# Metric for Estimating Congruity between Quantum Images +
## 1 Introduction  
Quantum image processing (QIP), which is an emerging sub-discipline that seeks to extend traditional (i.e., classical or digital) image processing tasks to (or using resources from) the quantum computing realm [1], has attracted a lot of attention from practitioners that come from divergent backgrounds, notably those from the quantum computing, computer science, and engineering fields [2]. The first published material relating quantum computers to image processing can be traced to the work [3] by [PERSON] in 1996, wherein the use of \"analogue\" quantum computing hardware to describe image analysis was mooted. This was followed by efforts to design quantum algorithms based on unstructured picture (data set) in [4] and [5]. Optics-based descriptions for quantum imaging then became very prominent [6], while the idea of a more general quantum signal processing framework was presented in [7]. However, it is the exploratory work by [PERSON] and [PERSON] [8], in 2003, that is oftencredited as the bellwether of what has transmuted into today's description of QIP, albeit with a slightly different acronym--quip. This was closely followed by [PERSON] and [PERSON]'s Qubit Lattice description for quantum images [9]. Years later, [PERSON] [10] proposed the Real Ket representation to encode quantum images. The third and final representation, completing the trio of representations that are collectively regarded as the pioneers of today's version of the QIP sub-discipline, is [PERSON] et al. 's flexible representation for quantum images (FRQI) that was proposed in 2010 [11] and later revised in 2011 [12]. Since then, the impressive development of the QIP sub-area is widely attributed to the FRQI representation because most of the QIP literature are either focused on its use in applications such as quantum watermarking [13], quantum movie [14], image data base search [15], etc. ; or the extension and modification of the FRQI representation to encode different types of images [16]. Overall, the objective of QIP is to utilise quantum computing technologies to capture, manipulate and recover quantum images in different formats and for different purposes. Detailed reviews on QIP can be found in [17; 18; 19]. Most of the papers cited above are, however, focused on covering the latter two objectives of QIP, i.e., capturing and manipulating the images, while the former (i.e., encoding an image) is scantily covered [20]. Even so, to the best of our knowledge, only [16] provided readers with insights into likely technologies to visualise transformed images resulting from operations on quantum images [17]. Meanwhile, interest in image and video security applications such as watermarking [13; 21], encryption and decryption [22] has continued to increase. To the best of our knowledge, all the aforementioned applications of QIP utilise the digital peak-noise-to-signal-ratio (PSNR) image quality measure to benchmark and validate their approaches. This is mainly attributed to the absence of a quantum-based metric equivalent to the PSNR. The closest attempt to quantify the extent that two or more images are alike is [PERSON] et al. 's probability-based image similarity score technique in [15; 22]. Most of the researchers are content with adopting the classical PSNR image quality measure to assess likeness between two or more quantum images. The quantum mechanical nature of the quantum information carrier--the qubit--however, specifies a, sometimes confounding, procedure for processing quantum information, considering which, we argue, in this study, that these available classical metrics are insufficient and/or ill-suited to effectively quantify the fidelity between two or more quantum images. Therefore, the absence of an effective metric for evaluating congruity between quantum images had remained an unexplored research area within the QIP sub-discipline until now. In recent work [1], [PERSON] et al. proposed a wholly quantum-based image fidelity metric (or simply QIFM) to assess \"likeness\" between quantum images.