# Metric for Estimating Congruity between Quantum Images +
## 5 Concluding Remarks  
While remarkable progress has been made in the QIP sub-discipline over the last few years, it has still continued to rely on classical techniques, notations, and formulations, notably the PSNR  
Figure 9: Comparison between PSNR and QIFM for watermarked images. (**A**) PSAU watermark logo; (**B**) original Lena image; (**C**) original Blonde Lady image; (**D**) watermarked version of Lena image; (**E**) watermarked version of Blonde Lady image. image quality measure, to evaluate fidelity between quantum images. Incidentally, recent studies have proven that the PSNR metric as an unreliable image quality metric even in some digital applications. Contending that such a metric does not conform with the intricacies inherent to quantum computing and QIP technologies, in this study, we have appraised the need for, and formulated a wholly quantum-based metric, the quantum image fidelity metric (QIFM), which could be used to efficiently assess the congruity between two or more quantum images. The proposed QIFM metric is designed as a pixel difference-based image quality measure that can be effectively implemented using available QIP representations for quantum images; while, only basic quantum computing resources and some useful data collated from classical versions of the reference and test image pairs are required to manipulate and assess the fidelity between the images. We simulated the implementation of our image quality metric (using classical computing resources) on a dataset of popularly used real images and our results suggest that in comparison with the traditional PSNR image quality measure the QIFM metric is a better tool to convey the notion of congruity. Since it is conceived for use on the quantum computing framework, the QIFM metric is equipped to withstand the intricacies of such hardware, whilst the digital PSNR metric is ill-suited for such hardware implementation. Similarly, in comparison with the only known quantum image similarity technique (i.e., [PERSON] et al. 's probability-based score) [15; 23], the ancilla-driven attribute of our proposed QIFM makes it a more efficient and prudent image quality measurement tool. Additionally, by using a statistical analysis we established that the proposed QIFM metric had a better correlation with digital image quality assessments of congruity than the other quantum image quality measures. Besides being a wholly quantum-based measure, the proposed QIFM image quality metric requires only \(362n\) basic quantum gates and \(26m\) Toffoli gates to implement. Additionally, the entire framework requires \(m+1+2^{n}\) qubits and \(m\times 2^{2(n-s)+4}\) ancillary measurement qubits in order to evaluate the fidelity of any predetermined \(2^{s}\times 2^{s}\) ROIs in a strip consisting of \(2^{m}\) FRQI images, each of size \(N=2^{n}\times 2^{n}\). The formulation of the QIFM metric, in itself, is the first step in ensuring that applications sensitive to the peculiarities of quantum computation are formulated for effective quantum image processing (QIP). In on-going work, we are assessing how to further reduce the overall cost of the proposed metric both in terms of its encoding (based on photonic quantum computing hardware implementation) and in terms of its overall execution. In later work, we will utilise the resulting improved version of the QIFM metric to better assess the fidelity between different pairs of quantum images, such as those realised when a quantum image is watermarked, make-up frames required for quantum movie aggregation, etc. [15; 22; 23]. The present formulation of the QIFM metric and its improved version, when realised, have the potential to improve the accuracy quantum image quality assessment and QIP applications in general. The authors are grateful to [PERSON] for his effort in typing the initial draft of this paper and [PERSON] for his contributions with some aspects of the experiments reported in the study.