# Quantum Compressive Sensing: Mathematical Machinery, Quantum Algorithms, and Quantum Circuitry
## 6 Demonstration
### Description of Model  
We consider a qualitative model of forest foliage that may be measured by overhead LIDAR imaging missions, such as the Concurrent Artificially-intelligent Spectrometry and Adaptive Lidar System (CASALS) satellite currently under development at NASA. Our model consists of two categories: a relatively young forest (Figure 7a), in which the canopy is still rising, and a mature forest (Figure 7b), in which the canopy has attained its maximum height and underbrush is measurably dense. We emphasize that this model is purely qualitative, and contrived solely for the sake of demonstrating our Quantum Compressive Sensing protocol on a tractably-sized system. In particular, the resolution attainable in real-world LIDAR data is multiple orders of magnitude greater than in the model presented here. The model is sampled by randomly selecting the class (young or mature), then perturbing each pixel in the prototype images of Figure 7 with Gaussian noise representing variation in the distribution of foliage. We construct three separate training sets sampling sixteen images each, shown in Figure 8.  
Figure 7: Prototypical images in a qualitative model for young (**a**) and mature (**b**) foliage. Training and validation signals are generated as random perturbations on one of these two images. Using the \"quantum average\" training protocol discussed in Section 4, each training set is mapped deterministically onto a pure quantum state \(\ket{\Psi}\), depending only on the parameter \(p\) used for the qubit mapping (Equation (2)). We consider an \"optimal\" \(p\) for each training set which _maximizes_ the information entropy \(S\) of the resulting probability distribution \(P(z)\). Figure 9 shows the dependence of \(S\) on the parameter \(p\) for each training set. We select \(p=0.5\) as a \"default\" choice and plot the probabilities \(P(z)\equiv\abs{\langle z|\Psi\rangle}^{2}\) of measuring each basis vector \(\ket{z}\) for each training set in Figure 10a, while Figure 10b plots the new probability distributions when the \(\ket{\Psi}\) are formed using the optimal \(p\). The gray bars mark the probabilities of the \"ideal\" training set constructed precisely from the prototype images in Figure 7. In all three cases, the \"quantum average\" successfully produces results which are qualitatively similar to one another and to the ideal, although we can see in Figure 9 that averaging over random perturbations from the ideal has the effect of smoothing out potentially interesting features in the overall signal space. Figure 8: Three training sets generated from the prototype signals shown in Figure 7.  
Figure 9: Information entropy of the training set as a function of the “midpoint” \(p\) used in Equation (2). The ideal curve considers a set of images which are split evenly between the two prototype signals. Throughout this section, we will assess the accuracy of our Quantum Compressive Sensing protocol using a measure we refer to as the \"Relative Log Likelihood\" (RLL), derived in Appendix A. The RLL is a function of both the probability \(P(z)\) that a binary image \(z\) is generated by a given signal \(y\), and the information entropy \(S\) intrinsic to that signal. To help visualize it, we plot the RLL of all possible \(z\) with respect to the signals \(y\) corresponding to the prototype images in Figure 7, using both the default \(p\) (Figure 10(a)) and the optimal \(p\) (Figure 10(b)). In general, a higher RLL indicates greater success, but we consider any RLL relatively close to zero (even negative values) to be \"good\". For example, the binary image corresponding to \(z=3\) is \(|000011\rangle\), evidently corresponding to a tall, thick canopy (reading the left-most bit as closest to the ground), consistent with a mature forest. Its RLL value (using optimal \(p\)) is 0.51. The binary image \(|000010\rangle\) corresponding to \(z=2\) is also consistent with a mature forest and has an RLL of \(-1.24\).