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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

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