A Comparative Study of Quantum Haar Wavelet and Quantum Fourier Transforms for Quantum Image Transmission

Abstract

Quantum communication has achieved significant performance gains compared to classical systems but remains sensitive to channel noise and decoherence. These limitations become especially critical in quantum image transmission, where high-dimensional visual data must be preserved with both structural fidelity and robustness. In this context, transform-based quantum encoding methods have emerged as promising approaches, yet their relative performance under noisy conditions has not been fully explored. This paper presents a comparative study of two such methods, the quantum Fourier transform (QFT) and the quantum Haar wavelet transform (QHWT), within an image transmission framework. The process begins with source coding (JPEG/HEIF), followed by channel coding to enhance error resilience. The bitstreams are then mapped into quantum states using variable qubit encoding and transformed using either QFT or QHWT prior to transmission over noisy quantum channels. At the receiver, the corresponding decoding operations are applied to reconstruct the images. Simulation results demonstrate that the QFT achieves superior performance under noisy conditions, consistently delivering higher Peak Signal-to-Noise Ratio (PSNR), Structural Similarity Index Measure (SSIM), and Universal Quality Index (UQI) values across different qubit sizes and image formats compared to the QHWT. This advantage arises because QFT uniformly spreads information across all basis states, making it more resilient to noise. By contrast, QHWT generates localized coefficients that capture structural details effectively but become highly vulnerable when dominant coefficients are corrupted. Consequently, while QHWT emphasizes structural fidelity, QFT provides superior robustness, underscoring a fundamental trade-off in quantum image communication.

1. Introduction

Image transmission has become a critical component of modern communication systems, with applications ranging from medical imaging and remote sensing to real-time video streaming. In conventional systems, compression and error correction techniques are employed to ensure efficient delivery of high-quality images. However, classical image transmission faces inherent limitations when operating under noisy or bandwidth-constrained environments. Channel noise, signal attenuation, and error propagation often result in visible distortions and loss of image fidelity. This problem is particularly critical in compressed image [1,2] transmission, since compression removes redundancies and represents information using fewer bits. As a result, even small errors in the transmitted bitstream can propagate during decoding, leading to significant degradation of reconstructed image quality. Moreover, as data rates continue to increase, classical approaches struggle to simultaneously guarantee efficiency, robustness, and scalability.

Quantum communication [3] has emerged as a promising alternative to overcome these limitations. By leveraging the principles of superposition [4] and entanglement [5,6], quantum systems provide fundamentally new ways to encode and transmit information that differ from classical approaches. In particular, quantum superposition allows multiple states to be represented simultaneously within a compact quantum register, thereby increasing noise resilience. Furthermore, quantum entanglement can be exploited to enhance correlation between transmitted qubits, which may improve resilience against certain types of channel impairments. Unlike classical methods, which rely heavily on redundancy and error correction codes to mitigate transmission errors, quantum techniques inherently distribute information across multiple basis states. This not only facilitates more compact data representation but also provides a degree of robustness, as the impact of noise or decoherence may be spread across the encoded state rather than concentrated in a few critical symbols. These unique advantages have motivated growing interest in the development of quantum image-transmission frameworks that aim to push beyond the performance boundaries of traditional classical systems.

An initial approach to quantum image transmission is single-qubit encoding, where each qubit directly represents a single classical bit in the time domain [7]. While this approach demonstrates feasibility with relatively low circuit complexity, it lacks scalability, as the number of required qubits grows proportionally with image size and the available parallelism remains underutilized. To overcome these limitations, multi-qubit encoding [8] schemes extend the representation by allowing multiple bits to be jointly encoded within several qubits. This provides a more compact representation, improves the utilization of quantum resources, and enhances resilience by distributing information across multiple qubits rather than relying on individual ones. Both single-qubit and multi-qubit schemes form the time-domain family of methods, which laid the foundation for quantum image transmission but remain constrained in scalability, compression efficiency, and noise resilience. These challenges have motivated the transition toward transform-based approaches, where methods such as the quantum Fourier transform (QFT) [9] and the quantum Haar wavelet transform (QHWT) [10] operate in the frequency and wavelet domains respectively, offering more efficient and robust solutions for quantum image transmission.

The QFT plays a central role in quantum information processing due to its ability to distribute information globally across basis states. In the context of image transmission, this property can enable efficient encoding where image data is represented in the frequency domain, ensuring that information is spread uniformly throughout the quantum state. This global distribution is highly advantageous for achieving compact representation, efficient reconstruction, and improved transmission robustness. In addition, the QHWT can provide a localized representation that captures both coarse and fine structural details of the images. By mapping image information into multi-resolution components, Haar encoding efficiently can preserve edges, textures, and high-frequency features that are vital for image quality. This hierarchical decomposition makes QHWT particularly effective for structured image analysis and transmission where detail preservation is essential. Together, QFT and QHWT represent two powerful transform-based approaches to quantum image transmission: one emphasizing global frequency-domain encoding, and the other focusing on localized structural encoding. Their complementary characteristics make them strong candidates for advancing noise-resilient quantum image communication systems.

While the QFT has been widely recognized for its effectiveness in quantum communication, the QHWT has thus far been confined to quantum image representation, compression, and structural analysis, without exploration in the domain of image transmission. Therefore, this work makes two highly novel contributions: first, it pioneers the use of QHWT as an encoding method specifically for quantum image transmission; and second, it delivers the first rigorous comparative study of QFT and QHWT-based frameworks under identical experimental conditions. By extending QHWT into the transmission domain and directly benchmarking it against QFT, this study not only highlights the fundamental differences between global and localized transform-based encoding, but also provides practical, application-driven insights into the design of noise-resilient quantum communication systems.

By addressing these research gaps, this study moves beyond theoretical discussion and into practical system design. Therefore, we develop a novel quantum image transmission framework to compare the performance of the QFT and the QHWT. The process begins with classical source coding, where input images are compressed using joint photographic experts group (JPEG) or high-efficiency image file format (HEIF) to reduce redundancy while preserving visual quality. To further enhance resilience against transmission errors, optional channel coding such as polar codes may be applied, adding controlled redundancy for error correction after noisy transmission. The encoded bitstream is then segmented according to the desired qubit size and mapped into quantum states, allowing both single-qubit and multi-qubit representations to be investigated. Once in quantum form, the states undergo a transform-based encoding stage: in the QFT framework, information is distributed globally across basis states to produce a frequency-domain representation, while in the QHWT framework, the image data is decomposed into hierarchical coefficients that capture both coarse and fine structural details. The encoded states are then transmitted through a noisy quantum channel that introduces noise and decoherence. At the receiver, the corresponding inverse quantum Fourier transform (IQFT) or inverse quantum Haar wavelet transform (IQHWT) is applied, followed by quantum-to-classical conversion to recover the transmitted bitstream. Finally, channel decoding (if applied) and source decoding regenerate the image, and the quality of the reconstructed images is evaluated using the peak signal-to-noise ratio (PSNR), the structural similarity index measure (SSIM) [11], and the universal quality index (UQI) [12], thereby providing an error-resilient quantum communication framework.

The key innovations of this research can be summarized as follows.

• Extension of the QHWT into the domain of quantum image transmission as an encoding method, marking its first application beyond image representation and compression.
• Systematic comparative analysis of QFT- and QHWT-based transmission frameworks under identical experimental conditions.
• Evaluation of performance at the image level using PSNR, SSIM, and UQI, rather than relying solely on quantum state fidelity.
• Provision of practical insights for the design of noise-resilient quantum communication systems tailored to image data.

The structure of this paper is as follows. Section 2 provides a review of existing approaches in quantum communication with an emphasis on quantum image representation and transform-based methods. Section 3 describes the proposed image transmission framework and outlines how QFT and QHWT are integrated into the encoding and decoding process. Section 4 reports the simulation results, comparing the two transforms in terms of image quality under noisy channel conditions. Finally, Section 5 summarizes the main findings and highlights potential future directions to advance noise-resilient quantum image transmission.

2. Related Works

The development of quantum communication [13] is viewed as having steadily evolved from fundamental cryptographic tools to broader applications in information transfer. The earliest progress is marked by the introduction of quantum key distribution (QKD) [14], through which the intrinsic properties of quantum mechanics are demonstrated to provide information-theoretic security unattainable by classical systems [15,16,17,18,19]. Protocols such as BB84 and E91 are presented as concrete realizations of QKD and have been experimentally validated in both fiber-based and free-space settings [20,21]. These developments have laid the groundwork for the first operational quantum networks. Beyond QKD, significant advances have been achieved in quantum teleportation [22,23,24,25] and quantum repeater architectures [26], through which unknown quantum states can be transferred between distant users without direct particle transmission, thereby extending the feasible communication distance and reliability of quantum links. As the field matures, attention is increasingly directed toward media-oriented quantum communication, where quantum techniques are being investigated for their ability to protect and transmit visual data. In this direction, quantum states have been shown to be employed to securely process and transmit images and videos, thus opening new possibilities for multimedia applications in quantum networks [27,28,29,30].

Building on the foundations of secure quantum communication, early efforts in efficient and high-quality quantum image transmission are focused on superposition-based encoding strategies [7]. The simplest method is single-qubit encoding, in which each classical bit is directly mapped to a single qubit state. This approach is extended to both image and video transmission [7,31], where it is supported by the development of quantum error correction (QEC) [32,33,34,35] mechanisms to mitigate noise and is applied in multi-input multi-output (MIMO)-based quantum communication frameworks [36] to increase reliability and throughput. Despite these advances, scalability is inherently limited, since the number of qubits required grows linearly with image size and the available parallelism is underutilized.

To address these issues, multi-qubit superposition encoding is introduced [8], by which multiple classical bits are jointly represented within several qubits. In this way, more compact data representation is achieved, quantum resources are more efficiently utilized, and resilience is enhanced, as information is distributed across multiple qubits rather than being tied to an individual one. While these methods are demonstrated to establish the feasibility of quantum image transmission and are regarded as valuable proof-of-concept implementations, they are still classified as time-domain techniques, since information is encoded directly in the computational basis without transformation into alternative domains. As a result, they are constrained in scalability, compression efficiency, and robustness under noise, and these inherent limitations are widely recognized as having motivated the transition toward transform-based approaches such as the QFT and the QHWT.

The transition from time-domain methods is marked by the adoption of the QFT as a more powerful tool for quantum image processing and communication [37]. Initially, QFT is employed in the context of quantum image compression, where its ability to distribute information globally across basis states is utilized to achieve compact representations and efficient data reduction [9]. Building on this foundation, the use of QFT as a direct encoding method for image transmission is explored, with its frequency-domain representation being leveraged to improve resilience against channel noise and enhance reconstruction quality [38]. Through these early applications, the potential of QFT is demonstrated not only as a mathematical tool but also as a practical framework for transform-based quantum image communication.

In parallel, the QHWT is recognized as another important transform-based technique, particularly suited for representing structural details in images [39]. Unlike the global frequency distribution achieved by QFT, QHWT is characterized by a localized, multi-resolution decomposition through which both coarse and fine image features are captured. This property is regarded as making it especially effective for tasks such as quantum image representation, compression, and structural analysis, where edge preservation and hierarchical encoding are required. However, despite these advantages, the QHWT has not yet been explored for direct quantum image transmission, and its role within communication frameworks remains unexamined. An overview of the developed research on QFT and QHWT, along with their advantages and limitations, is given in

Table 1. Recent advancements in QFT and QHWT for image compression and transmission.

Reference	System	Advantages	Limitations
[9]	Quantum image compression	Efficient encoding and decoding and high data reduction	Limited to compression, hardware dependency and scalability challenges in practice
[38]	Multi-qubit frequency-domain transmission	Enhanced scalability and noise resilience	Increased system complexity
[39]	QHWT + Qsobel quantum watermarking	Ensures high invisibility and robustness with reversible embedding	Lack of evaluation under noisy transmission, scalability, and hardware feasibility
[40]	QFT-based quantum image compression	Provides efficient frequency-domain data reduction	Limited to compression with scalability and hardware constraints
[41]	QFT-based quantum image compression	Provides efficient frequency-domain data reduction	Limited to compression
[42]	QWT for dimensionality reduction	Enables efficient handling and reconstruction of high-resolution data	Lacks evaluation under noisy transmission
[43]	QHWT-based quantum image compression	Provides higher PSNR, faster runtime, and better detail preservation	lacks evaluation under noisy transmission, limited by hardware immaturity, encoding dependency, and narrow benchmarking scope

.

Therefore, the novelty of this research lies in the presentation of the first comparative study of QFT-based and QHWT-based frameworks for quantum image transmission. While QFT has already been applied for compression and direct encoding, QHWT has thus far remained limited to representation and compression tasks, with no prior work extending it into the transmission domain. By benchmarking QHWT against QFT under identical experimental conditions, this study not only evaluates their relative strengths but also provides the first application-oriented insights into how global frequency-domain encoding and localized wavelet-based encoding perform in practical transmission scenarios. This comparative analysis establishes a foundation for developing noise-resilient quantum communication systems tailored to image data.

3. Methodology

The overall design of the transform-based quantum communication system is shown in Figure 1. The model combines classical image compression, multi-qubit quantum encoding, and robust error-handling to enable end-to-end transmission. A distinctive aspect of this framework is its use of QFT and QHWT. Using both transforms, the system can represent global spectral information through QFT while capturing localized structures through QHWT, making it effective for images with diverse levels of detail.

The framework is designed to be both scalable and flexible. A single logical quantum state can encode between one and eight qubits, with the upper limit chosen to align with the 8-bit pixel range (0–255), thereby ensuring efficient mapping of pixel values to quantum states. While this work adopts eight qubits as a practical upper bound, the approach is not inherently constrained and can be extended to higher qubit counts as permitted by hardware capabilities and application requirements.

For clarity, the operation of the proposed framework is explained step by step, with each functional block in Figure 1 discussed in detail in the following subsections.

3.1. Input Image

The proposed framework is designed to be input-agnostic, meaning it can process images regardless of their size, resolution, or content. To evaluate performance under realistic conditions, we utilize 100 images from the Microsoft COCO dataset [44]. This dataset is particularly suitable as it covers a wide spectrum of natural scenes, from simple low-texture backgrounds to cluttered environments containing multiple interacting objects.

Such diversity introduces significant real-world complexity, as images differ not only in spatial resolution but also in illumination, texture richness, edge density, and semantic content. These variations translate into different levels of spatial information (SI) and structural complexity, posing challenges that closely mirror those encountered in practical communication scenarios. By using this representative image set, the system can be rigorously assessed for robustness and adaptability across a broad spectrum of real-world conditions.

3.2. Source Encoder and Decoder

Before transmission, each input image is compressed using widely adopted source coding standards, such as JPEG [1] and HEIF [2]. These codecs reduce statistical and perceptual redundancy, generating binary representations that are more bandwidth-efficient. The degree of compression is governed by the quantization parameter (QP), which directly influences the size of the resulting bitstream. Lower QP values apply stronger compression, leading to shorter bitstreams, while higher QP values preserve more detail at the cost of producing longer bitstreams. To maintain uniform bandwidth usage across different multi-qubit configurations, QP values are carefully tuned so that the quantum encoded streams remain comparable in size. This balancing step ensures that performance comparisons between quantum encoding modes are not biased by variations in source compression.

On the receiver side, the transmitted quantum streams are decoded with the corresponding JPEG or HEIF decoders. The reconstructed images are then subjected to objective quality assessment, enabling a systematic evaluation of how well the proposed system maintains visual fidelity through the complete communication process.

3.3. Channel Encoder and Decoder

The communication framework employs polar codes [45] as the chosen channel coding strategy. Polar codes have gained recognition as one of the most advanced error correction techniques in the classical domain [46]. They are built on the concept of channel polarization, which allows them to approach channel capacity when decoded using successive cancellation. Compared with well-known coding schemes including low-density parity check (LDPC) [47], turbo [48], and Reed–Solomon codes [49], polar codes demonstrate superior performance in many practical conditions.

In this work, a code rate of 1/2 is selected. This means that for every block of information bits, an equal number of redundancy bits are introduced. The choice provides a balanced compromise between error protection and transmission efficiency, allowing the system to maintain reliability without consuming excessive bandwidth. At the receiver, the polar decoder reconstructs the transmitted data from the quantum-decoded bitstreams, which are then delivered to the subsequent source decoding stages.

The fundamental reason for the deliberate exclusion of QEC is the inherent domain incompatibility between conventional QEC methods and the transform-domain architecture. QEC codes are designed to operate in the time domain against local errors, assuming sequential qubit processing. Conversely, our system utilizes the QFT and QHWT, which encode information across spectral or frequency components. Directly embedding time-domain QEC into this framework would not only demand significant computational resources and qubit overhead but would also require a complex, incompatible restructuring of syndrome measurement and decoding to handle the resulting non-local, correlated error patterns, thus contravening the core goal of maintaining low system complexity and high efficiency.

3.4. Quantum State Encoder

The quantum state encoder provides the essential link between classical binary inputs and their quantum state representations. Its function is to transform conventional bitstreams into quantum states that can later undergo quantum transformations and be transmitted across noisy channels. The quantum state encoding process proceeds in two main stages: mapping individual bits to qubits and constructing multi-qubit state vectors.

3.4.1. Representing Classical Bits as Qubits

A binary digit $b\in \{0,1\}$ is first represented in the computational basis of a single qubit. When $b=0$, it corresponds to the state $|0\rangle$, and when $b=1$, it corresponds to the state $|1\rangle$. These two states span the Hilbert space ${\mathcal{H}}_2$ of a qubit and can be written in vector form as shown in Equations (1) and (2).

$$|0\rangle =\left(\begin{array}{c}1\\ 0\end{array}\right)$$

(1)

$$|1\rangle =\left(\begin{array}{c}0\\ 1\end{array}\right)$$

(2)

This mapping embeds classical information directly into a quantum-compatible representation, preparing it for further quantum processing.

3.4.2. Multi-Qubit State Construction

For multi-qubit encoding, classical bits are grouped into blocks of size n, where n ranges from 1 to 8 depending on the encoding configuration. Each block can be expressed as a binary sequence, as shown in Equation (3), and its corresponding quantum state is obtained by combining the individual qubits through the tensor product, as given in Equation (4).

$$b_1{b}_2\dots b_n,b_i\in \{0,1\}$$

(3)

$$|b_1{b}_2\dots b_n\rangle =|b_1\rangle \otimes |b_2\rangle \otimes \dots \otimes |b_n\rangle$$

(4)

For single-qubit encoding, the encoded quantum states are similar to those shown in Equations (1) and (2). For example, when the encoding size is $n=2$, the four possible input pairs are mapped to the corresponding quantum states, as shown in Equations (5)–(8).

$$|00\rangle =|0\rangle \otimes |0\rangle =\left(\begin{array}{c}1\\ 0\end{array}\right)\otimes \left(\begin{array}{c}1\\ 0\end{array}\right)={\left(\begin{array}{cccc}1& 0& 0& 0\end{array}\right)}^T$$

(5)

$$|01\rangle =|0\rangle \otimes |1\rangle =\left(\begin{array}{c}1\\ 0\end{array}\right)\otimes \left(\begin{array}{c}0\\ 1\end{array}\right)={\left(\begin{array}{cccc}0& 1& 0& 0\end{array}\right)}^T$$

(6)

$$|10\rangle =|1\rangle \otimes |0\rangle =\left(\begin{array}{c}0\\ 1\end{array}\right)\otimes \left(\begin{array}{c}1\\ 0\end{array}\right)={\left(\begin{array}{cccc}0& 0& 1& 0\end{array}\right)}^T$$

(7)

$$|11\rangle =|1\rangle \otimes |1\rangle =\left(\begin{array}{c}0\\ 1\end{array}\right)\otimes \left(\begin{array}{c}0\\ 1\end{array}\right)={\left(\begin{array}{cccc}0& 0& 0& 1\end{array}\right)}^T$$

(8)

In general, for a block size of n, the resulting state vector resides in a Hilbert space of dimension $2^n$. This exponential scaling highlights the efficiency of quantum encoding, as several classical bits can be compactly represented within a single high-dimensional quantum state. Similarly, an n-qubit quantum state can be systematically constructed by taking the tensor product of n individual single-qubit states.

3.5. Quantum Fourier Transform (QFT)

The QFT is a key operation that maps quantum states from the computational or time domain into the frequency domain. For the transformation to be valid, the size of the QFT operator must match the dimension of the state vector on which it acts. Since an n-qubit system spans a Hilbert space of size $2^n$, the state vector has dimension $2^n\times 1$, and the corresponding QFT operator must therefore be a unitary matrix of dimension $2^n\times 2^n$.

As an illustration, a system of $n=2$ qubits produces a $4\times 1$ state vector, requiring a $4\times 4$ QFT operator. Similarly, for $n=3$, the Hilbert space has dimension 8, and the QFT operator is an $8\times 8$ unitary matrix. This scaling applies directly to any number of qubits.

3.5.1. General Definition

For an n-qubit register with dimension $N=2^n$, the QFT is represented by the unitary operator $F_N$. Its action on a computational basis state $|x\rangle$ produces a frequency-domain superposition given by Equation (9).

$$F_N|x\rangle ={\displaystyle \frac{1}{\sqrt{N}}}\sum _{k=0}^{N-1}{e}^{2\pi i{\displaystyle \frac{xk}{N}}}|k\rangle$$

(9)

Here, the input basis state $|x\rangle$ is expanded as a uniform superposition of all basis vectors $|k\rangle$, with complex phase factors $e^{2\pi i{\displaystyle \frac{xk}{N}}}$ encoding frequency information. The normalization factor $1/\sqrt{N}$ guarantees unitarity by preserving the norm of the state.

3.5.2. Matrix Form of QFT

The QFT matrix $F_N$ can be expressed explicitly as shown in Equation (10).

$$F_N={\displaystyle \frac{1}{\sqrt{N}}}\left(\begin{array}{ccccc}1& 1& 1& \dots & 1\\ 1& \omega & {\omega }^2& \dots & {\omega }^{N-1}\\ 1& {\omega }^2& {\omega }^4& \dots & {\omega }^{2(N-1)}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 1& {\omega }^{N-1}& {\omega }^{2(N-1)}& \dots & {\omega }^{{(N-1)}^2}\end{array}\right)$$

(10)

where $\omega$ is the primitive N-th root of unity defined in Equation (11).

$$\omega =e^{{\displaystyle \frac{2\pi i}{N}}}$$

(11)

For $n=2$, where $N=4$, the QFT operator simplifies to the $4\times 4$ matrix in Equation (12).

$$F_4={\displaystyle \frac{1}{2}}\left(\begin{array}{cccc}1& 1& 1& 1\\ 1& i& -1& -i\\ 1& -1& 1& -1\\ 1& -i& -1& i\end{array}\right)$$

(12)

3.5.3. Example: Two-Qubit Transformation

To illustrate the effect of the QFT, consider the two-qubit basis states $|00\rangle$, $|01\rangle$, $|10\rangle$, and $|11\rangle$. Applying $F_4$ to each of these inputs produces distinct frequency-domain superpositions, as shown in Equations (13)–(16).

$$F_4|00\rangle ={\displaystyle \frac{1}{2}}{\left(\begin{array}{cccc}1& 1& 1& 1\end{array}\right)}^T={\displaystyle \frac{1}{2}}\left(|00\rangle +|01\rangle +|10\rangle +|11\rangle \right)$$

(13)

$$F_4|01\rangle ={\displaystyle \frac{1}{2}}{\left(\begin{array}{cccc}1& i& -1& -i\end{array}\right)}^T={\displaystyle \frac{1}{2}}\left(|00\rangle +i|01\rangle -|10\rangle -i|11\rangle \right)$$

(14)

$$F_4|10\rangle ={\displaystyle \frac{1}{2}}{\left(\begin{array}{cccc}1& -1& 1& -1\end{array}\right)}^T={\displaystyle \frac{1}{2}}\left(|00\rangle -|01\rangle +|10\rangle -|11\rangle \right)$$

(15)

$$F_4|11\rangle ={\displaystyle \frac{1}{2}}\left(\begin{array}{cccc}1& -i& -1& i\end{array}\right)={\displaystyle \frac{1}{2}}\left(|00\rangle -i|01\rangle -|10\rangle +i|11\rangle \right)$$

(16)

These results show how the QFT transforms time-domain basis states into frequency-domain representations, each carrying unique phase relationships. More generally, each column of the QFT matrix corresponds to a distinct computational basis state. This property makes the QFT central to many quantum algorithms and a powerful tool for quantum communication, where encoding in the frequency domain enables efficient manipulation and transmission of information.

3.6. Inverse Quantum Fourier Transform (IQFT)

In the decoding stage, the first task is to identify the dimension of the received quantum state, since this determines the appropriate size of the IQFT operator. Once the correct dimension is established, the IQFT is applied to revert the state from the frequency basis back into its time-domain representation. The resulting states are then measured in the computational basis, allowing them to be mapped to classical bits and thereby reconstructing the transmitted binary sequence.

3.6.1. Mathematical Formulation of Decoding in the Noiseless Case

Formally, the IQFT is defined as the Hermitian conjugate of the QFT matrix, as expressed in Equation (17).

$${\mathbf{F}}_{\mathrm{IQFT}}={\mathbf{F}}_{\mathrm{QFT}}^{†}$$

(17)

Let the received quantum state in the frequency domain be denoted by $|{\mathrm{\Psi }}_f\rangle \in {\mathbb{H}}^{2^n}$, where n is the number of encoded qubits and ${\mathbb{H}}^{2^n}$ is the Hilbert space describing the system. The decoding process applies the IQFT operator to $|{\mathrm{\Psi }}_f\rangle$ to recover its time-domain form, as described in Equation (18).

$$|{\mathrm{\Psi }}_t\rangle ={\mathbf{F}}_{\mathrm{IQFT}}|{\mathrm{\Psi }}_f\rangle$$

(18)

Here, $|{\mathrm{\Psi }}_t\rangle$ represents the reconstructed state in the time domain. The final stage consists of a projective measurement, which collapses $|{\mathrm{\Psi }}_t\rangle$ onto the computational basis, thereby producing a deterministic sequence of classical bits corresponding to the original data.

3.6.2. Mathematical Formulation of Decoding in the Presence of Noise

When the transmission channel introduces noise, the received quantum state will not perfectly align with the expected QFT basis states. Let the perturbed frequency-domain vector be denoted as $|{\mathsf{\Phi }}_f\rangle$, expressed in Equation (19).

$$|{\mathsf{\Phi }}_f\rangle =|{\mathrm{\Psi }}_f\rangle +|\nu \rangle$$

(19)

where $|\nu \rangle$ represents the noise contribution, assumed to have small magnitude with $\parallel |\nu \rangle \parallel \ll 1$.

To recover the most likely transmitted symbol, the decoder performs a nearest-neighbor search among the set of ideal QFT basis states, $\mathcal{B}={\left\{|{\mathrm{\Psi }}_{fj}\rangle \right\}}_{j=0}^{2^n-1}$. The estimated index $j^{\prime }$ is chosen to minimize the Euclidean distance, as defined in Equation (20).

$$j^{\prime }=arg\underset{j}{min}∥|{\mathsf{\Phi }}_f\rangle -|{\mathrm{\Psi }}_{fj}\rangle ∥$$

(20)

The closest matching basis state $|{\mathrm{\Psi }}_{f{j}^{\prime }}\rangle$ is then taken as the transmitted frequency-domain symbol. This state is passed through the IQFT operator to reconstruct the corresponding time-domain representation, as expressed in Equation (21).

$$|{\tilde{\mathrm{\Psi }}}_t\rangle ={\mathbf{F}}_{\mathrm{IQFT}}|{\mathrm{\Psi }}_{f{j}^{\prime }}\rangle$$

(21)

Finally, a projective measurement of $|{\tilde{\mathrm{\Psi }}}_t\rangle$ produces the estimated classical bit sequence $\hat{\mathbf{b}}\in \{0,1\}$.

In this approach, the recovery process does not depend on examining individual amplitudes or phases. Instead, it leverages the global structure of the received quantum state by comparing entire vectors in the frequency domain, which improves robustness to channel distortions.

3.7. Quantum Haar Wavelet Transform (QHWT)

The QHWT is a unitary transformation that, similar to the QFT, provides an alternative way of encoding quantum states. Unlike the QFT, which distributes information globally, the QHWT decomposes signals into localized low-frequency (approximation) and high-frequency (detail) components. This makes it especially effective for data such as images, where both global and local structures are important.

3.7.1. General Definition

The QHWT is derived from the classical Haar transform, which can be represented by orthogonal Haar matrices of dimension $N=2^n$. For the single-qubit case ($N=2$), the QHWT operator is given by Equation (22).

$$H_2={\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right)$$

(22)

This operator averages and differences the components of a two-element vector, and in the quantum setting, it acts as a basic building block of the QHWT.

3.7.2. Recursive Structure of Quantum Haar Matrices

For higher dimensions, the QHWT matrix $H_N$ is defined recursively. In the case of $N=4$ (two qubits), the transform is constructed using the base operator $H_2$ from Equation (22) together with the identity matrix $I_2$ from Equation (23).

$$I_2=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$$

(23)

The recursive definition for $H_4$ is given in Equation (24).

$$H_4={\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{c}{H}_2\otimes \left[11\right]\\ I_2\otimes [1-1]\end{array}\right)$$

(24)

This recursive construction expands explicitly to the form given in Equation (25).

$$H_4={\displaystyle \frac{1}{2}}\left(\begin{array}{cccc}1& 1& 1& 1\\ 1& 1& -1& -1\\ \sqrt{2}& -\sqrt{2}& 0& 0\\ 0& 0& \sqrt{2}& -\sqrt{2}\end{array}\right)$$

(25)

More generally, for n qubits ($N=2^n$), the QHWT matrix $H_{2^n}$ can be constructed recursively from $H_{2^{n-1}}$ as shown in Equation (26).

$$H_{2^n}={\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{c}{H}_{2^{n-1}}\otimes \left[\begin{array}{cc}1& 1\end{array}\right]\\ I_{2^{n-1}}\otimes \left[\begin{array}{cc}1& -1\end{array}\right]\end{array}\right)$$

(26)

where $H_{2^{n-1}}$ and $I_{2^{n-1}}$ can be constructed by taking the tensor product of the base matrices $H_2$ and $I_2$, respectively, repeated $(n-1)$ times.

3.7.3. Example: Two-Qubit Transformation

Applying $H_4$ to the two-qubit computational basis states $\{|00\rangle ,|01\rangle ,|10\rangle ,|11\rangle \}$ produces the transformed vectors shown in Equations (27)–(30).

$$H_4|00\rangle ={\displaystyle \frac{1}{2}}{\left(\begin{array}{cccc}1& 1& \sqrt{2}& 0\end{array}\right)}^T$$

(27)

$$H_4|01\rangle ={\displaystyle \frac{1}{2}}{\left(\begin{array}{cccc}1& 1& -\sqrt{2}& 0\end{array}\right)}^T$$

(28)

$$H_4|10\rangle ={\displaystyle \frac{1}{2}}{\left(\begin{array}{cccc}1& -1& 0& \sqrt{2}\end{array}\right)}^T$$

(29)

$$H_4|11\rangle ={\displaystyle \frac{1}{2}}{\left(\begin{array}{cccc}1& -1& 0& -\sqrt{2}\end{array}\right)}^T$$

(30)

In these results, the first two entries represent approximation (low-frequency) components, while the last two entries capture detail (high-frequency) components, clearly demonstrating the localized decomposition performed by the QHWT.

3.8. Inverse Quantum Haar Wavelet Transform (IQHWT)

The IQHWT is used at the decoder to reconstruct the time-domain quantum state. Since $H_N$ is a unitary operator, its inverse is obtained by taking the conjugate transpose, as shown in Equation (31).

$$H_N^{-1}=H_N^{†}$$

(31)

3.8.1. Mathematical Formulation of Decoding in the Noiseless Case

In the absence of channel noise, the received state is identical to the transmitted wavelet-domain vector. Let the received state be denoted as $|{\mathrm{\Psi }}_h\rangle \in {\mathbb{H}}^{2^n}$. The IQHWT is then applied to recover the time-domain state, as expressed in Equation (32).

$$|{\mathrm{\Psi }}_t\rangle =H_N^{†}|{\mathrm{\Psi }}_h\rangle$$

(32)

The resulting state $|{\mathrm{\Psi }}_t\rangle$ is subsequently measured in the computational basis, producing the original classical bit sequence without error.

3.8.2. Mathematical Formulation of Decoding in the Presence of Noise

When noise is introduced during transmission, the received vector deviates from the ideal quantum Haar basis states. Let the noisy received state be written as in Equation (33).

$$|{\mathsf{\Phi }}_h\rangle =|{\mathrm{\Psi }}_h\rangle +|\nu \rangle$$

(33)

where $|\nu \rangle$ is a perturbation vector with $\parallel |\nu \rangle \parallel \ll 1$.

To infer the transmitted state, the decoder performs a nearest-basis search over the quantum Haar basis set $\mathcal{H}={\left\{|{\mathrm{\Psi }}_{hj}\rangle \right\}}_{j=0}^{2^n-1}$. The estimated index $j^{\prime }$ is selected according to Equation (34).

$$j^{\prime }=arg\underset{j}{min}∥|{\mathsf{\Phi }}_h\rangle -|{\mathrm{\Psi }}_{hj}\rangle ∥$$

(34)

Once the closest match $|{\mathrm{\Psi }}_{h{j}^{\prime }}\rangle$ is determined, it is taken as the transmitted state. Applying the IQHWT operator reconstructs the time-domain signal, as shown in Equation (35).

$$|{\tilde{\mathrm{\Psi }}}_t\rangle =H_N^{†}|{\mathrm{\Psi }}_{h{j}^{\prime }}\rangle$$

(35)

Finally, projective measurement of $|{\tilde{\mathrm{\Psi }}}_t\rangle$ yields the estimated classical bitstream $\hat{\mathbf{b}}\in \{0,1\}$.

3.9. Quantum Channel

To assess the reliability of the proposed transform-based quantum image transmission framework, the propagation of quantum information is modeled through a noisy channel. In this study, five representative error mechanisms are considered: bit-flip, phase-flip, depolarizing noise, amplitude damping, and phase damping [31]. Together, these processes provide a sufficiently comprehensive abstraction of real-world quantum channel behavior, as they capture the dominant sources of decoherence, state inversion, and energy loss typically encountered in practical quantum systems.

3.9.1. Bit-Flip Noise

A bit-flip error corresponds to the exchange between $|0\rangle$ and $|1\rangle$, occurring with probability ${ϵ}_x$. The evolution of a quantum state $\mathrm{\Psi }$ under this channel is described in Equation (36), where ${\sigma }_x$ is the Pauli-X matrix.

$${\mathcal{N}}_x\left(\mathrm{\Psi }\right)={ϵ}_x{\sigma }_x\mathrm{\Psi }{\sigma }_x^{†}+(1-{ϵ}_x)\mathrm{\Psi }$$

(36)

3.9.2. Phase-Flip Noise

In phase-flip errors, the sign of the $|1\rangle$ component is inverted while populations remain unchanged. The effect of this noise process, with probability ${ϵ}_z$, is given in Equation (37), where ${\sigma }_z$ denotes the Pauli-Z operator.

$${\mathcal{N}}_z\left(\mathrm{\Psi }\right)={ϵ}_z{\sigma }_z\mathrm{\Psi }{\sigma }_z^{†}+(1-{ϵ}_z)\mathrm{\Psi }$$

(37)

3.9.3. Depolarizing Noise

Depolarization drives the state towards maximum uncertainty by replacing the density operator with the maximally mixed state. With error probability ${ϵ}_d$, this process is represented in Equation (38), where ${\sigma }_x,{\sigma }_y,{\sigma }_z$ are the Pauli matrices.

$${\mathcal{N}}_d\left(\mathrm{\Psi }\right)=(1-{ϵ}_d)\mathrm{\Psi }+{\displaystyle \frac{{ϵ}_d}{3}}\left({\sigma }_x\mathrm{\Psi }{\sigma }_x^{†}+{\sigma }_y\mathrm{\Psi }{\sigma }_y^{†}+{\sigma }_z\mathrm{\Psi }{\sigma }_z^{†}\right)$$

(38)

3.9.4. Amplitude Damping

Amplitude damping models irreversible energy loss such as photon leakage or spontaneous emission. For a decay probability ${ϵ}_a$, the channel is described by Equation (39), with Kraus operators $K_0,K_1$ defined in Equation (40).

$${\mathcal{N}}_a\left(\mathrm{\Psi }\right)=K_0\mathrm{\Psi }{K}_0^{†}+K_1\mathrm{\Psi }{K}_1^{†}$$

(39)

$$K_0=\left(\begin{array}{cc}1& 0\\ 0& \sqrt{1-{ϵ}_a}\end{array}\right),K_1=\left(\begin{array}{cc}0& \sqrt{{ϵ}_a}\\ 0& 0\end{array}\right)$$

(40)

3.9.5. Phase Damping

Phase damping destroys quantum coherence without altering the populations of $|0\rangle$ and $|1\rangle$. With error probability ${ϵ}_{\varphi }$, the channel action is expressed in Equation (41), using Kraus operators $L_0,L_1$ defined in Equation (42).

$${\mathcal{N}}_{\varphi }\left(\mathrm{\Psi }\right)=L_0\mathrm{\Psi }{L}_0^{†}+L_1\mathrm{\Psi }{L}_1^{†}$$

(41)

$$L_0=\left(\begin{array}{cc}1& 0\\ 0& \sqrt{1-{ϵ}_{\varphi }}\end{array}\right),L_1=\left(\begin{array}{cc}0& 0\\ 0& \sqrt{{ϵ}_{\varphi }}\end{array}\right)$$

(42)

3.9.6. Composite Channel Model

The combined effect of all noise sources is modeled as a convex mixture, as shown in Equation (43).

$${\mathcal{N}}_{\mathrm{tot}}\left(\mathrm{\Psi }\right)={ϵ}_x{\mathcal{N}}_x\left(\mathrm{\Psi }\right)+{ϵ}_z{\mathcal{N}}_z\left(\mathrm{\Psi }\right)+{ϵ}_d{\mathcal{N}}_d\left(\mathrm{\Psi }\right)+{ϵ}_a{\mathcal{N}}_a\left(\mathrm{\Psi }\right)+{ϵ}_{\varphi }{\mathcal{N}}_{\varphi }\left(\mathrm{\Psi }\right)$$

(43)

3.9.7. Noise Parameterization via SNR

To relate the error probabilities to physical channel quality, we parameterize them using the signal-to-noise ratio (SNR). The definition of SNR in decibels is given in Equation (44), where $P_s$ denotes the signal power and $P_n$ the noise power.

$$\mathrm{SNR}\left(\mathrm{dB}\right)=10{log}_{10}\left({\displaystyle \frac{P_s}{P_n}}\right)$$

(44)

The aggregate channel error probability ${ϵ}_{\mathrm{tot}}$ is modeled as an inverse function of the linear SNR, as expressed in Equation (45).

$${ϵ}_{\mathrm{tot}}=min\left(1,{\displaystyle \frac{1}{1+{10}^{\mathrm{SNR}/10}}}\right)$$

(45)

To distribute ${ϵ}_{\mathrm{tot}}$ among the five noise processes, random weights $r_i$ are drawn from a uniform distribution, as defined in Equation (46), and normalized according to Equation (47).

$$r_1,r_2,r_3,r_4,r_5\sim \mathcal{U}(0,1)$$

(46)

$$[{ϵ}_x,{ϵ}_z,{ϵ}_d,{ϵ}_a,{ϵ}_{\varphi }]=\left({\displaystyle \frac{r_i}{{\sum }_{j=1}^5{r}_j}}\right)·{ϵ}_{\mathrm{tot}}$$

(47)

This allocation satisfies the normalization condition shown in Equation (48).

$${ϵ}_x+{ϵ}_z+{ϵ}_d+{ϵ}_a+{ϵ}_{\varphi }={ϵ}_{\mathrm{tot}}$$

(48)

By combining these canonical noise processes into the unified model of Equation (43), the channel description accounts for bit inversions, phase disturbances, depolarization, energy decay, and coherence loss [50]. This abstraction mirrors realistic quantum communication environments, where multiple error sources act simultaneously [51]. Consequently, the performance evaluation of the proposed scheme is grounded in a practically meaningful and experimentally relevant noise model.

3.10. Quantum State Decoder

After applying the IQFT or IQHWT, the time-domain state $|{\psi }_t\rangle$ is measured in the computational basis. This projective measurement collapses the quantum state to one of the basis vectors $|b_1{b}_2\dots b_n\rangle$, directly yielding the classical bit sequence $\hat{\mathbf{b}}\in {\{0,1\}}^n$. This process is summarized in Equation (49).

$$\hat{\mathbf{b}}=\mathcal{M}\left(|{\psi }_t\rangle \right)$$

(49)

where $\mathcal{M}(·)$ denotes measurement in the computational basis. The recovered bit sequence is then passed to the classical channel decoder (polar decoder) and finally to the source decoder (JPEG/HEIF) to reconstruct the original image or video frame.

4. Results and Discussion

The performance of the proposed transform-based quantum image transmission frameworks is evaluated using both QFT and QHWT under identical simulation settings. Results are reported as average values across the test dataset, providing a representative measure of overall system behavior. Image reconstruction quality is assessed using three widely adopted objective metrics: PSNR, SSIM, and UQI, which together capture complementary aspects of pixel-level fidelity, structural similarity, and perceptual quality. For completeness, two source encoding formats, JPEG and HEIF, are examined, and the experiments are conducted under both uncoded transmission and channel-coded transmission using polar codes.

The results for JPEG and HEIF image formats are presented in the following subsections under both uncoded and channel-coded conditions. In all graphs, the QFT-based framework is plotted using solid lines with circular markers, while the QHWT-based framework is plotted using dashed lines with square markers. The legends further distinguish the encoding size: F1–F8 denote QFT with 1 to 8 qubit encoding, and H1–H8 denote QHWT with 1 to 8 qubit encoding.

4.1. JPEG Image Transmission Under Uncoded Scenario

In the uncoded transmission scenario, JPEG-compressed images are transmitted directly through the quantum channel without the aid of error correction. The results presented in Figure 2 show that for single-qubit encoding, the performance difference between QFT (F1) and QHWT (H1) is negligible, as both transforms rely on the same underlying encoding matrix representation. However, as the encoding size increases, both frameworks demonstrate improved robustness against noise, with QFT consistently outperforming QHWT. At the highest encoding size of eight qubits, QFT (F8) is observed to achieve up to a 3 dB maximum channel SNR gain compared to QHWT (H8), clearly demonstrating its superior resilience against channel noise. Importantly, since no error correction is applied, these results reflect the inherent strengths and limitations of the two transforms themselves, independent of channel coding schemes.

From the pixel-domain perspective (PSNR), as illustrated in Figure 2a, the QFT-based framework demonstrates a stronger ability to preserve intensity values under noisy conditions. While QHWT also benefits from increased qubit size, it suffers greater fidelity loss compared to QFT, particularly in the low-to-medium SNR regime. This confirms that QFT is better suited for pixel-level accuracy. When considering structural fidelity (SSIM), as illustrated in Figure 2b, QFT again demonstrates an advantage by retaining edges and textures more effectively across the full SNR range. QHWT performance improves with higher encodings, but fine structural details remain more vulnerable to distortion. These results suggest that QFT provides a stronger safeguard for structural similarity, which is critical in image interpretation. In terms of perceptual quality (UQI), as illustrated in Figure 2c, QFT achieves higher scores across nearly all SNR conditions.

It is important to note that all evaluations are performed under fixed bandwidth constraints. Consequently, smaller qubit encodings achieve higher maximum values of PSNR, SSIM, and UQI, but lack resilience in noisy conditions. In contrast, higher encodings reduce peak quality, but offer improved robustness, enabling reliable image reconstruction even at lower SNRs. This trade-off illustrates how encoding size influences the balance between maximum achievable quality and noise tolerance.

4.2. JPEG Image Transmission Under Channel-Coded Scenario

With the introduction of classical channel coding using polar codes, a marked improvement in JPEG image transmission performance is observed across all evaluation metrics, enhancing both fidelity and robustness against channel noise. As observed in the PSNR curves (Figure 3a), the addition of coding provides a consistent 3–4 dB channel SNR gain compared to the uncoded case. This implies that the same reconstruction fidelity can be obtained at substantially lower channel SNR values, which is particularly valuable in bandwidth or power-constrained systems where high SNR may not be achievable.

From the pixel-domain perspective (PSNR), polar coding shifts the degradation threshold toward lower SNR values, allowing images to retain much higher intensity fidelity at noisy channel conditions that would otherwise yield noticeable artifacts. This demonstrates the capacity of polar codes to suppress pixel-level distortions, thereby extending the usable operating range of both frameworks. For structural similarity (SSIM), the benefits are equally clear. As shown in Figure 3b, the inclusion of channel coding helps preserve edges, textures, and fine structural details in the reconstructed images, even at low-to-mid SNRs. This highlights the ability of polar codes to protect higher-order dependencies in image structure, ensuring that semantic content remains intact. In terms of perceptual quality (UQI), as illustrated in Figure 3c, performance stabilizes throughout the SNR range by applying polar coding. While uncoded systems exhibit sharp drops in perceptual quality at low SNRs, channel-coded transmission enables smoother degradation, with QFT consistently achieving the highest UQI values. QHWT also benefits significantly from error correction, though its perceptual quality remains below that of QFT.

Importantly, the relative performance between the two transforms is consistent with the uncoded case. QFT continues to outperform QHWT across all encoding sizes, and this margin widens as encoding size increases. For instance, at eight-qubit encoding, QFT achieves up to an additional 3 dB advantage over QHWT, reflecting its stronger inherent resilience to quantum noise and better alignment with polar coding.

Overall findings indicate that polar coding delivers a substantial robustness advantage for both frameworks, shifting the effective performance curves by 3–4 dB across all metrics. However, QFT leverages these gains more effectively, achieving superior pixel fidelity, structural preservation, and perceptual quality. These findings emphasize the importance of combining resilient quantum-domain transforms such as QFT with powerful classical error-correcting codes to realize practical, high-fidelity quantum image transmission systems.

4.3. HEIF Image Transmission Under Uncoded Scenario

For the uncoded transmission of HEIF-compressed images, the performance profile resembles that of JPEG uncoded scenario, but with important distinctions. Because HEIF employs a more advanced and efficient source coding strategy, it preserves image details more effectively at equivalent bitrates. This results in higher maximum values of PSNR, SSIM, and UQI compared to JPEG under the same bandwidth constraints. In other words, while both formats are subject to noise-related degradation, HEIF begins with a higher baseline image quality, giving it a distinct advantage in overall reconstruction fidelity.

From the pixel-domain perspective (PSNR), as illustrated in Figure 4a, QFT consistently delivers better noise resilience than QHWT. At the single-qubit level, the performance difference is negligible, as both transforms rely on the same encoding matrix. However, as the encoding size increases, the gap between QFT and QHWT becomes more evident. Higher encoding sizes allow both transforms to resist channel noise more effectively, but QFT retains higher PSNR across the SNR range. This reflects its stronger capacity to preserve subtle variations in pixel intensities under noisy conditions, which is particularly important for applications requiring accurate reproduction of fine image details.

When considering structural fidelity (SSIM), the benefits of HEIF become even more pronounced. As shown in Figure 4b, HEIF images demonstrate superior preservation of textures, edges, and other structural features compared to JPEG. The QFT framework consistently achieves higher SSIM values, maintaining robust structural similarity across a broad range of SNR values. QHWT also improves with increasing encoding size, but compared to QFT, its ability to retain structural information remains weaker, particularly at low-to-mid SNR levels where noise most strongly affects image integrity. From a perceptual quality perspective (UQI), HEIF once again surpasses JPEG, achieving higher maximum values under the same conditions (Figure 4c). This highlights the efficiency of HEIF in retaining perceptually significant information. Between the two transforms, QFT maintains a clear advantage, producing more stable perceptual quality across noisy environments.

It is important to emphasize that the results indicate no significant performance difference between JPEG and HEIF in terms of noise resilience, as the robustness of both formats is determined primarily by the choice of quantum transform rather than the source encoder. The inherent advantage of HEIF lies in its advanced compression efficiency, which provides higher maximum achievable quality values compared to JPEG under identical conditions. Higher qubit encoding sizes continue to improve resilience to channel noise but at the cost of reduced peak quality, reflecting the fundamental trade-off between robustness and maximum fidelity. Across all evaluations, QFT consistently achieves superior reconstruction quality compared to QHWT, underscoring its effectiveness for uncoded image transmission in noisy quantum channels.

4.4. HEIF Image Transmission Under Channel-Coded Scenario

When channel coding with polar codes is introduced, the performance of HEIF image transmission improves significantly across all metrics. As seen in the PSNR curves (Figure 5a), both QFT- and QHWT-based frameworks benefit from an additional 3–4 dB SNR gain compared to the uncoded scenario. This gain translates to achieving the same reconstruction quality at noticeably lower channel SNR levels, effectively enhancing the robustness of the system. Because HEIF already provides superior baseline compression efficiency compared to JPEG, the addition of channel coding further strengthens its ability to preserve pixel accuracy, resulting in higher maximum PSNR values overall.

From the perspective of structural fidelity (SSIM, Figure 5b), polar coding enables the system to sustain high levels of structural similarity even under severe noise. HEIF’s advanced compression retains more edge and texture information, and with error correction, both transforms show clear improvements over the uncoded case. However, QFT maintains consistently higher SSIM values across all encoding sizes, especially in mid-to-low-SNR regions where the channel is most challenging. Although QHWT performance improves with higher encoding sizes, QFT consistently achieves better results, and the difference between the two methods remains noticeable.

For perceptual quality (UQI, Figure 5c), the combined effect of HEIF’s efficient source coding and polar-based error correction proves particularly strong. Even at relatively low SNR, the UQI values remain high, showing that the perceived quality of images is well preserved. QFT continues to outperform QHWT in this scenario, with the margin becoming more pronounced at higher encoding sizes. Importantly, at the highest encoding size (n = 8), the QFT-based framework achieves up to 3 dB additional channel SNR gain beyond QHWT, reinforcing its advantage in noisy environments.

It is important to note that the benefits of channel coding are achieved without altering the inherent properties of the quantum transforms themselves. Polar coding primarily enhances resilience against noise by reducing the effective bit error rate, which translates into higher PSNR, SSIM, and UQI values across the SNR range. This indicates that both QFT- and QHWT-based frameworks not only achieve superior robustness but also preserve image quality, thereby rendering the trade-off between fidelity and resilience significantly more favorable under channel-coded conditions.

Overall, HEIF with polar coding delivers the highest overall image quality among all tested configurations. The advanced source coding efficiency of HEIF ensures higher baseline quality than JPEG, while polar coding provides robust error resilience. QFT consistently outperforms QHWT across all qubit encoding sizes, particularly at higher dimensions, where its inherent resilience allows it to leverage channel coding more effectively. These findings confirm that a QFT-based HEIF framework, combined with polar channel coding, offers a highly promising solution for reliable quantum image transmission under practical noisy channel conditions.

4.5. Sample of Decoded Images

To complement the quantitative analysis with PSNR, SSIM, and UQI metrics, representative decoded images are presented for JPEG image formats under the uncoded scenario. These samples are presented to provide a direct visual comparison between QFT- and QHWT-based transmission frameworks.

At SNR = −9 dB with eight-qubit encoding, both QFT (F8) and QHWT (H8) decoded images exhibit noticeable degradation compared to the original reference (Figure 6a), caused by the combined effects of quantization and channel noise. The degradation is more severe in the QHWT-based system, highlighting its weaker resilience under highly noisy conditions. As shown in Figure 6b, the QFT output demonstrates greater robustness, retaining clearer edges, sharper object boundaries, and more consistent luminance. In contrast, the QHWT result (Figure 6c) exhibits stronger distortions, including visible blurring and block-like artifacts, particularly in texture-rich regions. These results confirm that QFT provides superior resilience against noise, even under highly challenging channel conditions. At this SNR level, images encoded with other qubit sizes are not successfully decoded due to excessive bit errors.

At SNR = −6 dB, the decoded images exhibit noticeable improvement compared to the −9 dB case, as the lower noise level enables better recovery of image details. At this SNR level, images encoded with seven qubits are also successfully decoded. The QFT outputs (F8 and F7) (Figure 7a,c) exhibit sharper textures and more faithful structural preservation. At SNR = −6 dB, the QHWT-decoded image with eight-qubit encoding (H8) (Figure 7b) achieves a quality level comparable to the QFT counterpart. However, for seven-qubit encoding (H7) (Figure 7d), the QHWT-decoded image fails to preserve details as effectively, appearing less clear and more distorted compared to the QFT result. Furthermore, the eight-qubit encoding provides stronger noise resilience, achieving more stable image reconstruction than the seven-qubit case, though at the expense of a slight reduction in maximum achievable quality. These results demonstrate that QFT-based transmission consistently outperforms QHWT under identical qubit sizes, and that increasing the qubit encoding size enhances robustness against noise, even under low-SNR channel conditions.

4.6. Comparative Advantages of QFT in Noisy Quantum Image Transmission

The consistent advantage of the QFT over the QHWT is primarily attributed to the way each transform structures and distributes encoded information across the quantum state space. QFT produces a global frequency-domain representation in which all basis states contribute to the encoding of each data point. This uniform spreading of information ensures that noise or distortions in any single coefficient do not catastrophically affect localized features, but instead, the error is averaged across the system. This property is particularly beneficial in multi-qubit encodings, where the dimensionality of the Hilbert space grows exponentially and the preservation of global coherence becomes critical. By contrast, QHWT decomposes information hierarchically, with localized coefficients capturing specific scales or features. While this hierarchical structure can be efficient for certain compression tasks, it makes the representation more vulnerable to noise: if one coefficient or sub-band is corrupted, its impact propagates strongly to the reconstructed output. Consequently, QHWT tends to degrade more rapidly under high-noise conditions. The performance gap between the two transforms becomes most evident at higher qubit encodings (such as 7 or 8 qubits), where QFT exploits its global coherence properties to deliver up to 3 dB channel SNR gains over QHWT, as reflected in the reported results.

Importantly, it has been established that the QFT-based scheme consistently outperforms both quantum approaches in the time domain and classical transmission systems [38]. Here, by directly benchmarking QFT against QHWT, we further confirm that the QFT approach not only achieves higher PSNR, SSIM, and UQI values, but also produces reconstructions that are perceptually sharper and structurally more faithful to the original images. These findings underline the superior noise resilience and scalability of QFT for quantum image transmission, particularly under realistic channel impairments.

4.7. Complexity and Hardware Realizability

The computational complexity and hardware requirements of the QFT and QHWT differ significantly in both asymptotic scaling and practical implementation. QFT primarily consists of Hadamard and controlled-phase rotation gates. Its gate complexity scales as $O\left(n^2\right)$ for exact circuits, while approximate variants reduce this to $O(nlogn)$ by omitting rotations with negligible impact on fidelity. The circuit depth grows linearly due to sequential application of controlled rotations. QFT requires precise long-range interactions and high-fidelity gates, making it challenging to implement. Nevertheless, it is experimentally mature, with extensive compiler support and hardware demonstrations on superconducting and trapped-ion platforms. Approximate QFT circuits enable practical execution on near-term quantum processors.

In contrast, QHWT is built from hierarchical Hadamard and SWAP operations to perform local averaging and differencing. Its gate complexity scales linearly as $O\left(n\right)$, and the circuit depth is also linear but typically shallower than QFT due to high parallelism. The localized operations make QHWT well-suited for hardware with limited connectivity or shorter coherence times. However, multi-level Haar decompositions may require ancillary qubits and additional synchronization overhead, which can complicate large-scale implementations. Experimental realizations of QHWT remain limited and mostly at early stages.

In terms of hardware realizability, QFT is already practical on current hardware when using approximate circuits, whereas QHWT can be implemented efficiently for small- to medium-scale transforms. Over the medium term, improvements in coherence, gate fidelity, and compiler optimizations are expected to enable larger-scale implementations of both transforms. In summary, QHWT offers minimal gate overhead and reduced circuit depth, making it well-suited for hardware-constrained environments, whereas QFT provides higher spectral resolution, greater robustness against noise, and extensive experimental validation, making it the more effective choice for quantum image transmission on current and near-future devices. For high-resolution images, including 4K, the advantages of QFT remain significant as hardware capabilities mature. These comparisons are summarized in

Table 2. Comparison of QFT and QHWT in terms of computational complexity, circuit depth, and hardware realizability.

Aspect	QFT	QHWT
Primary Gates	Hadamard, Controlled-Phase	Hadamard, SWAP
Gate Complexity	$O\left(n^2\right)$ (exact), $O(nlogn)$ (approx.)	$O\left(n\right)$
Circuit Depth	Linear, deeper due to sequential rotations	Linear, shallower, supports parallelism
Connectivity	Long-range/global	Local/nearest-neighbor
Ancilla Requirement	None	May be needed for multilevel decomposition
Hardware Suitability	Trapped-ion, photonic, high-connectivity superconducting	Grid-based superconducting or solid-state processors
Experimental Maturity	High	Limited/early-stage demonstrations
Near-Term Realizability	Feasible with approximate circuits	Feasible for small-scale implementations
Medium-Term Realizability	Scalable with error mitigation and compiler support	Scalable with improved hardware and multilevel designs

.

4.8. Scalability

Scalability is a critical factor in assessing the practicality of QFT- and QHWT-based frameworks for quantum communication. While QHWT circuits offer linear growth in resources, their reliance on SWAP operations introduces routing overhead that may offset these gains on certain hardware platforms. In contrast, QFT circuits exhibit quadratic scaling but primarily rely on Hadamard and controlled-phase gates, which are native and low-error in many NISQ devices. As a result, QFT maintains stronger scalability in practice, particularly when high spectral resolution and fidelity are required under noisy transmission conditions. Thus, QHWT is advantageous for lightweight implementations, whereas QFT offers more robust scalability for large-scale, high-fidelity quantum image transmission.

4.9. Simulation-Based Evaluation: Strengths and Applications

In this study, simulations serve as the cornerstone for evaluating the proposed quantum image transmission frameworks, as large-scale fault-tolerant quantum hardware is not yet available. This approach allows for systematic exploration of the impact of channel noise, qubit encoding sizes, source formats, and error-protection schemes in a controlled and repeatable environment. Similarly to the early development of classical communication systems, where simulation-based investigations provided the foundation before practical hardware emerged, quantum communication research can leverage simulation to pave the way for scalable hardware realizations and real-world deployments. The use of simulations enables detailed stress-testing under extreme conditions that current experimental setups cannot yet support, offering insights into both the strengths and limitations of QFT- and QHWT-based methods.

From an application perspective, the proposed systems demonstrate strong potential in domains where data integrity and visual fidelity are paramount under bandwidth and noise constraints. These include high-quality multimedia transmission (e.g., quantum video streaming), remote sensing and satellite imaging, deep-space communication, and long-distance quantum networking, where environmental noise and channel impairments are unavoidable. By showcasing robustness through detailed simulations, this study not only validates the feasibility of transform-based quantum image transmission but also provides a roadmap for future hardware implementations tailored to these emerging application areas.

5. Conclusions and Future Work

5.1. Conclusions

In this study, a novel transform-based quantum image transmission framework is developed and evaluated using both QFT and QHWT. A systematic comparison is performed under uncoded and polar-coded scenarios with JPEG and HEIF source formats, and reconstruction quality is assessed using PSNR, SSIM, and UQI metrics. The results show that while the performance gap between QFT and QHWT is minimal for single-qubit encoding, the advantage of QFT becomes increasingly pronounced as the qubit encoding size grows. With eight-qubit encoding, QFT achieves up to a 3 dB channel SNR gain in image quality compared to QHWT, demonstrating its superior robustness for high-dimensional encodings. Incorporating polar-coded channel coding further enhances performance for both transforms, providing an additional 3–4 dB SNR gain across all qubit sizes. Increasing the qubit encoding size improves noise tolerance but reduces the maximum achievable image quality due to bandwidth limitations, highlighting a trade-off between fidelity and robustness. HEIF consistently outperforms JPEG due to its more efficient source compression, delivering higher baseline quality under both uncoded and coded transmission scenarios. Representative decoded images confirm that QFT better preserves perceptual quality under severe noise, whereas QHWT experiences faster degradation. Overall, this study provides practical insights into designing noise-resilient quantum communication frameworks and underscores the importance of transform selection, encoding size, and classical channel coding for high-quality image transmission over quantum channels. The findings indicate that QFT-based frameworks are particularly suitable for scenarios where robustness against channel impairments is critical.

5.2. Future Work

This study highlights several avenues for future research. A natural extension is toward video transmission, where the framework can be applied to compressed bitstreams generated by the versatile video coding (VVC) or high-efficiency video coding (HEVC) standards to enable scalable quantum video communication. Hardware-oriented validation also represents a key direction, with QFT and QHWT circuits mapped onto near-term NISQ devices to assess scalability, gate depth, error accumulation, and optimization strategies. Beyond single-transform approaches, hybrid designs, such as combining QFT and QHWT or employing adaptive transform selection guided by channel state information or image content, can further enhance robustness and efficiency. The integration of advanced error-resilient QEC methods beyond polar codes, particularly those structured for multimedia data, offers another promising research path. Finally, application-specific implementations in areas such as satellite imaging, quantum networking, and high-fidelity multimedia sharing would provide valuable testbeds where both bandwidth efficiency and resilience to channel impairments are critical. Collectively, these directions can advance the development of practical, scalable, and high-performance quantum communication systems for multimedia applications.