# Quantum Compressive Sensing: Mathematical Machinery, Quantum Algorithms, and Quantum Circuitry
## 6 Demonstration
### Quantum Compressive Sensing  
We now present the results of our Quantum Compressive Sensing protocol, applied to our qualitative model. We begin each trial by preparing the Born machine \(|\Psi\rangle\), using one of the training sets constructed above, or the ideal based on the prototype images in Figure 7. We then generate a random signal \(y\) from the forest model, and a random \(m\times n\) sensing matrix \(A\). We calculate \(x=Ay\), and then implement one of the protocols presented in Section 5. The protocol ends by measuring all qubits, yielding a binary image \(z\). Each trial is repeated 1024 times for the same signal \(y\) and sensing matrix \(A\) (but generally different \(z\)) to obtain the median RLL relating \(z\) and the original signal \(y\). Finally, this process is repeated for 32 different \(y\) and \(A\) to obtain a distribution of median RLL (we use the same \(y\) and \(A\) for all experiments). Throughout this section, we plot the inter-quartile ranges of these distributions as a function of the number of measurements \(m\). If our Quantum Compressive Sensing is successful, we should observe an increase in median RLL as \(m\) increases. Figure 11(a) shows our results when directly applying the Gaussian operator in Equation (12) with \(\sigma=0.5\) to each of the training sets, constructed using the default \(p\). Each sensing matrix \(A\) is a binary matrix with equal probability between 0 and 1, consistent with the micro-mirror LIDAR design presented in [PERSON] et al. (2019). We remind the reader that we are applying the Gaussian operator for this demonstration, but this can in principle be done in a quantum way with Quantum Imaginary Time Evolution. Hereafter, we label this projection protocol as QITE. This protocol will clearly be successful: At \(m=0\), we are essentially using our Born machine for generative sampling of our model, generating a binary image \(z\) which is consistent with either a young or mature forest, but with no preference based on the original signal \(y\). Those \(z\) which happen to match the class (that is, young or mature) of \(y\) are likely to have acceptable RLL, but we are just as likely to generate \(z\) with very poor RLL. As \(m\) increases, however, the projection step \"teaches\" the Born machine more and more about the original signal \(y\), enabling us to properly distinguish between the two classes and even select an optimal image among them. Figure 12: Median relative log likelihood as a function of measurements, after applying the QITE protocol over 1024 trials. (**a**) contrasts performance for each training set, while (**b**) contrasts different choices of the “midpoint” \(p\). At \(m=0\) in (**a**), our Born machine is being utilized for generative sampling of our model. As \(m\) increases, the projection step “teaches” the Born machine about the original signal \(y\), allowing for a distinction between the two classes and optimal image selection. In (**b**), we see that the default choice \(p=0.5\) produces better results and we thus utilize \(p=0.5\) for all our experiments. In Figure 12b, we investigate how our choice of \(p\) in constructing the trained Born machine \(|\Psi\rangle\) influences results. We focus on just the ideal training set and continue to use the QITE protocol with \(\sigma=0.5\). In our simulations, the default choice \(p=0.5\) turns out to produce better results, at least for this choice of experimental parameters. Therefore, for the remainder of this section, we set \(p=0.5\) for all experiments. This opens the question whether an alternative choice of \(p\) may improve results further. For example, while we justify the choice of a single \(p\) for the entire image by thinking of it as a normalization on the measuring apparatus, we might equally well optimize \(p\) for each individual pixel. We leave this line of inquiry as a direction for future research. The QITE projection protocol is unique in that it is deterministic, in the sense that the state of the Born machine \(|\Psi\rangle\) is fixed prior to sampling.