# Metric for Estimating Congruity between Quantum Images +
## 4 Experimental Validation of the QIFM Metric
### QIFM (\(F\)) versus [PERSON]'s Similarity Score (\(S\))
However, notwithstanding the aforementioned calibration to [PERSON]'s original similarity score, the revised similarity score (i.e., the _SIM_ values) attributes a similarity of 0.52 for the Lena-Inverted Lena reference and test image pair (i.e., pair A-B), which is obviously a poor correlation with explicit lexical descriptions of congruity (or likeness) between the two images). In contrast, to manifest itself as a better measure of congruity, the proposed QIFM metric attributes 62% fidelity between the pair, which is a better explicit lexical description of likeness between that pair. Similar shortcomings of both the original formulation of [PERSON]'s similarity score (\(S\)) and its recalibrated wider range, normalised version--the _SIM_ score--are clearly seen in the Couple-Trees pair (i.e., pair J-P), which has high similarity scores of \(S_{J,P}=0.9\) and \(SIM_{J,P}=0.7\). These values seem illogical since a mere visual inspection reveals that the two images are clearly dissimilar. In contrast, a fidelity value of \(F_{J,P}=52\%\) that is obtained using the QIFM metric, which clearly infers less congruity (likeness) between the two images. Additionally, a 52% fidelity clearly resembles the expectations of the explicit lexical description of the congruity (or likeness) between these two images (i.e., the Couples and Trees images in Figure 8). As noted earlier in Section 2, the most pertinent shortcoming of [PERSON]'s similarity score (\(S\)) is that it has high resource demands because its recovery (measurement) procedure is built entirely on the use of destructive measurements, which are carried out in each instance that the similarity between two images is assessed. To clearly illustrate this high cost, let us assume that the measurements to quantify the similarity of the images in our strip are carried out in the order indicated by the serial numbers in Table 1 (i.e., S/No. in the first column)--starting with the first pair (i.e., pair A-A in S/No. 1) and terminating with the Bridge-Aeroplane pair (i.e., pair O-K in S/No. 23)--it is instructive to note that the entire content of the strip will be lost upon the first measurement, i.e., after recovering the similarity between the first pair (i.e., \(S_{A,A}\) in S/No. 1). In other words, to execute the experiments narrated in this section (and quantify the similarity scores for all the 23 pairings of the reference and test images in Table 1) we need to prepare 23 versions of the same strip, i.e., one to recover the similarity of each reference and test image pair. In cost terms, to execute [PERSON]'s image similarity score a total of \(C_{s}=c^{n}\times 2^{m}H\) resources will be required. Here, \(m\) is the number of qubits required to encode the images that make up the strip; \(c\) is the number of pairs whose similarity is being measured and \(H\) accounts for the cost of resources needed to prepare and initialise the \(2^{m}\)-images long the strip each of \(2^{n}\times 2^{n}\) size and executing the same number of measurements, i.e., \(2^{m}\) measurement operations to obtain the similarity between the reference and test images in the pair. To illustrate this cost in terms of our experiments, quantifying [PERSON] et al. 's probability-based similarity score (\(S\)) for the 16-image long strip requires a total cost of \(C_{s}=23^{8}\times 2^{3}H\). Here, we should further clarify that \(H\) represents the cost associated with the peculiar needs of the technology (i.e., whether ion-trap, photonic, etc. hardware that is) used to prepare and initialise the 16-image long strip [1]. In contrast to this, the proposed QIFM fidelity 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 \(2^{s}\times 2^{s}\) ROI in a strip consisting of \(2^{m}\) FRQI images, each of size \(N=2^{n}\times 2^{n}\) (as discussed earlier in this section).