Quantum Simulation of Discretized Harmonic Oscillator on IBM Quantum Computer

Abstract

Here, we conduct a quantum simulation of a particle in a harmonic oscillator potential using a digital quantum simulator provided by IBM quantum experience platform. The simulation is carried out in two spatial dimensions and the algorithm used is generalized for n-spatial dimensions which can be used to simulate n-dimensional harmonic oscillator. We implement the unitary operator on an arbitrary quantum state to show that the probability amplitudes of position oscillate in time. We propose a quantum circuit to effectuate the unitary operator and design it on the simulator. The proposed circuit is then generalized for a n-qubit system that can be used to realize more meticulous simulations.

Full text

Quantum Simulation of Discretized Harmonic Oscillator on IBM Quantum Computer
Valay K. Jain,1, ∗Bikash K. Behera,2, †and Prasanta K. Panigrahi2, ‡
1Department of Physics, Bennett University, Greater Noida, India
2Department of Physical Sciences,
Indian Institute of Science Education and Research Kolkata, Mohanpur 741246, West Bengal, India
Here, we conduct a quantum simulation of a particle in a harmonic oscillator potential using a
digital quantum simulator provided by IBM quantum experience platform. The simulation is carried
out in two spatial dimensions and the algorithm used is generalized for n-spatial dimensions which
can be used to simulate n-dimensional harmonic oscillator. We implement the unitary operator on
an arbitrary quantum state to show that the probability amplitudes of position oscillate in time. We
propose a quantum circuit to effectuate the unitary operator and design it on the simulator. The
proposed circuit is then generalized for a n-qubit system that can be used to realize more meticulous
simulations.
I. INTRODUCTION
The quantum harmonic oscillator1is one of the most
significant and ubiquitous model studied rigorously in
quantum mechanics. It is a quantum treatment appli-
cable to any oscillating system with a typically discrete
energy spectrum. Diatomic molecular vibrations are of-
ten imitated as 2-bodied quantum harmonic oscillator.
Such treatments are conducted in three dimensions and
include more realistic morse-potential for rotating and vi-
brating molecules2. The quantum mechanical harmonic
chain of N identical atoms can be used to create sim-
ple quantum mechanical models of lattice. Various such
models lead to the picture of free-quasi particles called
‘phonons’3. In quantum chemistry, many helium like
atoms can also be explained using Hook’s method. This
method can be used to measure population density of
excited atoms4. A harmonic oscillator system customar-
ily has a non-zero vacuum state giving rise to zero point
energy when we quantify electric fields5. This gives rise
to quantum fluctuations in free space which in turn gives
rise to stimulated emissions which play a crucial role in
the fields of quantum optics and lasers.
Feynman in 1982 drew the world’s attention towards sim-
ulation of quantum systems. He proposed the idea that
a computer can act as a quantum mechanical simulator6.
Over the years there has been a rapidly growing inter-
est in using quantum computation to study the time
evolution of Hilbert spaces7. For large Hilbert spaces
the standard computation techniques prove to be in-
adequate. Various instances of unattainable solutions
have also shown up in simulating co-related quantum
systems8. There are significant open problems in impor-
tant areas, such as high temperature superconductivity,
where progress is slow because we cannot adequately test
our models or use them to make predictions. This expo-
nential scaling of Hilbert space can only be matched by
the increasing possible permutations which make a quan-
tum system twice as memory-full with the addition of
each qubit. Recent upsurge of quantum computation, in-
formation and quantum technologies aim to exploit these
inherent properties of nature. Computational complexi-
ties pose restrictions on density functional theory9widely
used in quantum mechanical modelling10. In this paper,
we show how a system of total 14 qubits is well capable of
observing the time evolution of a particle in a harmonic
oscillator potential for two spacial dimensions.
The rest of the paper is organized as follows: Sec. II
revisits the harmonic oscillator problem which is analyt-
ically solvable and discuss it’s solution along with some
of it’s properties. In Sec. III, the discretization of the
problem is discussed, explaining why finite elements and
finite dimensional operators are important for computa-
tion. The aim of our simulation is to observe the time
evolution of harmonic oscillator position eigenstates. Sec.
IV deals with the simplification of the unitary operator
along with a remarkable observation which allows us to
generalize the simulation to arbitrarily higher dimensions
along with a oversimplification of the quantum circuit.
This section takes us from analytic matrices to realizable
quantum gates. Sec. Vis completely dedicated to simu-
lation where the 2-qubit, 4-qubit and n-qubit algorithm
is explained in detail. The quantum circuit required to
implement the simulation has been generalized and dis-
cussed in Sec. VI. The detailed results for 2-qubit and
4-qubit simulation are provided in Sec. VII. We conclude
in Sec. VIII with future directions.
II. THE HARMONIC OSCILLATOR PROBLEM
The well-known harmonic oscillator potential for two
spatial dimensions is given by
V(x) = mω2
2(x2+y2)
where both x, y ∈(−∞,∞). The corresponding Hamil-
tonian is given by
ˆ
H=ˆp2
2m+mω2
2(x2+y2) (1)
The analytic solution to the time independent
Schrodinger equation is given by11 Ψ(x, y, t)
= Σ∞
nx=0Σ∞
ny=0(mω
π~)1
21
2nn!Hnx(ζ)Hny(β)e−ζ2/2e−β2/2U(t)

2
where ζ=pmω
~xand β=pmω
~yrespectively and Hn
are the n-th order Hermite polynomials. The unitary
operator of any system gives it’s time evolution
U(t) = exp(−ιEnt
~) (2)
It can be observed that the unitary operator suggests the
probability amplitude of the particle oscillation in time.
An elegant simulation of this can be found in Ref.12.En
are the ‘allowed’ energy values of the particle. It is only
for these energy eigenvalues that the individual energy
eigenstates are normalizable, therefore physically realiz-
able. These energy eigenvalues are given by
En= (nx+ny+ 1)~ω
. States corresponding to different energy eigenvalues are
orthogonal to each other and satisfy
Z∞
−∞
ψjψkdxi= 0 ; ∀xi
. A simplier way to solve this problem is by factoring the
Hamiltonian into two non-commuting operators given by,
a−
i=1
√2( ˆxi2+ ˆpi2) ; a+
i=1
√2( ˆxi2−ˆpi2)
. Here we assume ~, m and ωto be unity for the sake of
reducing mathematical clutter and i ∈[1,2] co-respond
to the spatial dimensions x and y. These operators are
famously known as the annihilation and the creation op-
erators which act as wonderful tools to realize states cor-
responding to unit energy lower or unit energy higher
index in the given energy spectrum. The normalized ex-
pressions for these are given as
a+ψj=pj+ 1 ψj+1 ;a−ψj=pj ψj−1
. The time independent Schrodinger equation in terms
of the annihilation and creation operators can be written
as
~ω(a+
1a−
1+a+
2a−
2+ 1)ψ=Eψ
.
Our interests however, lie in position eigenfunctions
corresponding to ‘discrete’ position eigenvalues. The
reason for discretization will be explained subsequently.
These eigenfunctions of position can be found by solving
the eigenvalue equation for position operator
ˆx ψ =x0ψ
, where ˆxis the operator and x0is the eigenvalue. To sat-
isfy this, we need a function which on multiplying with
a variable gives the variable scaled by the number itself.
One such function that satisfies this property is the Dirac
delta function. To make an analogy with spectral decom-
position theorem, we can expand any arbitrary function
as a linear combination of Dirac-Delta functions corre-
sponding to different eigenvalues. All we need to do is
find the correct constants. We conclude therefore the
position eigenfunctions are Dirac-delta function
fx(x) = A δ(x−x0)
. For a discrete eigenspectrum the spectral decomposi-
tion will be given by,
Ψ(x) = ΣN
0cnδ(x−x0)
, where N is the total number of discrete elements taken
into account. According to the expansion postulate |cn|2
gives the ‘the probability of the state Ψ(x) to collapse
onto ψnupon a measurement. The coefficients can be cal-
culated by exploiting the orthogonality of different states
cn=Z−∞
∞
(ΣN
0cmψm)ψndxi
=Z−∞
∞
(Ψ(x)) ψndxi
III. DISCRETIZATION OF THE PROBLEM
In order to do any computation, we need to work with
finite number of elements which can yield results under
realistic time scales. However, the Hamiltonian given in
Eq. (1) allows for a continuous eigenspectrum of position.
To tackle this problem, we need to ‘discretize space’. This
becomes the essence of the simulation. It implies that we
have to account the probability amplitudes in elements
of finite width rather than each and every space point.
This creates a mesh in space which can be mapped over
our potential ‘bowl’. An intuitive picture can be grasped
by Figs. 1and 2.
The Hamiltonian for the ‘discrete harmonic oscillator’
can now be given by
ˆ
H=(ˆpd)2
2+((ˆxd)2+ (ˆyd)2)
2(3)
where ˆpdis the discrete momentum operator and ˆxd,ˆyd
are the discrete position operators in x and y spatial co-
ordinates respectively. If we consider N number of finite
elements in a space of x, y ∈[−L, L] then a mesh of
N2number of finite spatial elements can be created with
each mesh point corresponding to a particular eigenvalue
of x and y. Assuming the harmonic oscillator potential is
centred at [0,0] we can create the position operator in the
form of an N ×N matrix with all the position eigenvalues
lying along it’s diagonal13.

3
FIG. 1. Discrete two dimensional space. The colour bar indi-
cates the amount of stretch required to map these points to
harmonic oscillator potential.
ˆxd=rπ
2N







−N/2 0 0 . . 0
0 (−N/2) + 1 0 .0
0. . .
0. . 0
0. . 0
0. . . 0 (N/2) −1







(4)
All position operators act multiplicatively in Cartesian
co-ordinate space. The operator for (xd)2can also be
calculated directly by multiplying xdwith itself. We can
calculate the momentum operator in a similar manner.
However, the momentum operators do not act multiplica-
tively. Also we have already expanded our total wave-
function in an eigenspectrum of position eigenfunctions.
Calculating the momentum eigenvalues for each position
eigenfunction can be a laborious task for large systems. A
much more efficient approach needs to be taken to make
a discrete quantum Fourier transform of the wave func-
tion. This takes the wave function to momentum space
where the momentum operators act multiplicatively and
the momentum eigenvalues will be the same as the po-
sition eigenvalues for respective discretized space points.
An inverse discrete quantum Fourier transform can be
done to bring back the function into Cartesian-space.
The momentum operator can then be applied as a N ×
N matrix given by
pd= (Fd)−1·xd·Fd(5)
where Fdstands for standard discrete quantum Fourier
transform matrix14. Each element of the matrix can be
given by
[Fd]j,k =exp(ι2πjk/N)
√N
. This approach is ideal as implementation of Q.F.T us-
ing quantum circuits is generalized for n-qubits and only
gets more and more accurate for larger systems13. With
both the position and momentum operators at hand we
are now ready to simulate the unitary operator to study
the time evolution of our system.
IV. UNITARY OPERATOR
From the Schrodinger’s equation
ι∂Ψ
∂t =ˆ
HΨ
implying
Ψ(t) = Ψ(0)exp(−ιˆ
Ht)
. Therefore the unitary operator that we need to eval-
uate is U(t) = exp(−ιˆ
Hdt) where ˆ
Hdis the discretized
Hamiltonian given in Eq. (3). The unitary operator is
then given by
U(t) = exp(−ιt[(ˆpd)2
2+(ˆxd)2+ (ˆyd)2
2])
Ux(t) = exp[−ιt
2((Fd)−1·[xd]2·Fd)+[xd]2]
, where Ux(t) is the unitary operator in x dimension. The
position operator[xd] being a diagonal matrix can be ex-
panded using the matrix exponential. Also having a finite
number of elements it’s expansion will be convergent
exp(−ιt
2[A]) = I+ Σ∞
m=1(−ιt
2)m[A]m
m!(6)
where A is the corresponding operator matrix. Using the
matrix in this form, a time evolution can be performed
for any arbitrary state. Such an evolution can be carried
out using n-qubits in a quantum circuit. We generalize
our simulation as well as the quantum circuit for n-qubits
in the following section.
V. SIMULATION
Our simulation will be utilizing 4 qubits for each spa-
tial dimension. For 4 qubits we are taking into account
a 24= 16 dimentional Hilbert space. Each dimension of
the Hilbert space corresponds to eigenfunction of a spe-
cific position eigenvalue. Each basis state of the 4 qubit
system maps to an eigenfunction in the Hilbert space.
This is the essence of our simulation: the proba-
bility distribution of each qubit state obtained after im-
plementing the quantum circuit will correspond one to
one with the probability distribution of each eigenfunc-
tion after the unitary operation. Allowing us to study
the time evolution of the particle in terms of the time
evolution of the qubit basis states. For two spatial di-
mensions our x, y ∈[−8,8] space discretizes into 16 ×16
= 256 individual mesh point. Each point corresponds to

4
FIG. 2. Mapping the mesh of discrete space over that har-
monic oscillator potential. This gives us an intuitive picture of
the potential energy of the particle at particular space points.
different eigenvalues in x and y spatial dimensions. We
can now evaluate the unitary operator in it’s matrix form
in order to construct quantum gates which can be later
implemented in the quantum circuit. According to Eq.
(4) the position operator is a diagonal matrix.
[xd] = rπ
32






−8 0 . . . 0
0 7 ....
.
.
.
.
....6 0
0. . . 0 7






Therefore the square of the operator will be a diagonal
matrix with all eigenvalues squared. The kinetic energy
operator will then be given by
[xd]0= [xd]2=π
32






64 0 . . . 0
0 49 ....
.
.
.
.
....36 0
0. . . 0 49






From Eq. (6), we can write the unitary operator for
potential energy part as
Uˆx(t) = I+ ( −ιt
2)[xd]0
1+( −ιt
2)2[xd]02
2! +( −ιt
2)3[xd]03
3! +....
A remarkable observation reveals each diagonal ele-
ment of U(t) makes an exact Taylor series expansion of
an exponential function with symmetry about the [8,8]
element which will be unity
Uˆx[1,1] 1.0+3.14it −4.963t2+ 5.17it3exp(3.14ιt)
Uˆx[2,2] 1.0−2.4it −2.89t2−2.32it3exp(−2.4ιt)
Uˆx[3,3] 1.0−1.77it −1.56t2+ 0.92it3exp(−1.77ιt)
.
.
..
.
..
.
.
Uˆx[15,15] 1.0−1.77it −1.56t2+ 0.92it3exp(−1.77ιt)
Uˆx[16,16] 1.0−2.4it −2.89t2+ 2.32it3exp(−2.4ιt)
(7)
Uˆx(t) =









e(ιt∗3.14) 0. . . . . . 0
0e(−ιt∗2.4) .
.
.
...........
.
.
.e(−ιt∗1.77) 0
0. . . 0e(−ιt∗2.4)









We can now implement these different phases in a
quantum circuit independently. Taking the [1,1] element
a common multiple which will act as a global phase, we
can write the unitary operator in the final form
Uˆx(t) = e(ιt∗3.14)









1 0 . . . . . . 0
0e(−ιt∗5.54) .
.
.
...........
.
.
.e(−ιt∗4.91) 0
0. . . 0e(−ιt∗5.54)









The benefit of this approach lies in the first element
being unitary. The first basis of our four qubit system
|0000iis incapable of picking up any phase term inadver-
tently. The rest of the diagonal terms will provide phase
to the next 15 basis states in the sequence. The unitary
operator for the kinetic energy can be formulated in a
similar manner by making subsequent quantum Fourier
transforms as explained in Eq. (5). The complete unitary
transformation can be given by
U(t) = exp[(−ιt
2)( ˆpx2+ ˆpy2+ ˆx2+ ˆy2)] = Uˆx·Uˆy·Uˆpx·Uˆpy
(8)
Both spatial dimensions being orthogonal to each other
will follow the same mathematical structure indepen-
dently. The position and momentum operators for or-
thogonal spacial dimensions commute. Hence we can
generalize our algorithm to calculate the unitary oper-
ator for arbitrary spacial dimensions. The calculation of
unitary operator for n qubits is given in the appendix
below. We propose a generalized circuit to carry out the
unitary operator for any arbitrary state using any arbi-
trary number of qubits, therefore making the approach
complete.
VI. QUANTUM CIRCUIT
The complete unitary transformation given in Eq. (8)
can be implemented by making a circuit for Uˆx(t) and
Uˆp(t) in series. Since the Uˆx(t) only adds phase factors
we will first implement the Uˆp(t). The quantum circuit
for quantum Fourier transform can be executed using a
series of Hadamard and control-phase rotation gates. An
efficient circuit for 4-qubit system is given in Fig. 3,

5
FIG. 3. An efficient quantum circuit for implementing the
quantum Fourier transform.
where the phase of individual rotation gates can be cal-
culated using
cU1n=2π
2n
The single qubit control-rotation gate would be matri-
ces of the form
cU1n=1 0
0e2πι/2n(9)
For implementing the inverse quantum Fourier trans-
form the mirror image of the direct Q.F.T circuit with
conjugate phases at each c-rotation gate can be used (Fig.
4). In the middle of [(Fd)]−1and [Fd)] lies the chief op-
erator [xd] which can be implemented using a series of
Toffolli and control-rotation gates. The crucial algorithm
that can be executed is explained as follows: for each 4-
qubit state we use a control gate for the qubit in |1istate
and an anti-control for the qubit in the |0istate. Three
ancilla qubits along with three cc-not gates are used to
infer the states of all 4 qubits. This series of control and
anti-control gates will simultaneously trigger only for a
specific sequence of |0iand |1istates assuring a particu-
lar 4-qubit state has infiltrated. Now we can trigger the
c-rotation gate calibrated to the phase corresponding to
that particular 4-qubit state. This phase rotation can be
performed at anyone of the qubits in |1istate. A mirror
sequence of Toffolli gates after the c-rotation gate can
be used to reverse all effects prior to our filtering. This
makes the arbitrary 4-qubit state ready for the next se-
ries of Toffolli gates while the crucial phase information
has also been imparted. Each set of filter prepares the
state for the next set of filter and a series of 15 such fil-
ters can be used recursively to implement all 15 phase
rotations to achieve complete Uˆx(t). A filter for the first
state |0001ihas been shown in Fig. 5.
A simpler circuit can be designed by realizing the same
logic for a 2-qubit system. Analogues to the 4-qubit sys-
tem the unitary matrix for a 2-qubit system will also
have diagonal elements as the squared position eigenval-
ues. Without making any approximation, we observe the
elements form a Taylor exponential series.
FIG. 4. The mirror image of Q.F.T circuit which conjugate
phases to implement the inverse quantum Fourier transform.
FIG. 5. A filter for the |0001istate which detects the state of
all 4 qubits then performs a rotation on the 4th qubit. The
series of Toffolli gates after cU1 reverses all changes done to
the initial state.
U[2]
ˆx(t) = 


e−ιt∗0.785 000
0e−ιt∗0.196 0 0
0 0 1 0
0 0 0 e−ιt∗0.196


(10)
We can take out the first element as a global phase.
This will give us the first element as unity.
U[2]
ˆx(t) = 


1 0 0 0
0e−ιt∗0.981 0 0
0 0 e−ιt∗0.785 0
0 0 0 e−ιt∗0.981


(11)
The quantum Fourier transform in 2 dimensions can
be executed by two Hadamard and one control-rotation
gate. The inverse Q.F.T can be executed by it’s mir-
ror image circuit with conjugate phase in the rotation
gate. For two qubit system we will have 3 filters for
|01i,|10iand |11irespectively. A detailed figure is pro-
vided (Fig. 6) where two U3 gates can be used to ini-
tialize the system to any arbitrary state. We can use
the same logic to propose a circuit which is generalized
for n-qubits. Starting with the generalized circuit for
the Fourier transform15 for n qubits is given in Fig. 7.
The inverse quantum Fourier transform can be given by
the mirror image circuit and conjugate phases at the c-
rotation gates. The position operator [xd] for n-qubits

6
FIG. 6. Quantum circuit for implementing Uˆp(t) for a 2-qubit system.
FIG. 7. A generalized quantum circuit for quantum Fourier
transform of n-qubit system.
will be a diagonal matrix with 2nposition eigen values
each lying on the diagonal. The exponential expansion
of the unitary operator gives a Taylor exponential series
in each diagonal element. Inductively, each element will
have a constant phase difference and therefore the sys-
tem with n-parameters can be expressed using a single
parameter. Hence, without any approximation we can
create [xd] for a n-qubit system. The quantum circuit for
n qubits can be implemented using the same algorithm
proposed above; by placing control gates and anti-control
gates on |1iand |0irespectively. Then sending the state
information of nth qubit to the (n−1)th ancilla in order
to trigger the desired phase rotation gate on the required
qubit only. This phase rotation will happen only for a
particular state of n-qubits based on our filter configu-
rations. A series of mirrored Toffolli gates can be used
to undo all changes done by the filter. The qubits then
can proceed to the next filter afresh. A series of (n-1)
such filters are required to execute U[n]
ˆx. Another series
of (n-1) such filters when squeezed between the Q.F.T
and inverse Q.F.T gates can be used to execute U[n]
ˆp.
When we initialize our qubits to a particular state we
provide large probability to a particular eigenvalue of the
particle. According to our initialized state the position of
the particle can be considerably pin-pointed to any of the
corresponding mesh points. The quantum circuit then
simulates the unitary operator to give time evolution of
the particle in terms of time evolution of the probabil-
ity amplitude for different eigenvalues. These eigenval-
ues probabilities map one to one with the corresponding
FIG. 8. A representation of cross multiplication of proba-
bility amplitudes at t=T
4, where T is total time period of
oscillation.
probabilities of the qubit basis states. Multiple simula-
tions with the variation of time parameter tare carried
out until multiple identical oscillations of the particle are
observed. Each individual simulation is carried out on
ibmq qasm simulator with 8192 shots to give the best
possible ensemble. In this simulation we divide the space
of x, y ∈[−8,8] unit2into 256 mesh points. A table of
4-qubit states and their corresponding mesh points are
given in Eq. (12).
4-qubit state Mesh point
|0000ixi=−8
|0001ixi=−7
|0010ixi−6
.
.
..
.
.
|1111ixi= 7
(12)

7
VII. RESULTS
We are therefore capable of executing the uni-
tary operation of a particle in a harmonic oscil-
lator potential for any arbitrary number of qubits.
The more number of qubits we take the denser is
our spatial mesh resulting in further accuracy. We
also become capable of performing the unitary op-
eration for arbitrary spatial dimensions by executing
2n-1 qubit circuits in parallel for each dimension.
The results for a single oscillation of the particle for a
2-qubit and 4-qubit system are presented in Figs. 9,10
and 11,12 respectively where we initialize our state to
|0001i. A cross multiplication of both data sets presents
an intuitive picture for the probability amplitudes in two
spatial dimensions. One such representation is given in
Fig. 8. The URL for a simple animation based on our
data is cited below16.
VIII. CONCLUSION
We infer from the experimental results that the proba-
bility amplitude of a particle in a harmonic oscillator po-
tential oscillate in time mimicking the essence of a quan-
tum harmonic oscillator. The probability amplitudes of
two spatial dimensions are independent of each other
which was expected for two non-commuting variables.
The simulation helps us to understand the movement of
a particle in such a potential intuitively. The change
in probability amplitudes over time helps us comprehend
the concept of velocity in the quantum realm. Ultimately,
the simulation can be generalized for arbitrary spatial di-
mensions. The simulation can also be carried out using
arbitrary number of qubits to make it more meticulous.
ACKNOWLEDGMENTS
V.K.J. would like to thank Indian Institute of Science
Education and Research Kolkata for providing hospital-
ity during the course of the project. B.K.B. acknowl-
edges the support of IISER-K Institute Fellowship. The
authors acknowledge the support of IBM Quantum Ex-
perience for producing experimental results. The views
expressed are those of the authors and do not reflect the
official policy or position of IBM or the IBM Quantum
Experience team. V.K.J would like to thank Gaurav
Rudra Malik [17] for useful discussions and encourage-
ment.
∗vj7049@bennett.edu.in
†bkb18rs025@iiserkol.ac.in
‡pprasanta@iiserkol.ac.in
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theory
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son Prentice Hall (2004).
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schroeder/software/HarmonicOscillator.html
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17 Department of Physics, Institute of Science, Banaras
Hindu University, Varanasi, 221005, Uttar Pradesh, India
APPENDIX
The Unitary operator for a n-qubit system can be cal-
culated as follows;
The position operator for n-qubits will have 2n=Ndi-
mensional Hilbert space. The operator is given by Eq.
(4)

8
FIG. 9. Probability amplitudes over the first half of the Oscillation cycle for a 2-qubit simuation.

9
FIG. 10. Probability amplitudes over the second half of the Oscillation cycle for a 2-qubit simulation.

10
FIG. 11. Probability amplitudes over the second half of the Oscillation cycle for a 4-qubit simualtion.
ˆxd=rπ
2N







−N/2 0 0 . . 0
0 (−N/2) + 1 0 .0
0. . .
0. . 0
0. . 0
0. . . 0 (N/2) −1







(13)
The square of the operator will be given by
[ˆxd]0=π
2N







(−N/2)20 0 . . 0
0 ((−N/2) + 1)20.0
0. . .
0. . 0
0. . 0
0. . . 0 ((N/2) −1)2







(14)

11
FIG. 12. Probability amplitudes over the second half of the Oscillation cycle for a 4-qubit simulation.
The Unitary operator will then be evaluated as
Uˆx(t) = exp[−ι[xd]02t
2]
Which can be expanded using (Eq. (6)) as
exp(−ιt
2[xd]0) = I+ Σ∞
m=1(−ιt
2)m[xd]0m
m!(15)
After expanding we get
Uˆx(t) = I+ ( −ιt
2)[xd]2
1+ ( −ιt
2)2[xd]4
2! + ( −ιt
2)3[xd]6
3! +....
We ignore the π
2Nfactor for reducing mathematical
clutter and write the individual elements of the Unitary
matrix in the table below
Uˆx[1,1] 1 −ιt
2(−N
2)2+ι2t2
4
1
2(−N
2)4−ι3t3
8
1
6(−N
2)6+......
Uˆx[2,2] 1 −ιt
2(−N
2+ 1)2+ι2t2
4
1
2(−N
2+ 1)4+......
Uˆx[3,3] 1 −ιt
2(−N
2+ 2)2+ι2t2
4
1
2(−N
2+ 2)4+......
.
.
..
.
.
Uˆx[N , N] 1 −ιt
2(N
2−1)2+ι2t2
4
1
2(N
2−1)4+......
Clearly, each element is a Taylor exponential expansion
and the unitary matrix can be written as,

12
Uˆx(t) =






e(−ιt
2(−N
2)2)0. . . 0
0e(−ιt
2(−N
2+1)2).
.
.
........
.
..
0. . . 0e(−ιt
2(−N
2−1)2)






We may now substitute the value of N=4 for 2-qubit
system and N=16 for 4-qubit system to cross check the
results. We can also take out the first element as a com-
mon multiplicative factor which acts as the global phase.
This makes the first element unity as for N-dimensional
system the first basis |000....0iis incapable of picking up
any phase howbeit. We then observe a symmetry in the
mth and (N−m+ 2)th element. This is in accord with
the calculations shown for 2-qubit and 4-qubit systems
above.