse’ of the wavefunction. In other
words, the exact state of the system can be determined.
The function to be integrated on our quantum information processor is
 3 
sin 2 
x  . We use a three qubit system, one work bit and two function bits, hence M =
 2 
4. The function and the gate sequence to implement the integration is shown below.
1
work
bit
0
function bits
0
0
0
1
H
H
H
H
Controlled-NOT gate
Extract
amplitude
of
1 00
state
0  00  10  1  01  11 
The NMR sample used is alanine, which contains three spin - ½ carbon atoms.
These three carbon atoms rotate at slightly different frequencies under the applied
magnetic field. This is called the chemical shift, w and the values are shown in the figure.
This allows the spins to be addressed individually. In addition there is a J-coupling
between the spins due to shared electrons allowing the spins to interact with each other
and gives the ability to perform two-qubit gates.
The alanine system.
RF nutation
rate (radians)
The qubits are addressed by applying radio-frequency pulses that are specially
designed to implement a single unitary on any number of spins. A computer program
designs the pulses based on the parameters of the input spin system and the parameters:
power, duration, offset frequency and phase. The real and imaginary parts of a pulse that
implements the Hadamard gates on the function bits is shown here.
time
At room temperature an NMR system is in a highly mixed state. In order to create
an effective pure state it is necessary to perform a sequence of unitary and non-unitary
operations. The experimental density matrices of the initial state and ‘psuedo-pure’ state
are
Initial NMR state
Psuedo-pure state
The complete NMR experiment was done following the gate sequence shown
above: creation of a psuedo-pure state, Hadamard gates on the function bits, a controlledNOT gate which is the evaluation of f(a), a final set of Hadamards on the function bits.
The answer is read from the 1 00 state of the density matrix after the second set of
Hadamards. The density matrix at various points in the experiment are shown below
along with the correlation – a measure of how accurately the algorithm up to that point
has been performed.
Pseudo-pure state
projection = .98
Hadamard on function bits
correlation = .92
CNOT31
correlation = .97
Hadamard on function bits
correlation = .91
From the final density matrix we can now read out the value of the integral which
should be .5.
The 100 element gives the result of the integration.
100 element
Amplitude = .497