# Quantum Computation-Based Image Representation, Processing Operations and Their Applications

^{1}

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## Abstract

**:**

## 1. Introduction

- In Section 2, the basic knowledge and notations used in this paper are introduced first. In addition, the flexible representation of quantum images (FRQI), the general framework of all geometric transformations on FRQI quantum images and the efficient color transformations on FRQI quantum images are reviewed.
- In Section 3, the extension of FRQI quantum images that allows for processing color images, the multi-channel representation for quantum images (MCQI) is introduced. What is more, quantum circuits to realize color operations on the channel of interest, channel swapping and α blending of MCQI quantum images are presented.
- In Section 4, a method to compare multiple pairs of FRQI quantum images in parallel is discussed, where the similarities of the images are estimated through the probability distributions of the readouts from quantum measurements. It offers a significant speed-up in comparison to performing the same task on traditional computing devices by transforming multiple images in a strip simultaneously.
- In Section 5, an FRQI quantum image searching method is presented based on the parallel comparison method introduced in Section 4. It is achieved by using low computational resources, which are only a single Hadamard gate combined with m + 1 quantum measurement operations.
- In Section 6, we build on the pioneering watermarking and authentication strategy for FRQI quantum images, WaQI, to propose protocols that would facilitate the notion of watermarking MCQI color images. The proposed MC-WaQI is a double-key and double-domain watermarking strategy that is secure and flexible by utilizing QFT techniques and quantum measurements to watermark MCQI quantum images.
- In Section 7, similarly, the pioneering attempt to represent and produce movies on quantum computers (quantum movie) is extended to the multi-channel color image framework. Following this, we describe a video encryption and decryption protocol on quantum computers based on color information transformations on each frame.
- In Section 8, we conclude with some remarks on possible technologies and directions that practitioners in the area opine could be used to realize some of the FRQI-based frameworks reviewed in earlier sections. Finally, we offer a few concluding remarks.

## 2. A Flexible Representation of Quantum Images and Its Processing Operations

#### 2.1. Quantum Bits and Quantum Gates

^{⊗}

^{n}. Similarly, for two vectors, |u〉 and |v〉, the tensor product of them can be expressed as |u〉 ⊗ |v〉 and shortened as |u〉|v〉 or |uv〉.

^{k}× 2

^{k}unitary matrix, in addition, the number of qubits in the input of the gate has to be equal to the output end. The mission of the final step in quantum simulation is to convert the quantum information into the classical form, which is realized by analyzing the probability distributions of the readouts from the quantum measurement. To distinguish the probabilistic classical bit from a qubit, a double-line wire is adopted, as depicted in Figure 2. The probability p of a measurement result r occurring when state ψ is measured is $\langle \psi \mid {M}_{r}^{\u2020}{M}_{r}\mid \psi \rangle $. The state of the system after measurement, |ψ′〉, is:

#### 2.2. Flexible Representation for Quantum Images

_{n}

_{−1}y

_{n}

_{−2}· · · y

_{0}and the location along the horizontal axis by using the second n-qubit x

_{n}

_{−1}x

_{n}

_{−2}· · · x

_{0}. The FRQI state is a normalized state, i.e., |||I(θ)〉|| = 1, as given by:

#### Theorem 1

_{0}, θ

_{1}, · · ·, θ

_{22n−1}), (n ∈ N) of angles, there is a unitary transform ℘ that is composed of polynomial number of simple gates to turn quantum computers from the initialized state, |0〉

^{⊗2}

^{n}

^{+1}, to the FRQI state in Equation (7).

^{⊗2}

^{n}

^{+1}is changed to |H〉 and then to |I(n)〉, as presented in Figure 4.

^{⊗2}

^{n}, where I is the 2D identity matrix and H is the Hadamard gate, on |0〉

^{⊗2}

^{n}

^{+1}, produces the state |H〉,

_{y}(2θ

_{i}) (along the y-axis by the angle 2θ

_{i}) and controlled rotation matrices R

_{i}(i = 0, 1, · · ·, 2

^{2}

^{n}− 1),

_{i}is a unitary matrix, since ${R}_{i}{R}_{i}^{\u2020}={I}^{\otimes 2n+1}$. Applying R

_{k}and R

_{l}R

_{k}on |H〉 provides us that:

^{⊗2}

^{n}

^{+1}, to the FRQI state in Equation (7).

#### 2.3. Fast Geometric Transformations on FRQI Quantum Images

_{I}, on FRQI quantum images can be defined as in Equation (19),

^{2}

^{n}− 1 is the unitary transformation performing geometric exchanges on the basis of the position information |i〉.

^{2}N) for two-point swapping and O(logN) for the other operations [29].

#### 2.4. Efficient Color Transformations on FRQI Quantum Images

_{I}is the qubit gate to transform the color information in quantum image |I(n)〉 and |c(θ

_{k})〉 encodes the color information which is presented as:

## 3. Multi-Channel Quantum Images and Related Operations

#### 3.1. Multi-Channel Representation for Quantum Images

_{α}is set as zero to make the two coefficients constant (cos θ

_{α}= 1 and sin θ

_{α}= 0) to carry no information, which is discussed in [27,28].

_{1}, c

_{2}and c

_{3}) are color qubits that encode RGB color information for an image, and the remaining 2n qubits (y

_{n}

_{−1}, y

_{n}

_{−2}, . . . , y

_{0}and x

_{n}

_{−1}, x

_{n}

_{−2}, . . . , x

_{0}) are used to encode position information (Y-axis and X-axis) about pixels of a 2

^{n}× 2

^{n}image. A simple example of an MCQI quantum image with its quantum state is shown in Figure 8. The methods for storing and retrieving quantum images are discussed in [27,28,42]. Like FRQI, the MCQI state is also a normalized state, i.e., |||I(n)

_{mc}〉|| = 1, as given by:

- MCQI representation provides a solution using many fewer qubits to encode R, G and B channel information in normalized quantum states.
- MCQI makes it easier to design color image operators with much lower complexity.
- MCQI representation offers the potential possibility to design a quantum-cryptography-based color image watermarking algorithm.

#### 3.2. Channel of Interest Operator

_{X}operator is realized by using U

_{X}= C

^{2}R

_{y}(2θ) gate, where θ is the shifting parameter (or shifting angle). The calculation produces the result $\mid I{(n)}_{mc}^{X}\rangle $ of the application of CoI

_{X}on |I(n)

_{mc}〉, given as:

_{X}operator, shown as:

_{mc}〉 by shifting the θ angle on the R, G, B or α channel. The quantum circuits of U

_{X}(U

_{R}, U

_{G}, U

_{B}and U

_{α}) are C

^{2}R

_{y}(2θ) gates and are shown in Figure 9, and the C

^{2}R

_{y}(2θ) can be constructed from elementary gates (controlled rotation and CNOT gates), as shown in Figure 10.

#### 3.3. Channel Swapping Operator

_{Y}operator is completed by means of the CNOT gate or SWAP gate on c

_{2}and c

_{3}color qubits. The calculation produces the result $\mid I{(n)}_{mc}^{Y}\rangle $ of the application of CS

_{Y}on |I(n)

_{mc}〉, given as:

_{Y}operator, presented as:

_{mc}〉 by applying the CS

_{Y}operator, and specifically, quantum circuits of U

_{Y}(U

_{RG}, U

_{RB}and U

_{GB}) are shown in Figure 11. From Figure 12, at most, three quantum basic gates are utilized to build the CS operator.

#### 3.4. The α Blending Operator

^{n}× 2

^{n}image), as shown in Figure 13, where ${C}_{A}^{i}$ and ${C}_{B}^{i}$ are the color states of images A and B, respectively, which are defined as:

_{AX}and θ

_{BX}(X ∈ R, G,B, α) are angles encoding the color information of images A and B, respectively. Because, before blending, the two images are totally opaque, the initial value of ${\theta}_{A\alpha}^{i}$ and ${\theta}_{B\alpha}^{i}$ is zero. After storing the two images concurrently, two C

^{3}-rotation gates are applied on a

_{1}and color (c

_{1}, c

_{2}, c

_{3}) qubits, where the control operations are on a

_{1}, c

_{2}, c

_{3}and the rotation is on c

_{1}qubits. The computation procedure is shown as:

## 4. Parallel Comparison of Multiple Pairs of FRQI Quantum Images

#### 4.1. Representation of Strip Encoding Multiple FRQI Images

^{m}-ending FRQI quantum images lies in its ability to utilize the parallelism inherent to quantum computation in order to transform multiple images using very few quantum resources. The definition of the strip and its properties are introduced in this subsection.

#### Definition 1

^{m}FRQI quantum images, which is defined by:

_{s}(n)〉 is a FRQI quantum image as defined in Equation (7) at position |s〉, |c

_{s,i}〉 and |i〉 encode the information about the colors and their corresponding positions in the image |I

_{s}(n)〉.

^{m}quantum images. Each image in the strip is an FRQI state, while the combination of such states in the strip is best represented as a multiple FRQI or simply the mFRQI state. The mFRQI state can represent 2

^{m}quantum images using only m + 2n + 1 qubits, since all of the images are of the same size in this strip.

^{m}-ending FRQI quantum images encoded in a strip can be horizontally (or vertically) oriented [12,36]. The latter case, the vertically-oriented strip is implied in the ensuing discussion. Control-conditions on strip wires could control the image that is being processed, combined with the control-conditions from the position |y〉|x〉 to the color wire; every pixel in this strip can be accessed.

#### 4.2. Scheme to Compare Images in Parallel on Quantum Computers

_{k}(n)〉 and |I

_{t}(n)〉,

^{m}quantum images, parallel comparison of quantum images retrieves the similarities between 2

^{m}− 1 pairs of images in the strip simultaneously.

#### Definition 2

_{k}(n)〉 and |I

_{t}(n)〉, as defined in Equations (49) and (50), is given by:

_{k,i}and θ

_{t,i}represent the color information at position i of the two images, respectively.

#### Definition 3

_{k}(n)〉 and |I

_{t}(n)〉, as defined in Equations (49) and (50), is a function of pixel difference σ

_{k,t}at every position of the image given by:

_{k}〉, |I

_{t}〉) ∈ [0, 1].

- Preparation of the strip comprising 2
^{m}quantum imagesThe quantum images are prepared into FRQI states using their classical versions images. The color information, as well as the corresponding positions of every point in the classical version are integrated into the quantum state, and the 2^{m}quantum images being compared are combined to form a vertically-oriented strip. The routine involved in preparing FRQI quantum images and its extension to encode multiple FRQI quantum images as a single register, called the strip, are discussed thoroughly in [12,21,36]. - Comparison of quantum images through quantum operationsThe strip as prepared in the preceding period is transformed using geometric transformations [29] on all of the images in the strip. This transformation step combines with measurement operations that follow it to convert the quantum information into the classical form as probability distributions. Since measurements are known to destroy the superposition state in quantum systems [15], the strip has to be prepared n (n > 1) times in order to compare the similarity between two FRQI quantum images (in parallel).
- Observation of readouts from quantum measurementsThe readouts from the n quantum measurements build up a histogram that implicitly reflects the probability distributions. Extracting and analyzing these distributions gives information about the similarity values between the quantum images being compared. The strip preparation will be continued until min(P(|s
_{m}_{−1}, . . . , s_{0}〉)) ≥ δ, where min(P(|s_{m}_{−1}, . . . , s_{0}〉)) is the minimum of the probabilities of the readouts from the experiments and δ ∈ [0, 1] is a pre-set threshold, which can be read as the reasonable estimation for the similarity between two quantum images being compared.

#### 4.3. A Parallel Comparison of Multiple Pairs of Images in a Strip

^{m}images as defined in Definition 1 provides us a crucial condition to make it possible, because the operation on the strip wires can transform the information in every image simultaneously. The generalized circuit structure of comparing 2

^{m}− 1 pairs of quantum images in parallel is presented in Figure 16. By applying a Hadamard operation on the r−th strip wire in the circuit, s

_{r}, the mathematical expressions between the two images being compared are realized. The final step in the procedure consists of m measurements from which the similarity can be retrieved in each pair of images.

_{k}〉, and the (k + 2

^{r}) − th image, |I

_{k+2r}〉 (r is the index of s

_{r}in the circuit). Therefore, the mFRQI state of the strip when 2

^{m}− 1 pairs of images are compared is:

_{m}

_{−1}, . . . , s

_{r}

_{+1}, s

_{r}, s

_{r}

_{−1}, . . . , s

_{0}〉, s

_{r}∈ {0, 1}.

_{r}(expressed by H

_{r}|S(m, n)〉) transforms the state of the strip into:

_{m}

_{−1}, . . . , s

_{r}

_{+1}, 1, s

_{r}

_{−1}, . . . , s

_{0}〉 on s

_{r}is given by:

_{m}

_{−1}, . . . , s

_{r}

_{+1}, 0, s

_{r}

_{−1}, . . . , s

_{0}〉 and |s

_{m}

_{−1}, . . . , s

_{r}

_{+1}, 1, s

_{r}

_{−1}, . . . , s

_{0}〉 represent all of the images that are at the k − th and (k + 2

^{r}) − th position of the strip, respectively. In order to determine the similarity of every pair of images, the generalized representation of the probability of |I

_{k}

_{+2r}(n)〉 in the strip is given by:

_{k}(n)〉 and |I

_{k}

_{+2r}(n)〉 can be presented as:

_{k}〉 and |I

_{k}

_{+2r}〉 are the two images being compared in the strip, P

_{s}

_{r}(|k + 2

^{r}〉) is defined in Equation (60) and sim(|I

_{k}〉, |I

_{k}

_{+2r}〉) ∈ [0, 1]. The similarity between |I

_{k}〉 and |I

_{k}

_{+2r}〉, which are encoded in the strip comprising 2

^{m}images, is also determined in accordance with Definition 3, where $f({\sigma}_{k,t}^{0},{\sigma}_{k,t}^{1},\dots ,{\sigma}_{k,t}^{{2}^{2n}-1})={\scriptstyle \frac{1}{{2}^{2n}}}{\sum}_{i=0}^{{2}^{2n}-1}\text{cos\hspace{0.17em}}{\sigma}_{k,k+{2}^{r}}^{i}$.

#### 4.4. Comparison between Two Arbitrary Quantum Images and Sub-Blocks of Them

^{r}) − th images in the strip (r is the index of s

_{r}in the circuit), in a strip is relatively fixed. In order to compare two arbitrary quantum images and/or contents of their sub-blocks from a strip, some geometric transformation and control-conditions are applied to the quantum system. In this subsection, the more complicated cases of quantum image comparison are discussed, such as comparing arbitrary pairs of images, comparing sub-blocks from two images in a strip. The circuit structure for realizing such processes is presented in Figure 17.

_{I}〉 in the circuit); the operation G

_{S}that is applied on the strip wires is the geometric operation, which can swap two images in the strip when two arbitrary images are supposed to be compared. A notation “⊘”, to indicate for “0” or “1” control-condition, is adopted throughout the discussion. The additional control-conditions on either of the position axis (Y-axis or X-axis) are necessary in order to confine this Hadamard operation to the required sub-blocks from the images that are being compared. The operation G

_{I}is needed when the sub-blocks being compared are at different positions from the two images. The state in the circuit after applying the Hadamard gate on the r − th strip wire is transformed into H

_{r}G

_{s}|S

_{GI}〉, as shown in Figure 17.

_{k}〉 and |i

_{t}

_{→}

_{k}

_{+2r}〉 are the sub-blocks from the two images |I

_{k}〉 and |I

_{t}

_{→}

_{k}

_{+2r}〉 in the strip, |I

_{t}

_{→}

_{k}

_{+2r}〉 is the image from the position t to k + 2

^{r}using the geometric transformation on the strip wires, p is the area of the image |I

_{k}〉 or |I

_{t}

_{→}

_{k}

_{+2r}〉, q is the area of the sub-block |i

_{k}〉 or |i

_{t}

_{→}

_{k}

_{+2r}〉 in the two images and P

_{s}

_{r}(|k +2

^{r}〉) is the probability of the readouts in the measurements from the state |k +2

^{r}〉, as discussed in Equation (60), sim(|i

_{k}〉, |i

_{t}

_{→}

_{k}

_{+2r}〉) ∈ [0, 1].

#### 4.5. Simulation Experiment to Assess the Similarity of Quantum Images

_{0}〉 (labeled as $\mid {i}_{0}^{3}\rangle $) with the watermarked Lena image in the top right of the image in |I

_{3}〉 (labeled as $\mid {i}_{3}^{2}\rangle $); and the Man image (labeled as $\mid {i}_{1}^{3}\rangle $) with the processed Man image (labeled as $\mid {i}_{2}^{3}\rangle $) at the same position in |I

_{1}〉 and |I

_{2}〉, respectively. The enlarged versions of these four images that are being compared are presented in the bottom row of the Figure 18. For brevity, the four 1, 024 × 1, 024 images are indicated by only their labels |I

_{0}〉, |I

_{1}〉, |I

_{2}〉 and |I

_{3}〉 in the strip on the left in the same figure.

- Swap the position between |I
_{1}〉 and |I_{3}〉 using the C-NOT gate on the strip wires. - Swap the position of the watermarked Lena image with baboon in |I
_{3}〉. - Compare the two Lena images and two “Man” images in parallel by applying the Hadamard gate on s
_{0}with appropriate control-condition operations to confine the operation to the desired sub-blocks. - Observe the readouts from the quantum measurements to build up a histogram that can reflect the similarity of the two pairs of images.

## 5. Quantum Image Searching Based on Probability Distributions

#### 5.1. Representation of Z-Strip to Indicate Multiple FRQI Quantum Images

#### Definition 4

^{m}

^{+1}quantum images. The Z-axis differentiates the strip that 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 zFRQI state.

^{m}

^{+1}quantum images using only m + 2n + 2 qubits, since all of the images are of the same size on this Z-strip. The notation, “○” for “0” or “●” for “1” control-condition on the Z-axis or S-axis, is sufficient to specify any quantum image in the Z-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 zFRQI state, is shown in Figure 22.

#### 5.2. Image Searching on Quantum Mechanical Systems

^{m}+ 1 images in Definition 4 provides us a crucial condition to make the parallel comparison of quantum images possible, because the operation on the strip wires can transform the information in every image simultaneously. The generalized circuit structure of comparing 2

^{m}pairs of FRQI quantum images in Z-strip in parallel is presented in Figure 23.

_{m}

_{−1}· · · s

_{r}· · · s

_{0,}s

_{r}∈ {0, 1}, give the position of probabilities of the measurements on the Z-axis. According 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.

_{s}(n)〉 and |R

_{s}(n)〉. Therefore, the probability of state |0〉 on the Z-axis at position |s

_{m}

_{−1,}s

_{m}

_{−2,}· · ·, s

_{0}〉 is shown by:

#### Definition 5

_{s}

_{,}

_{i}, is defined by:

_{l}

_{,}

_{s}

_{,}

_{i}and θ

_{r}

_{,}

_{s}

_{,}

_{i}encode the color information at position i of the two images, which are at the s–th position of the Z-strip, respectively.

_{s}

_{,}

_{i}is decided by the probability of obtaining readout of one from the Z-axis, P

_{s}(|1〉), in the measurement. In addition, the similarity between the two images at the same position of Z-strip is given by:

_{s}(n)〉 and |R

_{s}(n)〉 are the two images being compared, P

_{s}(|1〉) is defined in Equation (74) and sim(|L

_{s}(n)〉, |R

_{s}(n)〉) ∈ [0, 1].

^{m}+1 quantum images being compared are combined as a Z-strip. Because of the superposition property of quantum computation, such work can be realized using only a few quantum resources. However, the Z-strip is prepared n(n > 1) times to compare the similarity between two quantum images in parallel, since a measurement would collapse the superposition state in the quantum system [15]. By analyzing the distributions, the similarity values between the quantum images being compared are provided, so that the image with the highest similarity to the particular test image could be retrieved as a result from the database.

#### 5.3. Simulation Experiments to Search Quantum Images from Database

_{0}(2)〉, |D

_{1}(2)〉, …, |D

_{63}(2)〉 and sixty-four |T(2)〉s is constituted as shown in Figure 25.

- The test images |T(2)〉 is prepared from the classical version using FRQI representation and integrated with a Z-strip state with the images |D〉 in the database.
- A Hadamard operation is applied on the Z-axis in order to compare the test image |T(2)〉 with |D
_{0}(2)〉, |D_{1}(2)〉, … and |D_{63}(2)〉. - The measurements that convert the quantum information to the classical form are used on the S-axis and Z-axis to distribute the readouts from which the histogram is built to reflect the similarity of the sixty-four pairs of images.

_{Z}are applied. A simulation of a single Hadamard gate and seven measurement operations are used to obtain the similarities for these 64 pairs of images based on the probabilities of getting the readouts on the Z-axis and S-axis in the measurements, as shown in the Figure 27. From the histogram, the image |D

_{37}(2)〉, which manifests the highest similarity value of 0.93 to the test image |T(2)〉, is retrieved as the searching result. It is testified by that the quantum image searching is based on the pixel difference between the test image and the images in the database.

## 6. Watermarking and Recovery Strategies for FRQI and MCQI Quantum Images

#### 6.1. Watermarking of Quantum Images Based on Restricted Geometric Transformations

- In the watermark blending step, it is on the basis of the classical version of the image-watermark pair completely.
- In the watermark circuit transformation step, the content retrieved from the blending step are transformed into applicable quantum circuit elements.

#### 6.2. Two-Tier Grayscale Version of the WaQI Protocol

_{α}and T

_{β}are confined to predetermined areas of the cover image using the control-conditions specified by I

_{Rl}and I

_{S}, respectively.

#### 6.3. A Watermark Strategy for Quantum Images Based on Quantum Fourier Transform

- Watermark image preprocessing.Generate two sequences of keys and then scramble the watermark image according to the image scrambling method introduced earlier.
- Execute QFT on the carrier image and obtain its Fourier coefficients.
- Embed the watermark image into the carrier image.If we only take the color information of a FRQI quantum image into consideration, we can assume the final revised watermark image is ${\sum}_{i=0}^{MN-1}{\omega}_{i}\mid i\rangle $, the carrier image is ${\sum}_{i=0}^{MN-1}{x}_{i}\mid i\rangle $ and the QFT of the carrier image is ${\sum}_{i=0}^{MN-1}{y}_{i}\mid i\rangle $. Therefore, the Fourier transform of the embedded carrier image is in the following form: ${\sum}_{i=0}^{MN-1}{y}_{i}^{\prime}\mid i\rangle ={\sum}_{i=0}^{MN-1}({y}_{i}+\alpha *{\omega}_{i})\mid i\rangle $, where α decides the embedded proportion (0 < α < 1).
- Execute the inverse QFT to obtain the embedded carrier image.

#### 6.4. A Duple Watermarking Strategy for Multi-Channel Quantum Images

- (1)
- Preprocessing procedure:
- Prepare for MCQI quantum images |I〉 and |W〉 from the classical version of them I and W.
- Create two watermark information |FW〉 and |SW〉 from the original watermark image |W〉 for the embedding into both frequency and spatial domains of the carrier image.
- Apply measurement operation on image |SW〉 to obtain retrieved image M.
- Generate the color information key (CIK) from image M by means of the encoding rule.
- Execute operations on |SW〉 using CIK to get image |SW′〉.
- Compose image |FW′〉 from image |FW〉.
- Scramble image |FW′〉 to obtain image |FW″〉 by applying the position information key (PIK) operation, and it is an optional operation to scramble image |SW′〉 to |SW″〉.
- Resize image |FW″〉 and |SW″〉 to get image |FW‴〉 and |SW‴〉.

- (2)
- Embedding procedure:
- Embed image |FW‴〉 into the QFT coefficients of the carrier image to transform image |I〉 to image |I′〉.
- Embed image |SW‴〉 into the spatial domain of image |I′〉 to generate image |I″〉.

- (3)
- Extraction procedure:
- Extract watermark image |RFW〉 from the frequency domain using PIK.
- Extract watermark image |RSW〉 from the spatial domain using CIK (probably with PIK, depending on the preprocessing procedure).

## 7. A Framework for Representing and Producing Movies on Quantum Computers

#### 7.1. Framework for Quantum Movie Representation and Manipulation

#### 7.2. Quantum Video Encryption and Decryption Protocol

^{m}MCQI quantum images (M-strip), which is formulated as:

_{s}(n)〉 is an MCQI quantum image, as defined in Equation (24) at position |s〉.

## 8. Likely Technologies to Realize Quantum Image Processing Applications and Concluding Remarks

#### 8.1. Photonic-Based Realization of Efficient FRQI Quantum Image Processing

#### 8.1.1. Photonic-FRQI Quantum Image Preparation

^{⊗2}

^{n}

^{+1}, to the ghost FRQI state, |H〉. Since both the vacuum state and the Hadamard operations (single qubit beam splitters) have been realized and implemented physically, it is trivial to assume that the ghost FRQI state, |H〉, for any n×n image can be realized. Figure 36 shows the execution of the $\mathscr{H}$ transformation to transform the vacuum state into a ghost FRQI state. The second part of the PPT theorem, the $\mathcal{R}$ transform, requires 2

^{2}

^{n}controlled-rotation or the generalized C

^{2}

^{n}(R

_{y}(2θ

_{i})) operation to accomplish this [26]. Multiple-qubit operations on quantum states are still challenging to realize using photonic or quantum technologies. Therefore, when restricted within the confines of today’s technologies and the requirements of the $\mathcal{R}$ transform, as discussed in [21], only a 2×2-pixel image, i.e., the case where n = 1, can be realized. The sub-circuit to execute the $\mathcal{R}$ transform for such a small-sized image (as shown in Figure 10 in Section 3.2) requires two controlled-rotation and two controlled-NOT gates. As specified by the PPT theorem, preparing a 2 × 2 FRQI quantum image requires a total of 40 simple quantum operations [21,26]. This limitation regarding the size of images we can realize from available technology is attributed to the lack of quantum optical elements to execute multiple-control condition operations. The best we can realize for larger-sized images is the intermediary ghost FRQI quantum image state arising from the application of the $\mathscr{H}$ transform on the initialized vacuum state as specified by the FRQI PPT theorem. Such a transformation to transform an n-qubit vacuum state to its intermediary or ghost FRQI state is depicted in Figure 36.

#### 8.1.2. FRQI Quantum Image Transformation on Photonic Quantum Computers

#### 8.1.3. Recovering Photonic-FRQI Quantum Image States

^{2}

^{n}− 1 spanning the entire pixels of the image. This actually conforms to what is obtained in classical image processing, wherein an image is described by a bunch of numbers: 0–1 for binary images; 0–255 for grayscale images; and so on.

^{2}

^{n}ancillary qubits were used to encode the color of every pixel in the input images. The interaction operation of the ancilla-driven quantum computing [44] model was then used to transfer the final state of the pixel (using the qubit) to its corresponding ancilla qubit. Destructive measurements to “see” these new states were then performed on these ancilla qubits. This way, the transformed state of the image was recovered without disturbing the content of the images encoded and transformed in the movie strip, as presented in Section 7. This technique facilitated the recovery of the content of a movie sequence comprising of the key, viewing and make-up frames. Even so, the content recovered was binary, 0–1, states [34]. Implementing this primal FRQI quantum image state recovery with today’s photonic quantum technologies is also a victim of the need for multiple control-condition operations. Actually, even with the realization of optical elements to execute multiple control-condition operations, as mentioned in the preceding discussions, the cost of implementing a read-out similar to the one enumerated in [12] appears very high. Therefore, recovering the final state of the FRQI quantum image on photonic quantum computers will pose the most difficult challenge when compared alongside the other two criteria we have adopted for realizing our photonic-FRQI quantum image processing application-specific device, i.e., the quantum image state preparation and manipulation criteria.

#### 8.2. Concluding Remarks

^{m}quantum images in the Z-strip, and only one qubit on the Z-axis can represent the images on both the left and right side of Z-strip.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 9.**The general quantum circuit of U

_{X}operations, including: (

**a**) U

_{R}; (

**b**) U

_{G}; (

**c**) U

_{B}; and (

**d**) U

_{α}.

**Figure 10.**C

^{2}R

_{y}(2θ) can be constructed from basic gates, CR

_{y}(thetas;), CR

_{y}(−thetas;) and CNOT gates.

**Figure 19.**Circuit structure for realizing the comparison in Figure 18.

**Figure 26.**Circuit structure for realizing the image searching in Figure 25.

**Figure 33.**The key, makeup and viewing frames, |F

_{m}〉, $\mid {\mathcal{F}}_{m}^{c}\rangle $ and |f

_{mq}〉, respectively, from m-shots in a video.

Image Comparison | Probability | Similarity |
---|---|---|

$\mid {i}_{0}^{3}\rangle ,\mid {i}_{3}^{2}\rangle $ | P_{s}_{0} (|01〉) = 0.004 | sim(|i_{0}〉, |i_{3}〉) = 0.936 |

$\mid {i}_{1}^{3}\rangle ,\mid {i}_{2}^{3}\rangle $ | P_{s}_{0} (|11〉) = 0.013 | sim(|i_{1}〉, |i_{2}〉) = 0.787 |

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Yan, F.; Iliyasu, A.M.; Jiang, Z. Quantum Computation-Based Image Representation, Processing Operations and Their Applications. *Entropy* **2014**, *16*, 5290-5338.
https://doi.org/10.3390/e16105290

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Yan F, Iliyasu AM, Jiang Z. Quantum Computation-Based Image Representation, Processing Operations and Their Applications. *Entropy*. 2014; 16(10):5290-5338.
https://doi.org/10.3390/e16105290

**Chicago/Turabian Style**

Yan, Fei, Abdullah M. Iliyasu, and Zhengang Jiang. 2014. "Quantum Computation-Based Image Representation, Processing Operations and Their Applications" *Entropy* 16, no. 10: 5290-5338.
https://doi.org/10.3390/e16105290