Journal of Quantum Information Science, 2012, 2, 55-60
http://dx.doi.org/10.4236/jqis.2012.23010 Published Online September 2012 (http://www.SciRP.org/journal/jqis)
Quantum Image Searching Based on Probability
Distributions
Fei Yan1, Abdullah M. Iliyasu1,2, Chastine Fatichah1, Martin L. Tangel1, Janet P. Betancourt1,
Fangyan Dong1, Kaoru Hirota1
1Department of Computational Intelligence and Systems Science, Tokyo Institute of Technology, Yokohama, Japan
2College of Engineering, Salman Bin Abdul-Aziz University, Al-Kharj, KSA
Received August 19, 2012; revised September 4, 2012; accepted September 10, 2012
ABSTRACT
A quantum image searching method is proposed based on the probability distributions of the readouts from the quantum
measurements. It is achieved by using low computational resources which are only a single Hadamard gate combined
with m + 1 quantum measurement operations. To validate the proposed method, a simulation experiment is used where
the image with the highest similarity value of 0.93 to the particular test image is retrieved as the search result from 4 × 4
binary image database. The proposal provides a basic step for designing a search engine on quantum computing devices
where the image in the database is retrieved based on its similarity to the test image.
Keywords: Quantum Computation; Image Processing; Quantum Image; Quantum Circuit; Image Searching;
Probability Distribution
1. Introduction
Research on quantum image processing has started from
proposals on quantum image representations such as Qubit
Lattice [1], Real Ket [2], and Flexible Representation of
Quantum Image (FRQI) [3]. On the basis of these image
representations, basic quantum operations can be realized
by applying the elementary gates such as Pauli-X and
Hadamard gates combined with appropriate quantum
measurements [4,5]. For example, several processing
transformations have been proposed based on the FRQI
representation such as the geometric transformations,
GTQI [6], and the CTQI [7], which focuses on the color
information. In addition, a method to analyze the similar-
ity between two FRQI quantum images of the same size is
suggested in [8], which advances a fundamental step to-
wards image searching on quantum mechanical systems.
Inspired by the image searching on conventional com-
puters, the research on quantum image searching is also
an indispensible field on quantum image processing. In
order to improve the limitation of the traditional search-
ing, e.g. only text based and time consuming, the quan-
tum image searching on the strength of the content of the
images can be executed in parallel to realize more effi-
cient computation.
A quantum image searching method is proposed whe-
reby an image could be retrieved as a search result from a
database based on the extent of its similarity in compari-
son with the particular test image. The searching result is
provided by the probability distributions from two types
of quantum measurements, the first of which, Z-axis
measurement, represents the similarity between two cur-
rent images being compared; the second type, S-axis
measurements, gives the position of the compare- ing
results in Z-axis measurement. Succinctly put, the main
contributions of this work include the analysis of the si-
milarity between multiple pairs of images simulta-
neously and the proposal of the whole scheme for the
image searching on quantum mechanical systems.
The searching process is based on “parallel compare-
son”, where 2m pairs of quantum images are compared in
parallel. In addition, the method is executed using low
computational resources in comparison with performing
the same task on traditional computing devices, since
only a single Hadamard gate as well as m + 1 quantum
measurement operations could transform the entire in-
formation encoding the quantum images in a strip simul-
taneously.
The Z-Strip is defined based on the flexible represent-
tation of quantum images (FRQI) in 2. The proposed
scheme to realize image searching on quantum mechani-
cal systems is presented in 3. The simulation experiment
and its discussion are shown in 4.
2. Representation of Z-Strip to Indicate
Multiple FRQI Images
For the quantum image processing, a good deal of opera-
C
opyright © 2012 SciRes. JQIS

F. YAN ET AL.
56
tions are done by relying on the corresponding applicai-
tons on classical image processing as reference [3,9]. The
flexible representation for quantum images, FRQI [3],
which is similar to the pixel representation for images on
conventional computers, captures the essential information
about the colors as well as the corresponding positions of
every point in an image and integrates them into a quan-
tum state having its formula in (1),

2
21
0
1,
2
n
ii
n
I
nc



i
(1)
cos0 sin 1,
ii i
c   (2)


2
0, π2,0,1,, 21,
n
ii
   (3)
where 0 and 1 are 2-D computational basis quan-
tum states, i, are 2n-D computa-
tional basis quantum states and 2
01 21
n, is
the vector of angles encoding colors. There are two parts
in the FRQI representation of an image;
2
0,1,, 21
n
i


,, , 

 
i
c and i
which encode information about the colors and corre-
sponding positions in the image, respectively.
A dexterous property of Z-strip representation encod-
ing 2m+1-ending FRQI images is its ability to utilize the
parallelism inherent to quantum computation in order to
transform multiple images using very few quantum re-
sources. The Z-strip representation is defined in Defini-
tion 1.
Definition 1 A Z-strip,

,
Z
mn , is a horizontal
combination of two strips [10], which are located on the
left and right side, respectively. The state of Z-strip is
defined by







21
0
12
,
101
2
m
ss s
m
Zmn
Ln Rns




,
(4)
where

s
Ln and

s
Rn are FRQI images as de-
fined in (5) and (6),

2
21
0,,
1,
2
n
silsi
n
Lnc i



 (5)

2
21
0,,
1,
2
n
sirsi
n
Rnc i



 (6)
,, ,,,,
cos0sin1 ,
lsi lsilsi
c   (7)
,, ,,,,
cos0sin1,
rsi rsirsi
c   (8)


,, ,,
,0,π2,
lsirsi
  (9)
2
0,1,, 21,0,1,, 21.
n
is
m

(10)
As seen in Figure 1, the size of a Z-strip in the repre-
sentation captures the input state comprising 2m+1 quan-
tum images. The Z-axis differentiates the strip which is
located on the left and the right position. Each image in
the Z-strip is an FRQI state while the combination of
such states in the Z-strip is best represented as a Z-FRQI
state.
The Z-FRQI state represents 2m + 1 quantum images us-
ing only m + 2n + 2 qubits since all of the images are of
the same size on this Z-strip. A notation “○” for “0” or
“●” for “1” control-condition on Z-axis or S-axis, is suf-
ficient to specify any quantum image in the Z-strip. In
addition, combining with the control-conditions from the
position
y
x to the color wire; every pixel in this
strip can be accessed. The representation also facilitates
the quantum operation to all the images in this strip.
An example that has two 2 × 2 images on both the left
and right side of the Z-strip, respectively, including its
circuit structure and Z-FRQI state is shown in Figure 2.
3. Image Searching on Quantum Mechanical
Systems
Inspired by the image searching on conventional com-
puter, quantum image searching from a database is also
an indispensable field in quantum image processing [11,
12]. A first step towards realizing that would be to pro-
pose a scheme so as to evaluate the extent to which two
or more images are similar to one another. The parallel
computation on quantum computer leads us to find a way
Figure 1. Circuit structure to encode the Z-strip input.
Figure 2. An example of Z-strip, its circuit structure and
Z-FRQI st a te.

F. YAN ET AL.
57




10
ss
PP
that comparing many pairs of images in parallel. The
proposal of the Z-strip comprising 2m+1 images in the
Definition 1 provides us a crucial condition to make the
parallel comparison of quantum images possible because
the operation on the strip wires can transform the infor-
mation in every image simultaneously. The generalized
circuit structure of comparing 2m pairs of FRQI quantum
images in parallel is presented in Figure 3.
1

, as they should.
Definition 2 Pixel difference in position i, ,
s
i

, is de-
fined by


,,,,,,
, 0,π2,
silsirsisi
   (16)
where ,,lsi

and ,,rsi

represent the color information
at position i of the two images which are at the sth posi-
tion of the Z-strip, respectively.
The input of this circuit is the Z-FRQI state as defined
in (4), a Hadamard gate, which maps the basis state 0
to

01 2 and 1 to

01 2, is applied
on the Z-axis to obtain the new mathematical expressions
between the two images being compared. The final step
in the circuit consists of m + 1 measurements from which
the similarity can be retrieved in each pair of images.
It is apparent that, arising from (15) and (16), the pixel
difference ,
s
i

is related to the probability of getting
readout of 1 from the Z-axis,

1
s
P, in the measure-
ment and


1
s
P will increase when pixel difference
increases. Furthermore, the similarity between the two
images, which is the function of the pixel differences at
every position, depends on

1
s
P as given by
When n experiments are performed, the measurement
results on the Z-axis follow a binomial distribution. The
probability of obtaining k readouts of 1 in n experiments
is given by the probability mass function
 


2
21
0,
2
1
,121 cos
2,
n
s
ss i
n
sim LnRnPsi



 
(17)
where


s
Ln and


s
Rn are the two images being
compared,

 
,1 nk
kk
n
PrXkC pp  (11)

1
s
P

is defined in (15), and





,0
where X is the incident that the result of measurement is
1, p is the probability of 1 when the results on the Z-axis
are measured, . 0,1, ,kn
,1sim LnRn
ss
Two special cases of the similarity between two quan-
tum images are listed as follows:
.
a) ifi

, ,π2
si


, th en
 

,0
ss
sim LnRn

,
two images are totally different;
Meanwhile, the measurement results on the S-axis,
10
,
mr
s
ss
 , give the position of
probabilities of the measurements on the Z-axis. Accord-
ing to the readouts on both the measurements, the
similarity between each pair of images on the Z-strip can
be assessed, from which the quantum image searching
can be realized.

0,1
r
s
b) ifi

, ,0
si


, then
 

,1
ss
sim LnRn

,
two images are exactly the same,
where i = 0, 1, ···, 22n − 1, ,
s
i

is the pixel difference at
position i as defined in Definition 2.
Corresponding to the circuit shown in Figure 3, the
state of quantum system after applying the Hadamard
gate on the strip wire can be shown in (12) and (13).
Obviously, the result of the measurement depends on
the disparities between


s
Ln and

s
Rn . The
probability of state 0 on the Z-axis at position
12 0
,,,
mm
s
s
 s is shown by


2
21
0,,,
21
11 cos
22
0.
n
,
s
ilsir
n
P 



 
si
(14)
In the same manner, that of state 1 on the same
wire is


2
21
0,,,
21
11
1cos
,
22
n
.
s
ilsirsi
n
P 



 
 (15)
Figure 3. Generalized circuit structure for parallel com-
parison of quantum images.
The probabilities of these two states sum up to 1,
  



 

2
21
2
21
0
2
0
12
01 01
1
,222
1
01,
2
m
m
ziss
m
is ss s
m
HZmnLn Rns
Ln RnLn Rns





 









(12)
where
  
2
21
0,,,, ,,,,
1coscos0 sinsin1.
2
n
ssilsirsilsirsi
n
Ln Rni  


 

 (13)

F. YAN ET AL.
58
Based on the comparison method and the probability
distributions introduced above, the scheme to accomplish
the image searching on quantum mechanical systems is
presented in Figure 4.
The quantum images are prepared from the classical
images using FRQI representation [3,8,10,13]. The color
information as well as the corresponding positions of
every point in the classical image is integrated into the
quantum state, and 2m+1 quantum images being com-
pared are combined as a Z-strip. Because of the superpo-
sition property of quantum computation, such a work can
be realized using only a few quantum resources.
The Z-strip prepared in the preceding period is trans-
formed using a gate array comprising of geometric,
GTQI [6], and color, CTQI [7], transformations on all the
images in the strip. For this particular application, the
transformations are built in a way to allow the recovery
of the pixel difference as defined in (16). This transfor-
mation unit combines with measurement operations that
follow it to convert the quantum information into the
classical form as probability distributions. The Z-strip is
prepared n (n > 1) times to compare the similarity be-
tween two quantum images in parallel since a measure-
ment would destroy the superposition state in the quan-
tum system [5]. Extracting and analyzing the distribu-
tions gives information that the similarity values between
the quantum images being compared, so that the image
with the highest similarity to the particular test image
could be retrieved as a result from the database.
The operation to search image on quantum mechanical
systems is realized by using only a single Hadamard gate
and several measurements. Such an image searching
scheme, however, can only be achieved on a classical
computer by comparing one pair of images at a time.
Hence, the proposed method offers a significant speed-up
compared to how it is performed using classical comput-
ing resources.
Figure 4. Block diagram of scheme to realize image search-
ing on quantum mechanical systems.
4. A Simulation Experiment to Search
Quantum Images from Database
A conventional desktop computer with Intel Core i7, 2
Duo 2.80 GHz CPU, 4GB RAM and 64bit operating
system is used to simulate the experiment. The simula-
tion experiment is based on linear algebra with complex
vectors as quantum states and unitary matrices as unitary
transformations using Matlab, and the program is en-
coded by means of equations as well as the definitions
that are introduced in earlier sections of this paper. The
purpose of this experiment is that to search the image
from a database which has the highest similarity with the
test image. An original database which includes
sixty-four (64) 4 × 4 binary image data is used, then the
Z-strip comprising of


02D,

12D, ,


63 2D,
and sixty-four (64)


2Ts is constituted as shown in
Figure 5.
The corresponding circuit structure to realize such an
image searching is presented in Figure 6. There are three
steps to achieve this comparison:
Figure 5. Image searching from database
D
.
Figure 6. Circuit structure for realizing the image searching
in Figure 5.

F. YAN ET AL. 59
Step 1. The test images


2T is prepared from the
classical version using FRQI representation and inte-
graed to a Z-strip state with the images D in the da-
tabse.
Step 2. A Hadamard operation is applied on the Z-axis
in order to compare the test image

2T with

02D,

12D, ,


63 2D.
Step 3. The measurements which convert the quantum
information to the classical form are used on the S-axis
and Z-axis to distribute the readouts from which the his-
togram is built to reflect the similarity of the 64 pairs of
images.
The circuit comprises of 12 qubits of which 6 are used
to address positions of the image, 1 qubit is reserved for
storing the information about the colors, and the remain-
ing qubits are prepared for representing the Z-strip wire
where the Hadamard gate and measurement MZ are ap-
plied. A simulation of a single Hadamard gate and 7
measurement operations are used to obtain the similari-
ties for these 64 pairs of images based on the probabili-
ties of getting the readouts on the Z-axis and S-axis in the
measurements as shown in the Figure 7. From the histo-
gram, the image


37 2D, which manifests the highest
similarity value of 0.93 to the test image

2T is re-
trieved as the search result. There is only one different
grid between the test image and

37 2D, which are
shown in Figure 8. It is testified from that the quantum
image searching is based on the pixel difference between
the test image and the images in the database.
The foregoing experiment provides the foundation for
the next step in quantum image processing based on the
FRQI representation. The results as indicated in this sec-
tion show that the quantum image searching on quantum
mechanical systems is feasible and practical. Further-
more, the target area to apply the proposed method is the
development of the search engine on quantum computing
devices.
5. Conclusions
The simulation experiment is performed to search for a
target image from an original database comprising of
sixty-four (64) binary images. There are 12 qubits which
encodes each image in the Z-strip and 7 quantum meas-
urements which are for converting the quantum informa-
tion to the classical form as probability distributions in
the circuit. According to the readouts from the measure-
ments, the similarity of each pair between the test image
and the images in the database is calculated. For the
simulation-based database used in this paper, the 38th
image,

37 2D, with the highest similarity value of
0.93 is retrieved as search result. It is concluded that the
more images in the database, the better the ability of the
proposed method. This is because m qubits on the strip
wires can represent 2m quantum images in the Z-strip,and
Figure 7. Similarities among different pairs of images in
Z-strip.
Figure 8. The test image


2T and the retrieved image


37
2D.
only one qubit on the Z-axis can represent the images on
both the left and right side of Z-strip. This further dem-
onstrates the low computational resources of the pro-
posed method compared to performing the same task on
traditional computing devices.
As for future work, the proposal will be applied on de-
signing a search engine on such quantum computing de-
vices that the image in the database is retrieved based on
its similarity to the test image. Most of the search engines
recently are only based on the text to realize the search-
ing. Even some searching is developed based on the con-
tent of the images. It is, however, usually time-consum-
ing. This work, which realizes the searching based on the
content of the images and is executed in parallel, pro-
poses a basic step for the quantum image searching, es-
pecially when a database comprising of a huge amount of
data is confronted.
6. Acknowledgements
This work is sponsored by the ASPIRE League (Asian
Science and Technology Pioneering Institutes of Re-
search and Education). The authors appreciate the kind
comments and professional criticisms of the anonymous
reviewer. The suggestions and advices received from
Messrs Phuc Q. Le and Bo Sun are also appreciated.
These various inputs have greatly enhanced the overall

F. YAN ET AL.
60
quality of the manuscript and opened numerous perspec-
tives geared toward improving the work in the future.
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