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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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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.