Content uploaded by Marina Lisnichenko
Author content
All content in this area was uploaded by Marina Lisnichenko on Sep 05, 2022
Content may be subject to copyright.
Content uploaded by Marina Lisnichenko
Author content
All content in this area was uploaded by Marina Lisnichenko on Sep 05, 2022
Content may be subject to copyright.
Content uploaded by Stanislav Protasov
Author content
All content in this area was uploaded by Stanislav Protasov on Feb 24, 2022
Content may be subject to copyright.
Quantum Image Representation: A Review
Marina Lisnichenko MSc 1* and Dr Stanislav Protasov PhD
1
MLKR Laboratory, Innopolis University, Universitetskaya,
Innopolis, 420500, Tatarstan Republic, Russia.
*Corresponding author(s). E-mail(s):
m.lisnichenko@innopolis.university;
Contributing authors: s.protasov@innopolis.ru;
Abstract
Quantum programs allow to process multiple bits of information at the
same time, which is useful in multidimensional data handling. Images are
an example of such multidimensional data. Our work reviews 14 quan-
tum image encoding works and compares implementations of 8 of them
by 3 metrics: number of utilized qubits, quantum circuit depth, and
quantum volume. Our work includes a practical comparison of 2nˆ2n
images encoding, where nvaries from 1up to 8. We observed that Qubit
Lattice approach shows the minimum circuit depth as well as quantum
volume, Flexible Representation of Quantum Images (FRQI) utilizes the
minimum number of qubits. If to talk about variety of processing tech-
niques, FRQI and Novel Enhanced Quantum Representation (NEQR)
representations are the most fruitful. As far as quantum computers
are limited in qubit number, we concluded that almost all approaches
except Qubit Lattice are promising for the near future of quantum
image representation and processing. From the point of view of the
quantum depth, discrete methods showed the most appropriate result.
Keywords: quantum images, complexity, quantum algorithms, quantum
image processing
1
2Quantum Image Representation: A Review
1 Introduction
Quantum computation is a rapidly developing field. In 2021 private capital
investments in quantum computations exceeded $3B [1]. Due to the quan-
tum parallelism the information processing is performed potentially faster.
Acceleration of the calculations through parallelism is highly relevant to mul-
tidimensional data, including images. In the quantum programs, images are
usually represented in the same way as in classical machines – with pixel coor-
dinates and pixel intensities, but amplitude and phase encodings use different
physical parameters for these values. In this work, we compare 14 ways of image
encoding including methods: with quantum state amplitudes – amplitude-
angular (continuous), qubit binary – amplitude-state (discrete), mixed, and
phase image representations. We implemented 8 core representation techniques
(other methods are derived or equivalent to the implemented and share their
characteristics) for the practical comparison1. Our survey covers the following
quantum image representation methods:
•qubit lattice [2] (2003);
•real ket [3] (2005);
•flexible representation of quantum images - FRQI [4] (2011);
•multi-channel representation for images - MCRQI [5] (2011);
•novel enhanced quantum representation of digital images - NEQR [6]
(2013);
•quantum states for M colors and quantum states for N coordinates -
QSMC and QSNC [7] (2013);
•a simple quantum representation - SQR [8] (2014);
•normal arbitrary quantum superposition state - NAQSS [9] (2014);
•generalized quantum image representation - GQIR [10] (2015);
•quantum representation of multi wavelength images - QRMW [11] (2018);
•quantum image representation based on bitplanes - BRQI [12] (2018);
•order-encoded quantum image model - OQIM [13] (2019);
•quantum representation of indexed images and its applications - QIIR [14]
(2020);
•Fourier transform qubit representation - FTQR [15] (2020).
The review contains the information about quantum image representation
methods and applicable image processing techniques. To compare the methods,
we use 3 parameters:
•number of utilized qubits - each existing computer is limited in number of
qubits and this number defines the possibility of algorithm execution;
•circuit depth - length of the longest quantum gate path from the zero-
state to the end of the encoding procedure. The bigger the depth is, the
more errors affect on the output result quality. This metric is an analogy
of the classical time complexity;
•quantum volume - squared minimum between the circuit depth and num-
ber of qubits. This metric varies for different quantum machines, and
1github.com/UralmashFox/QPI
Quantum Image Representation: A Review 3
depends on the base gates. Quantum volume is an integral metric which
allows to evaluate a computing capability with a single quantity.
In conclusion 3we sum the observed information and make a statement
about future work.
2 Quantum image representation (QIR)
techniques
We identified four major ways of quantum image representation: continuous
amplitude representation, where pixel intensity is encoded with quantum
state amplitude p, corresponding to observation probability p2,ampli-
tude with binary intensity representation (discrete), mixed, and phase
intensity representation. Figure 1shows the suggested classification.
Continuous representation allows to use a single qubit for intensity or
coordinate encoding. This is the main advantage of the methods. Multiple
measurements are required to estimate the pixel intensity with high precision.
In the discrete intensity representation, oppositely, each state corresponds
to separate intensity or coordinate bit-value. Measurement result contains
accurate data expansion of a single pixel data.
Mixed representations either store pixel coordinates discretely or do not
encode coordinate.
Phase encoding uses continuous representation, but in an XY Bloch sphere
projection plane.
In the following subsections, we describe each approach separately.
2.1 Mixed representation
The chapter describes quantum image representation algorithms based on
both discrete and continuous methods. It’s common for mixed methods where
continuous encoding is used for the pixel intensity and the pixel location is
represented discretely. We also include Qubit Lattice and SQR algorithms,
however these methods do not have a specific coordinate encoding procedure.
2.1.1 Qubit Lattice
Venegas-Andraca and Bose [2] did a major preparatory work in the quantum
image representation and processing. The paper describes the basic quantum
definitions and measured results interpretation. Proposed image representa-
tion is na¨ıve and consists of literally copying the classical representation into
quantum. Authors suggest to use Ryrotation gate to set each pixel’s inten-
sity. Therefore, the number of utilized qubits is 22nwhere 2nˆ2nis an image
size. Figure 2b shows an example of encoding the image with pixel intensities
t0,125,200,255u.
Coordinate qubits absence and quantum circuit simplicity are the strong
sides of the encoding method. Due to simplicity, authors of quantum convolu-
tional neural networks papers actively utilize this representation method (or
4Quantum Image Representation: A Review
Quantum image representation
Amplitude Phase
Continuous Discrete
Mixed
SQR, Qubit attice
FRQI, MCRQI, Real ket
QSMC & QSNC
NAQSS, OQIM
NEQR, QUALPI, GQIR
QRMW, BRQI, QIIR
FTQR
Fig. 1: Quantum image representation (QIR) classification
Fig. 2: (a) Classical image and (b-f) quantum image representations
modified) even if it is not evidently claimed [16], [17], [18]. At the same time,
classical image processing based on Qubit Lattice did not spread. The approach
has strong negative sides such as big number of used qubits and small number
of known processing methods. However, for the sake of justice, it is the first
formulation of quantum image storing.
2.1.2 FRQI
Authors [4] use the continuous amplitude encoding with intensity-to-amplitude
representation.
|Ipαqy “ 1
2n
22n´1
ÿ
i“0pcosαi|0y ` sinαi|1yqb | iy(1)
where |Ipαqy is a quantum image representation, αis a parameter responisble
for intensity and equal to a half of Ryrotation angle, |iyrepresents the pixel
coordinate binary expansion. The greater αis, the closer pixel intensity to the
Quantum Image Representation: A Review 5
maximum. Therefore, the precision of the intensity estimation depends on the
number of circuit executions.
Figure 2c shows the image with pixel intensities t0,125,200,255u. The
image representation code is available in our repository.
The FRQI implies a huge number of processing algorithms. For example,
the paper presents processing algorithms that affect intensity, coordinate, or
both intensity and coordinate. The first processing group changes all the pixel
intensities, the second group changes intensity at some locations, the last group
”targets information about both color and position as in Fourier transform”.
Moreover, multi-channel expansion [19], image compression, line detection [4],
binarization, histogram computing, histogram equalization [20], global and
local translation designs [21] are also available for FRQI. The image repre-
sentation technique maintains comprehensive processing such as information
hiding [22], Richardson-Lucy image restoration [23], multilevel segmentation
[24], hybrid images creating [25]. Additionally, FRQI helps to find correla-
tion property of multipartite quantum image [26], implement image fusion
[27], encrypt images via algorithm based on Arnold scrambling and wavelet
transforms [28].
At the same time, FRQI has the following drawbacks:
•due to the single intensity qubit ”some digital image-processing operations
for example certain complex color operations”, are impossible (such as
”partial color operations and statistical color operations”) [6].
•circuit depth is Op24nqfor 2nˆ2nimage [6].
FRQI is beneficial for the applications with limited qubit number and does
not demand high intensity precision. FRQI supports broad spectrum of quan-
tum image processing techniques useful for basic image processing. However
the continuous intensity representation approach may limit such processing
algorithms as edge detection and texture features.
2.1.3 MCRQI
The authors of [5] apply FRQI 2.1.2 approach to the RGBα images, where α
is a transparency channel. The difference is in the number of qubits used to
encode an intensity. Authors represent the multichannel image as follows:
|Ipθqy “ 1
2n
22n´1
ÿ
i“0|ci
RGBα yb | iy(2)
where |ci
RGBα yis a color state and equals to:
|ci
RGBα y “ 1
2ˆ„cos θRi |0y
sin θRi |1yȷb | 00y ` „cos θGi |0y
sin θGi |1yȷb | 01y`
„cos θBi |0y
sin θBi |1yȷb | 10y ` „cos θαi |0y
sin θαi |1yȷb | 11y˙.
(3)
6Quantum Image Representation: A Review
Multi-channel representation encodes intensities P r0,255swith angles
θRi, θGi , θBi , θαi P r0,π
2svia uniform scaling. As far as image has several
channels, authors say about applicability of the one-channel operations for
each of them. Autors [29] developed a chromatic framework for quantum
movies and provided frame-to-frame and color of interest operations, sub-
block swapping. Yan et al. described audio-visual synchronisation in quantum
movies using MCRQI [30]. Sun et al. proposed channel of interest realization,
channel-swapping and αblending operations [31]. Hu et at. all proposed image
encryption based on FRQI modification [28]. The MCRQI has the same pros
and cons as the FRQI. Additionally, to get access to each color layer channels
r, g, b has to be measured separately.
2.1.4 SQR
The authors [8] combine Qubit Lattice image representation with normalized
pixels and NEQR (see 2.3.1) approach of max and minimum pixel values.
Instead of pixel intensity authors use ”energy” term to show the relation to
the infrared images. Their paper describes the Qubit Lattice approach [2] with
modifications. Authors of the Qubit Lattice converted intensities directly into
angles, while in SQR intensities go through a normalization step first. Suppose,
Eij is an energy value detected at the position i, j,Emminimum energy value,
EMmaximum energy value. The normalized energy value is ˜
Eij “Eij ´Em
EM´Em.
˜
Eij determines Ryrotation angle via the following expression:
θij “2 arcsin ˜
Eij (4)
To encode complete image information, 10 qubits store EMand Emenergy
values. Totally, to encode an image of size 2nˆ2nalgorithm needs 2nˆ2n`10
qubits.
Authors provide clear explanation of how to convert energy into quantum
representation and why is it so intuitive. Due to narrow representation spec-
ification, the SQR did not become widespread in terms of processing. Paper
describes global and local operations, “retrieval of marked information” (kind
of segmentation). However the approach is still ”heavy” in qubits.
2.1.5 Real Ket
Latorre [3] attempted to separate a pixel’s coordinates and intensity into quad-
rants. The first step of the encoding is to split image into 4 parts (upper-left,
upper-right, lower-left, and lower-right). Next, the same step (splitting) repeats
until each block contains a quad of pixels. A pair of qubits describes the grid-
square pixel coordinate and additional qubit’s amplitude holds intensity. This
method, as well as FRQI, intensively uses multicontrol Rygated (Figure 3a),
but Real Ket does not have a dedicated intensity qibit. In theorem 8 authors
of [32] proved that decomposition of one such k-qubit controlled gate requires
2kCNOT gates.
Quantum Image Representation: A Review 7
Overall, the algorithm requires 2n`1 qubits, where 2nˆ2nis an image
size. The final state equation is the following:
|Ψ2nˆ2ny “ ÿ
i1,...,in“1,...,4
cin,...,i1|in, ..., i1y.(5)
where |Ψyis a quantum image, cipixel intensity, inis a representation of
the pixel’s location corner in the 4 ˆ4 grid. Figure 3b gives graphical pixels’
coordinates representation for the 4 ˆ4 image.
(a) Multicontrol Ry gate
i2“1i2“2
i2“3i2“4
i1“1i1“2i1“1i1“2
i1“3i1“4i1“3i1“4
i1“1i1“2i1“1i1“2
i1“3i1“4i1“3i1“4
(b) Real Ket image encoding
Fig. 3: (a) Multicontrol Ry gate (gate consists of 3 control qubits that rotate
the last one – target – qubit for an angle α) and (b) Real Ket coordinate
estimation.
Authors of the paper describe the image compression algorithm based on
Fourier transform. Ma et al. proposed quantum Radon transform based on
Real Ket encoding [33]. The similar image representation is used for low-rank
tensor completion [34].
2.2 Pure continuous representation
Representation algorithms of the current chapter utilize only continuous
amplitude encoding for both pixel intensity and coordinates.
2.2.1 QSMC and QSNC
Suppose the image consists of mdifferent pixel intensities taken from Mpos-
sible (if an image is an 8-bit gray-scaled, M“256), and contains N“2nˆ2n
pixels. The authors [7] split image representation into 3 parts: quantum reg-
ister for mcolors (QSMC), quantum register for Ncoordinates (QSNC),
and image storing. Authors provide the following intensity representation
algorithm, which we reproduced in our repository.
If |wiyis a pixel intensity representation and |uiyis a pixel coordinate
representation, then full pixel representation is |ψiy “| wiyb | uiy.
The paper proves the reliability of the method from the mathematical
prospects. Moreover, the authors provide a quantum image compression algo-
rithm and image segmentation based on extended Grover search [35].The
8Quantum Image Representation: A Review
method was not broadly utilized in quantum image processing and we could
not find any publications based on it.
The main weakness of the method is the continuous approach to coordinate
representation. When the number of pixels becomes bigger, even a small shift in
probabilities affects the outcome. For example, IonQ QPU’s Rygate precision
is 10´3πradians [36], what makes impossible to encode coordinates of image
larger than 512 ˆ512. In this case the coordinate encoding error becomes
significant and coordinate estimation precision decreases.
2.2.2 OQIM
The authors [13] provide a way to store classical images by intensity sorting.
Firstly, authors map the ordered pixel intensities present in the image to num-
bers. Then the continuous representation defines pixel color and coordinate (θ
and ϕrespectively). Figure 2d provides an example of the 2 ˆ2.
The authors’ idea is to utilize a single qubit for both intensity and coor-
dinate. An additional qubit controls the mode of representation (coordinate
or intensity). Thus, for the image represented in the figure 2a the following
equation holds:
|Iy “ 1
23{2”
`pcos θ0|0y ` sin θ0|1yq | 0y`pcos ϕ0|0y ` sin ϕ0|1yq | 1y˘b | 00y`
`pcos θ1|0y ` sin θ1|1yq | 0y`pcos ϕ1|0y ` sin ϕ1|1yq | 1y˘b | 01y`
`pcos θ2|0y ` sin θ2|1yq | 0y`pcos ϕ2|0y ` sin ϕ2|1yq | 1y˘b | 10y`
`pcos θ3|0y ` sin θ3|1yq | 0y`pcos ϕ3|0y ` sin ϕ3|1yq | 1y˘b | 11yı(6)
Authors propose the histogram specification process that allows to shift
the image histogram. Authors provide ”image comparison between two images
and multiple images..., the parallel quantum searching” [37]. Paper also covers
the multidimensional OQIM case. Basic algorithm implementation is available
in our repository.
2.2.3 NAQSS
Authors’ encoding approach [9] is similar to the QSMC and QSNC (section
2.2.1). Encoding starts with color representation. If colorirepresents the ith
pixel color, then colors tcolor1, color2, ..., colorMu- a set of all possible colors
for the image. Therefore rotation angle ϕi“πpi´1q
2pM´1qdefines ith state rotation
angle. For the RGB image, each color value is i“Rˆ256ˆ256`Gˆ256`B`1
what takes range from color1“ p0,0,0qto color16777216 “ p255,255,255q.
The coordinate representation is the following. Suppose data has kdimen-
sions (for flat image we assume k“2). Then, each pixel sets in the following
state:
Quantum Image Representation: A Review 9
|ψϕy “
2n´1
ÿ
i“0
ai|v1y | v2y... |vky(7)
where i“i1...ijij`1...il...im...inis a binary expansion of the coordinate
v1“i1..ij,v2“ij`1...iland vk“im...inare the expansions of each coordinate
separately.
The approach bases on a normalised intensity representation:
θ“ai
bř2n´1
y“0a2
y
,(8)
where aiis an intensity of current pixel iand ř2n´1
i“0θ2
i“1.
Authors suggests to use image cropping and extend circuit by 1 qubit. The
equation explains the cropped quantum image:
|Ψy “
2n´1
ÿ
i“0
θi|v1y | v2y... |vky | χiy(9)
where |χiy “ cosγi|0y ` sinγi|1y,γi”represents the serial number
of the sub-image which contains the pixel corresponding to the coordinate
|v1y | v2y... |vky”. Simply – index of a sub-image.
The idea of image cropping index allows to retrieve a target sub-image
(segmentation) from a quantum system. Additionally, a single qubit is a cheap
way of color representation. Li et al. described image encryption based on nor-
mal arbitrary superposition state [38]. Based on geometric transformations of
multidimensional color images based on normal arbitrary superposition state
authors provide such transformations as two-point swapping, flip (unclud-
ing local), orthogonal rotation, and translation [39]. Zhou and Sun proposed
multidimensional color images similarity comparison for QNASS encoded
images [40].
2.3 Pure discrete representation
Representation algorithms of this chapter utilizes only discrete amplitude
encoding for both pixel intensity and coordinates. Methods described here
utilizes the multcontrol CX-gate as in the figure 4.
Fig. 4: multi-control CX gate. Gate consists of 3 control qubits that apply
NOT operation on the last one - target - qubit
10 Quantum Image Representation: A Review
2.3.1 NEQR
Authors [6] suggest to use full intensity binary expansion for image encoding.
The most powerful thing of this explicitly encoded approach is the possibility
to read intensity determinedly. The image with the intensity range 2q(where
qequals 8 in case of 256 intensity levels) is encoded in the following way:
|Iy “ 1
2n
2n´1
ÿ
Y“0
2n´1
ÿ
X“0
q´1
â
i“0|Ci
Y X y | Y X y(10)
Where |Ci
Y X yand |Y X yare intensity and coordinate expansions
respectively. The same principle INCQI authors use in their paper [41].
Figure 2e represents an example of the image encoded with NEQR. The
algorithm code is available in the repository.
Apart from determinism, the NEQR has several advantages. The paper
says about ”partial color operations and statistical color operations” [6]. The
authors of the described paper split all possible image operations into 3 groups:
•Complete Intensity Operations (CC): changes pixel’s intensity in a whole
image or it’s area.
•Partial Intensity Operations (PC): changes intensity of pixels in the
certain gray-scaled range.
•Color Statistical Operation (CS): computes intensity statistics.
Authors extended approach to the log-polar images in QUALPI [42]
(figure 2f shows the encoding example). Additionally, such comprehensive
processing as complement intensity transformation and upscaling are also
described in the book [19].
Next, strong point of the NEQR is a more efficient circuit compression
algorithm. The Espresso Boolean expressions compression [43] is a base for this
algorithm. While FRQI algorithm compression ratio is 50% of utilized gates,
NEQR allows to reduce the number of gates by 75%.
The negative side appears from the determinant approach. Each intensity
expansion leads to utilizing as much qubits as the length of the expansion.
Thus, the quantum circuit depth increases proportionally to the image inten-
sity resolution. However image size increases the number of utilized qubits only
logarithmically.
As a result, NEQR is useful for reliable intensity representation, supports
large images and comprehensive processing. The approach is less applicable for
data with high intensity resolution. However this encoding approach is boun-
tiful in terms of processing techniques. For instance, identification of desired
pixels in an image using Grover’s quantum search algorithm [44], edge extrac-
tion based on Laplacian operator [45], scaling [46], least squares filtering [47],
morphological gradient [48], edge extraction based on classical Sobel opera-
tor [49], quantum image histogram [50], edge extraction based on improved
Prewitt operator [51], mid-point filter [52] dual-threshold quantum image
segmentation [53], steganography based on least significant bit [54], erosion
Quantum Image Representation: A Review 11
and dilation [55]. Finally, the same encoding method is used for LSB-Based
quantum audio watermarking [56].
0 128 255
00 01 10 11
0
1
Fig. 5: Non-square image, empty cells are expanding to powers of 2
2.3.2 GQIR
The authors [10] refer to the NEQR approach (section 2.3.1) and suggest
an approach to represent nonsquare images. Figure 5represents the possible
nonsquare image with binary intensity expansions 00000000,10000000, and
11111111 respectively for each pixel.
As far as 3 qubits are able to represent up to 4ˆ2 images, therefore 5 states
out of 8 are redundant. To keep measured image shape the same as input’s,
the authors do not encode pixels without intensity value and leave them to
have black color with 0 qubit amplitude. This advantage allows to represent
images of an arbitrary shape.
The authors suggest image up-scaling based on the nearest-neighbor
method. In addition, Zhu et al. provided encryption scheme [57], Zhou and
Wan implemented image scaling based on bilinear interpolation [58], Zhang et
al. extended approach to floating-point quantum representation and proposed
two rows interchanging and swap operations [59]. We implemented GQIR
representation method in our repository.
2.3.3 QRMW
The authors [11] provide the representation approach for multichannel images.
Each channel corresponds to a specific wavelength. QRMW allows ”to encode
the color values corresponding to the respective wavelength channel of the
pixels in the image”.
The equation below defines the intensity of each pixel located in the x, y, λ
coordinates (where λrepresents wavelength channel index):
fpλ, y, xq “ c0
λyx , c1
λyx , ..., cq´2
λyx , cq´1
λyx ,
fpλ, y, xqPr0,2q´1s, ck
λyx P r0,1s,(11)
where 2q´1 is the maximum amplitude of any wavelength. Then, the
QRMW image representation is the following:
12 Quantum Image Representation: A Review
|Iy “ 1
?2b`n`m
2b´1
ÿ
λ“0
2n´1
ÿ
y“0
2m´1
ÿ
x“0|fpλ, y, xqyb | λyb | yxy,(12)
where bdefines the number of channels. As a results, the method contains
the extended realization of NEQR (2.3.1). In addition, the authors describe
image compression, complete and partial color operations, and position oper-
ations. The same authors presented the edge detection algorithm [60] and
extended their encoding approach to the multichannel audio representation
[61].
2.3.4 BRQI
The authors [12] suggest to split image into nbitplanes where nis a resolution
of the pixel intensity. Encoding proceeds through availability of the current
bitplane in the pixel with current coordinates.
Suppose an example of the gray-scaled 2 ˆ2 image with the following pixel
intensities: t0,125,200,255u. The binary expansion for pixel-wise intensity is
the following: 00000000,01111101,11001000,11111111. Taking bits from
each expansion, one can receive 8 bitplanes as in the figure 6.
(1) LSB (2) (3) (4) (5) (6) (7) (8) MSB
Fig. 6: Bitplane image representation, each letter corresponds to the number
of biplane. The first and the last bit for each bitplane have special names ”the
least significant bit” (LSB) and ”the most significant bit” (MSB) respectively.
This paper was inspired by general NEQR approach and defines jth
bitplane as follows:
|Ψjy “ 1
?2n
2n´k´1
ÿ
x“0
2k´1
ÿ
y“0|gpx, yqy | xy | yy(13)
where j- denotes the bitplane index pj“0, ..., 7q,gpx, yq P t0,1ushows
presence or absence of a bit in jth bitplane. As a result, the qubit scheme
consists of X-, Y-coordinate qubits, bitplane qubits and 1 target qubit, showing
availability of the bitplane’s current bit. The encoding method is implemented
in our repository.
BRQI supports processing operations: bitplane interchange, translation
and intensity operation. Heidari et al. proposed selective encryption for the
BRQI encoded information [62]. Recently, Khorrampanah et al. proposed
RGB images encryption [63]. The work of Mastriani, 2015 [64] deserves spe-
cial attention, as it describes Quantum Boolean (QuBo) image denoising
based on image bitplane splitting. The same paper explains the Classical-to-
quantum (C2QI) and Quantum-to-classical interfaces (Q2CI). Later Mastriani
Quantum Image Representation: A Review 13
proves [65] reliability of the bitplane splitting approach in term of C2QI and
Q2CI.
To conclude, the BRQI is a promising method in terms of image representa-
tion and processing. At the same time, the technique is built upon the NEQR,
thus the number of qubit remains in dependence on the intensity resolution.
2.3.5 QIIR
Indexed images consist of 2 tables: combination of the channel values (first
table) and pointers to the palette matrix (second table). An example of such
an image is in the figure 7. Authors [14] encode both image’s tables (|QDatay
and |QMap y) using NEQR.
(a) real image
0 3
0 15
(b) Quantum Data
Matrix
R G B
0
. . .
3
.
.
.
15
255 117 255
210 109 25
0 185 83
(c) Quantum Palette
Matrix
Fig. 7: QIIR image representation
2.3.5.1 Quantum Data Matrix
In order to encode 2nˆ2nQData matrix, authors apply NEQR 2.3.1 approach.
If pY, X qis a cell location, IY X is a cell value, then expression is the following:
|QDatay “ 1
2n
2n´1
ÿ
Y“0
2n´1
ÿ
X“0|IY X yb | Y X y(14)
2.3.5.2 Quantum Palette Matrix
The authors use the same NEQR 2.3.1 approach to encode the color palette.
In case of RGB encoding 24 qubits are reserved: 8 for each color. Following
equation describes the pallete matrix encoding.
|QMap y “ 1
?2q
2q´1
ÿ
j“0|Cjyb | jy(15)
Authors provide several processing techniques such as Ryrotation on 90°,
cyclic shift, color inversion, color replacement, color look-up, steganography.
Besides the authors’ suggested processing techniques, no other images handling
procedures exist from the best of our knowledge.
The number of utilized qubits is a major disadvantage of the method, as
the data literally consist of 2 images.
14 Quantum Image Representation: A Review
2.4 Phase representation. FTQR
Grigoryan and Aganyan offer phase-based image encoding [15].
|q
fy “ 1
?N
N´1
ÿ
k“0
eiαfk|ky(16)
where α“2π{1024. The term eiαfk|kyallows to map classical intensity
values p0,255qinto imaginary plane px`i¨yq. Here fkis in input signal in
the coordinate kwhich expands to xand y. In this way the representation is
following:
|q
fy “ 1
?MN
M´1
ÿ
m“0
N´1
ÿ
n“0
eiαfn,m |n, my(17)
Authors do not provide any processing techniques but extend the approach
up to multidimensional images. The image representation did not take share
in the processing techniques research due to its novelty from our opinion.
The method is purely phase based, and for measurement it needs phase-to-
amplitude transformation. This fact implies additional computation resources.
3 Discussion and conclusion
We reviewed 14 method of image encoding and propose to categorize them
according to classification in figure 1. Most of them have the common spe-
cific disadvantage, which comes from the quantum computing nature: it is
impossible to measure all the pixels in a single trial. For continuous methods,
the intensity estimation precision depends on the number of trials. Dis-
crete methods allow to derive information only about 1 pixel from 1 shot.
Figure 8represents the comparison of three metrics for the selected algorithms
implemented in our repository: circuit depth, qubit number, and quantum
volume.
2 4 8 16 32 64 128 256
100
101
102
103
104
105
106
107
Image side size
circuit depth, number of gates
BRQI
FRQI
QL
GQIR
NEQR
OQIM
QSMC
FTQR
(a) Circuit depth
2 4 8 16 32 64 128 256
100
101
102
103
104
105
Image side size
number of utilized qubits
BRQI
FRQI
QL
GQIR
NEQR
OQIM
QSMC
FTQR
(b) Qubits number
2 4 8 16 32 64 128 256
101
101.4
101.7
102
102.4
102.7
Image side size
Quantum volume
BRQI
FRQI
QL
GQIR
NEQR
OQIM
FTQR
(c) Quantum volume
Fig. 8: Metrics
Tables A1,A2,A3 with columns representing image side size show the
same metrics.
We observed that majority of techniques utilize a similar number of qubits,
however the cicruit depth shows the highest values for continuous and phase
Quantum Image Representation: A Review 15
representations and the lowest for the Qubit Lattice approach. The continu-
ous and phase representations are so heavy in depth that we could not execute
circuits for images bigger than 32 ˆ32. Moreover, for all approaches, we could
not succeed in real IBM-Washington [66] and IonQ [36] QPU simulation where
depth reaches 105for the images of size 4ˆ4 and 8 ˆ8 for each machine respec-
tively. This happened because of the multicontrolled rotation gate, which is a
base for all continuous and phase methods, is itself expensive for implementa-
tion. Following that, it is computationally expensive to simulate high resolution
images encoding, especially for continuous and approaches. Additionally, we
did not succeed to measure FTQR, and as a result, we were unable to check
whether representation works in the full encoding-decoding pipeline correctly.
However, FTQR shows the minimum growth of utilized qubits and quantum
volume numbers. Considering the number of qubits, Qubit Lattice is the most
inappropriate representation algorithm even for IBM-Washington, as qubits
can represent at most 8ˆ8 images. Discrete representations stand in between,
therefore, is the most promising group of image representation algorithms.
Each method has it’s own features. For example, some authors constructed
their methods to encode a specific type of image (log-polar, multi-wavelength,
indexed). Other authors made an accent on the representation way (based on
bitplanes, Fourier transform, order-encoding). While some authors describe
their approach in terms of quantum state normality (NAQSS), others operated
on color and coordinate differentiation (QSMC and QSNC). This fact might
explain the huge number of processing types for the NEQR 2.3.1 and FRQI
2.1.2 representations. Both of them are general, intuitive and appear as main
representative for the discrete and continuous groups respectively. Still, NEQR
”could perform the complex and elaborate color operation more conveniently
than FRQI” [67] due to obvious both – coordinate and pixel intensity encoding.
The image processing techniques significantly impact image representa-
tions. The majority of the discussed methods come with processing algorithms.
And yet, the overall stack of available processing is not full. Such comprehen-
sive tools as segmentation, feature extraction, machine learning, recognition
keep poor and imperfect review. At the same time, research in these branches
actively move these topics forward. Thus, opinion about quantum image
processing as a ”quantum hoax” [68] seems like a matter of time.
We assign our next work to the problems of the image processing in the
perspective of machine learning applications. Convolutions, pooling, statistics
calculation, and multi-image processing are in the scope of the future work. We
are also inspired for further research of the image representation algorithms.
Declarations
Code availability
The python code is available in the GitHub repository https://github.com/
UralmashFox/QPI.
16 Quantum Image Representation: A Review
Authors’ contributions
Marina Lisnichenko - had the idea for the article, performed the literature
search and data analysis. Stanislav Protasov - critically revised the work.
Data availability
Data sharing not applicable to this article as no datasets were generated or
analysed during the current study.
Ethics approval and consent to participate
Not applicable
Human and Animal Ethics
Not applicable
Consent for publication
Not applicable
Competing interests
The authors have no conflicts of interest to declare that are relevant to the
content of this article.
Funding
No funds, grants, or other support was received.
Acknowledgements
Not applicable
Authors’ information
Marina Lisnichenko received the M.S. in
robotics from Innopolis Univesity in 2021. She
is currently pursuing the PhD degree with the
Machine Learning and Knowledge Represen-
tation lab, Innopolis University. Her research
interests include quantum information process-
ing, quantum circuit compression, geographic
information processing.
Quantum Image Representation: A Review 17
Stanislav Protasov received the PhD degree
in the computer science from Voronezh State
University in 2013. He is currently an Asso-
ciate Professor with Innopolis University. His
research interests include applied machine
learning, information retrieval, applications of
quantum computing, quantum state prepara-
tion.
18 Quantum Image Representation: A Review
Appendix A Metrics tables
Table A1: Circuit depth
Image encoding 2 4 8 16 32 64 128 256
BRQI 50 206 738 3164 12400 49218 195902 784866
FRQI 467 8003 130307 2094083 – – – –
QL 2 2 2 2 2 2 2 2
GQIR 26 102 389 1491 6180 24511 98014 392236
NEQR 22 88 375 1513 6075 24410 97923 392281
OQIM 111 1983 32511 523263 8382466 – – –
QSMC 35 943 16063 260863 4189183 – – –
FTQR 5 158 2635 42813 869016 – – –
Table A2: Number of utilized qubits
Image encoding 2 4 8 16 32 64 128 256
BRQI 6 8 10 12 14 16 18 20
FRQI 3 5 7 9 11 – – –
QL 4 16 64 256 1024 4096 16384 65536
GQIR 10 12 14 16 18 20 22 24
NEQR 10 12 14 16 18 20 22 24
OQIM 4 6 8 10 12 – – –
QSMC 4 6 8 10 12 – – –
FTQR 4 6 8 10 – – – –
Table A3: Quantum volume
Image encoding 2 4 8 16 32 64 128 256
BRQI 36 64 100 144 196 256 324 400
FRQI 9 25 49 81 121
QL 4 4 4 4 4 4 4 4
GQIR 100 144 196 256 324 400 484 576
NEQR 100 144 196 256 324 400 484 576
OQIM 16 36 64 100 144 – – –
QSMC 16 36 64 100 144 – – –
FTQR 4 16 36 64 100 – – –
Quantum Image Representation: A Review 19
References
[1] Leprince-Ringuet, D.: Daphne Leprince-Ringuet: Quan-
tum Computing Is at an Early Stage. But Investors Are
Already Getting Excited. https://www.zdnet.com/article/
quantum-computing-is-at-an-early-stage-but-investors-are-already-getting-excited/
[2] Venegas-Andraca, S.E., Bose, S.: Storing, processing, and retrieving an
image using quantum mechanics. In: Quantum Information and Compu-
tation, vol. 5105, pp. 137–147 (2003). https://doi.org/10.1117/12.485960.
International Society for Optics and Photonics
[3] Latorre, J.I.: Image compression and entanglement. arXiv preprint quant-
ph/0510031 (2005). https://doi.org/10.48550/arXiv.quant-ph/0510031
[4] Le, P.Q., Dong, F., Hirota, K.: A flexible representation of quantum
images for polynomial preparation, image compression, and processing
operations. Quantum Information Processing 10(1), 63–84 (2011). https:
//doi.org/10.1007/s11128-010-0177-y
[5] Sun, B., Le, P.Q., Iliyasu, A.M., Yan, F., Garcia, J.A., Dong, F., Hirota,
K.: A multi-channel representation for images on quantum computers
using the rgbαcolor space. In: 2011 IEEE 7th International Symposium
on Intelligent Signal Processing, pp. 1–6 (2011). https://doi.org/10.1109/
WISP.2011.6051718. IEEE
[6] Zhang, Y., Lu, K., Gao, Y., Wang, M.: Neqr: a novel enhanced quantum
representation of digital images. Quantum information processing 12(8),
2833–2860 (2013). https://doi.org/10.1007/s11128-013-0567-z
[7] Li, H.-S., Qingxin, Z., Lan, S., Shen, C.-Y., Zhou, R., Mo, J.: Image
storage, retrieval, compression and segmentation in a quantum system.
Quantum information processing 12(6), 2269–2290 (2013). https://doi.
org/10.1007/s11128-012-0521-5
[8] Yuan, S., Mao, X., Xue, Y., Chen, L., Xiong, Q., Compare, A.: Sqr:
a simple quantum representation of infrared images. Quantum Infor-
mation Processing 13(6), 1353–1379 (2014). https://doi.org/10.1007/
s11128-014-0733-y
[9] Li, H.-S., Zhu, Q., Zhou, R.-G., Song, L., Yang, X.-j.: Multi-dimensional
color image storage and retrieval for a normal arbitrary quantum super-
position state. Quantum Information Processing 13(4), 991–1011 (2014).
https://doi.org/10.1007/s11128-013-0705-7
20 Quantum Image Representation: A Review
[10] Jiang, N., Wang, J., Mu, Y.: Quantum image scaling up based on
nearest-neighbor interpolation with integer scaling ratio. Quantum infor-
mation processing 14(11), 4001–4026 (2015). https://doi.org/10.1007/
s11128-015-1099-5
[11] S¸ahin, E., Yilmaz, I.: Qrmw: quantum representation of multi wavelength
images. Turkish Journal of Electrical Engineering & Computer Sciences
26(2), 768–779 (2018). https://doi.org/10.3906/elk-1705-396
[12] Li, H.-S., Chen, X., Xia, H., Liang, Y., Zhou, Z.: A quantum image
representation based on bitplanes. IEEE Access 6, 62396–62404 (2018)
[13] Xu, G., Xu, X., Wang, X., Wang, X.: Order-encoded quantum image
model and parallel histogram specification. Quantum Information Pro-
cessing 18(11), 1–26 (2019). https://doi.org/10.1007/s11128-019-2463-7
[14] Wang, B., Hao, M.-q., Li, P.-c., Liu, Z.-b.: Quantum representa-
tion of indexed images and its applications. International Journal of
Theoretical Physics 59(2), 374–402 (2020). https://doi.org/10.1007/
s10773-019-04331-0
[15] Grigoryan, A.M., Agaian, S.S.: New look on quantum representation of
images: Fourier transform representation. Quantum Information Process-
ing 19(5), 1–26 (2020). https://doi.org/10.1007/s11128-020-02643-3
[16] Oh, S., Choi, J., Kim, J.: A tutorial on quantum convolutional neural
networks (qcnn). In: 2020 International Conference on Information and
Communication Technology Convergence (ICTC), pp. 236–239 (2020).
IEEE
[17] Cong, I., Choi, S., Lukin, M.D.: Quantum convolutional neural net-
works. Nature Physics 15(12), 1273–1278 (2019). https://doi.org/10.
1038/s41567-019-0648-8
[18] Yang, C.-H.H., Qi, J., Chen, S.Y.-C., Chen, P.-Y., Siniscalchi, S.M., Ma,
X., Lee, C.-H.: Decentralizing feature extraction with quantum convo-
lutional neural network for automatic speech recognition. In: ICASSP
2021-2021 IEEE International Conference on Acoustics, Speech and Signal
Processing (ICASSP), pp. 6523–6527 (2021). IEEE
[19] Yan, F., Venegas-Andraca, S.E.: Quantum image processing (2020)
[20] Caraiman, S., Manta, V.: Image processing using quantum computing.
In: 2012 16th International Conference on System Theory, Control and
Computing (ICSTCC), pp. 1–6 (2012). IEEE
Quantum Image Representation: A Review 21
[21] Zhou, R.-G., Tan, C., Ian, H.: Global and local translation designs of
quantum image based on frqi. International Journal of Theoretical Physics
56(4), 1382–1398 (2017). https://doi.org/10.1007/s10773-017-3279-9
[22] Thenmozhi, S., BalaSubramanya, K., Shrinivas, S., Joshi, S.K.D., Vikas,
B.: Information hiding using quantum image processing state of art
review. Inventive Computation and Information Technologies, 235–245
(2021). https://doi.org/10.1007/978-981-33-4305-4 18
[23] Ma, H., He, Z., Xu, P., Dong, Y., Fan, X.: A quantum richardson–
lucy image restoration algorithm based on controlled rotation operation
and hamiltonian evolution. Quantum Information Processing 19(8), 1–14
(2020). https://doi.org/10.1007/s11128-020-02723-4
[24] Tariq Jamal, A., Abdel-Khalek, S., Ben Ishak, A.: Multilevel segmen-
tation of medical images in the framework of quantum and classical
techniques. Multimedia Tools and Applications, 1–14 (2021). https://doi.
org/10.1007/s11042-020-10235-7
[25] Farina, T.: Creating hybrid images using a quantum computer. PhD
thesis, UNION COLLEGE (2021)
[26] Sanchez, M., Sun, G.-H., Dong, S.-H.: Correlation property of multipar-
tite quantum image. International Journal of Theoretical Physics 58(11),
3773–3796 (2019). https://doi.org/10.1007/s10773-019-04247-9
[27] Liu, X., Xiao, D.: Multimodality image fusion based on quantum
wavelet transform and sum-modified-laplacian rule. International Journal
of Theoretical Physics 58(3), 734–744 (2019). https://doi.org/10.1007/
s10773-018-3971-4
[28] Hu, W.-W., Zhou, R.-G., Luo, J., Jiang, S.-X., Luo, G.-F.: Quantum
image encryption algorithm based on arnold scrambling and wavelet
transforms. Quantum Information Processing 19(3), 1–29 (2020). https:
//doi.org/10.1007/s11128-020-2579-9
[29] Yan, F., Jiao, S., Iliyasu, A.M., Jiang, Z.: Chromatic framework
for quantum movies and applications in creating montages. Frontiers
of Computer Science 12(4), 736–748 (2018). https://doi.org/10.1007/
s11704-018-7070-8
[30] Yan, F., Iliyasu, A.M., Jiao, S., Yang, H.: Audio-visual synchronisa-
tion in quantum movies. In: 2018 IEEE 5th International Congress on
Information Science and Technology (CiSt), pp. 274–278 (2018). IEEE
[31] Sun, B., Iliyasu, A.M., Yan, F., Sanchez, J.A.G., Dong, F., Al-Asmari,
A.K., Hirota, K.: Multi-channel information operations on quantum
22 Quantum Image Representation: A Review
images. Journal of Advanced Computational Intelligence and Intelligent
Informatics 18(2), 140–149 (2014)
[32] Shende, V.V., Bullock, S.S., Markov, I.L.: Synthesis of quantum-logic
circuits. IEEE Transactions on Computer-Aided Design of Integrated
Circuits and Systems 25(6), 1000–1010 (2006). https://doi.org/10.1145/
1120725.1120847
[33] Ma, G., Li, H., Zhao, J.: Quantum radon transform and its application.
arXiv preprint arXiv:2107.05524 (2021)
[34] Ding, M., Huang, T.-Z., Ji, T.-Y., Zhao, X.-L., Yang, J.-H.: Low-rank ten-
sor completion using matrix factorization based on tensor train rank and
total variation. Journal of Scientific Computing 81(2), 941–964 (2019).
https://doi.org/10.1007/s10915-019-01044-8
[35] Grover, L.K.: Quantum computers can search arbitrarily large databases
by a single query. Physical review letters 79(23), 4709 (1997)
[36] IonQ: Best Practices for Using IonQ Hardware: Https://ionq.com/best-
practices. https://ionq.com/best-practices
[37] Guanlei, X., Xiaogang, X., Xun, W., Xiaotong, W.: A novel quantum
image parallel searching algorithm. Optik 209, 164565 (2020). https://
doi.org/10.1016/j.ijleo.2020.164565
[38] Li, H.-S., Li, C., Chen, X., Xia, H.: Quantum image encryption based
on phase-shift transform and quantum haar wavelet packet transform.
Modern Physics Letters A 34(26), 1950214 (2019). https://doi.org/10.
1142/S0217732319502146
[39] Fan, P., Zhou, R.-G., Jing, N., Li, H.-S.: Geometric transformations of
multidimensional color images based on nass. Information Sciences 340,
191–208 (2016)
[40] Zhou, R.-G., Sun, Y.-J.: Quantum multidimensional color images sim-
ilarity comparison. Quantum Information Processing 14(5), 1605–1624
(2015). https://doi.org/10.1007/s11128-014-0849-0
[41] Su, J., Guo, X., Liu, C., Lu, S., Li, L.: An improved novel quantum
image representation and its experimental test on ibm quantum expe-
rience. Scientific Reports 11(1), 1–13 (2021). https://doi.org/10.1038/
s41598-021-93471-7
[42] Zhang, Y., Lu, K., Gao, Y., Xu, K.: A novel quantum representation
for log-polar images. Quantum information processing 12(9), 3103–3126
(2013). https://doi.org/10.1007/s11128-013-0587-8
Quantum Image Representation: A Review 23
[43] Brayton, R.K., Hachtel, G.D., McMullen, C., Sangiovanni-Vincentelli, A.:
Logic Minimization Algorithms for VLSI Synthesis vol. 2. Springer, ???
(1984)
[44] Iqbal, B., Singh, H.: Identification of desired pixels in an image using
grover’s quantum search algorithm. arXiv preprint arXiv:2107.03053
(2021). https://doi.org/10.48550/arXiv.2107.03053
[45] Fan, P., Zhou, R.-G., Hu, W.W., Jing, N.: Quantum image edge
extraction based on laplacian operator and zero-cross method. Quan-
tum Information Processing 18(1), 1–23 (2019). https://doi.org/10.1007/
s11128-018-2129-x
[46] Zhou, R.-G., Cheng, Y., Qi, X., Yu, H., Jiang, N.: Asymmetric scal-
ing scheme over the two dimensions of a quantum image. Quantum
Information Processing 19(9), 1–20 (2020). https://doi.org/10.1007/
s11128-020-02837-9
[47] Wang, S., Xu, P., Song, R., Li, P., Ma, H.: Development of high
performance quantum image algorithm on constrained least squares filter-
ing computation. Entropy 22(11), 1207 (2020). https://doi.org/10.3390/
e22111207
[48] Yang, J., Zhu, Y., Li, K., Yang, J., Hou, C.: Tensor completion from
structurally-missing entries by low-tt-rankness and fiber-wise sparsity.
IEEE Journal of Selected Topics in Signal Processing 12(6), 1420–1434
(2018). https://doi.org/10.1109/JSTSP.2018.2873990
[49] Fan, P., Zhou, R.-G., Hu, W., Jing, N.: Quantum image edge extraction
based on classical sobel operator for neqr. Quantum Information Process-
ing 18(1), 1–23 (2019). https://doi.org/10.1007/s11128-018-2131-3
[50] Heidari, S., Abutalib, M., Alkhambashi, M., Farouk, A., Naseri, M.:
A new general model for quantum image histogram (qih). Quantum
Information Processing 18(6), 1–20 (2019). https://doi.org/10.1007/
s11128-019-2295-5
[51] Zhou, R.-G., Yu, H., Cheng, Y., Li, F.-X.: Quantum image edge extraction
based on improved prewitt operator. Quantum Information Processing
18(9), 1–24 (2019). https://doi.org/10.1007/s11128-019-2376-5
[52] Ali, A.E., Abdel-Galil, H., Mohamed, S.: Quantum image mid-point filter.
Quantum Information Processing 19(8), 1–23 (2020). https://doi.org/10.
1007/s11128-020-02738-x
[53] Yuan, S., Wen, C., Hang, B., Gong, Y.: The dual-threshold quan-
tum image segmentation algorithm and its simulation. Quantum
24 Quantum Image Representation: A Review
Information Processing 19(12), 1–21 (2020). https://doi.org/10.1007/
s11128-020-02932-x
[54] Zhou, R.-G., Luo, J., Liu, X., Zhu, C., Wei, L., Zhang, X.: A novel quan-
tum image steganography scheme based on lsb. International Journal of
Theoretical Physics 57(6), 1848–1863 (2018). https://doi.org/10.1007/
s10773-018-3710-x
[55] Li, P., Shi, T., Lu, A., Wang, B.: Quantum circuit design for several mor-
phological image processing methods. Quantum Information Processing
18(12), 1–35 (2019). https://doi.org/10.1007/s11128-019-2479-z
[56] Nejad, M.Y., Mosleh, M., Heikalabad, S.R.: An lsb-based quantum
audio watermarking using msb as arbiter. International Journal of
Theoretical Physics 58(11), 3828–3851 (2019). https://doi.org/10.1007/
s10773-019-04251-z
[57] Zhu, H.-H., Chen, X.-B., Yang, Y.-X.: A multimode quantum image rep-
resentation and its encryption scheme. Quantum Information Processing
20(9), 1–21 (2021). https://doi.org/10.1007/s11128-021-03255-1
[58] Zhou, R.-G., Wan, C.: Quantum image scaling based on bilinear
interpolation with decimals scaling ratio. International Journal of
Theoretical Physics 60(6), 2115–2144 (2021). https://doi.org/10.1007/
s10773-021-04829-6
[59] Zhang, R., Xu, M., Lu, D.: A generalized floating-point quantum repre-
sentation of 2-d data and their applications. Quantum Information Pro-
cessing 19(11), 1–20 (2020). https://doi.org/10.1007/s11128-020-02895-z
[60] S¸ahin, E., Yilmaz, ˙
I.: A quantum edge detection algorithm for quantum
multi-wavelength images. International Journal of Quantum Information
19(03), 2150017 (2021). https://doi.org/10.1142/S0219749921500179
[61] S¸ahin, E., Yilmaz, ˙
I.: Qrma: quantum representation of multichannel
audio. Quantum Information Processing 18(7), 1–30 (2019). https://doi.
org/10.1007/s11128-019-2317-3
[62] Heidari, S., Naseri, M., Nagata, K.: Quantum selective encryption for
medical images. International Journal of Theoretical Physics 58(11),
3908–3926 (2019). https://doi.org/10.1007/s10773-019-04258-6
[63] Khorrampanah, M., Houshmand, M., Lotfi Heravi, M.M.: New method
to encrypt rgb images using quantum computing. Optical and
Quantum Electronics 54(4), 1–16 (2022). https://doi.org/10.1007/
s11082-022-03581-3
Quantum Image Representation: A Review 25
[64] Mastriani, M.: Quantum boolean image denoising. Quantum Infor-
mation Processing 14(5), 1647–1673 (2015). https://doi.org/10.1007/
s11128-014-0881-0
[65] Mastriani, M.: Quantum image processing? Quantum Information Pro-
cessing 16(1), 1–42 (2017). https://doi.org/10.1007/s11128-014-0881-0
[66] IBM: IBM-Washington Machine Details. https://quantum-computing.
ibm.com/services?services=systems&order=qubits%20DESC&view=
table&system=ibm washington
[67] Anand, A., Lyu, M., Baweja, P.S., Patil, V.: Quantum image processing.
arXiv preprint arXiv:2203.01831 (2022). https://doi.org/10.48550/arXiv.
2203.01831
[68] Ruan, Y., Xue, X., Shen, Y.: Quantum image processing: opportuni-
ties and challenges. Mathematical Problems in Engineering 2021 (2021).
https://doi.org/10.1155/2021/6671613
