# Quantum Compressive Sensing: Mathematical Machinery, Quantum Algorithms, and Quantum Circuitry

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## Abstract

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## 1. Introduction

## 2. Quantum Compressive Sensing

- Training
- Measurement
- Projection
- Sampling

## 3. Pixel–Qubit Mapping

## 4. Training

## 5. Projection

#### 5.1. Decomposition

#### 5.2. Rodeo Algorithm

#### 5.3. Quantum Imaginary Time Evolution

#### 5.4. Entangled Born Machine

## 6. Demonstration

#### 6.1. Description of Model

#### 6.2. Quantum Compressive Sensing

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MPS | Matrix Product State |

TNCS | Tensor Network Compressed Sensing |

LIDAR | Light Detection and Ranging |

CASALS | Concurrent Artificially-Intelligent Spectrometry |

NLL | Negative Log Likelihood |

SVD | Singular-Value Decomposition |

QITE | Quantum Imaginary Time Evolution |

NISQ | Noisy Intermediate-Scale Quantum |

RLL | Relative Log Likelihood |

## Appendix A. Relative Log Likelihood

## References

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**Figure 1.**Visualization of Equation (3) as a function of x when the “midpoint” $p=0.1,0.3,0.5,0.7,0.9$.

**Figure 2.**A simple quantum circuit to map an n-dimensional vector y onto n qubits. The quantum gate ${R}_{y}\left(\theta \right)$ is defined as $exp(-i\frac{\theta}{2}\widehat{Y})$, where $\widehat{Y}$ is the Pauli Y spin operator.

**Figure 3.**A quantum circuit to prepare the “quantum average” of the signals ${y}_{i}\equiv {U}_{i}|0\rangle $. The circuits ${U}_{i}$ are the state-preparation circuits given in Figure 2. The quantum gate H is the Hadamard gate. Dots on a wire represent controlled operations: a closed dot has a control value of |1〉 while an open dot has a control value of |0〉. Meters at the end of a circuit represent measurement.

**Figure 4.**General workflow diagram of the proposed protocol where it can be seen that the Pixel–Qubit mapping (e.g., Figure 2) occurs before undertaking training (e.g., Figure 3). The projection (e.g., Figure 5 and Figure 6) can then utilize the training and measurements which ultimately leads to the sampling.

**Figure 5.**A quantum circuit to perform Quantum Compressive Sensing with the Decomposition protocol. The input state $|\mathsf{\Psi}\rangle $ is a trained quantum state such as the one prepared by the circuit in Figure 3. Each quantum gate ${\theta}_{i}$ represents a Givens rotation, as implemented in the

`cirq`Python package. Meters represent measurement, both within and at the end of the circuit.

**Figure 6.**A quantum circuit to perform Quantum Compressive Sensing with the Rodeo algorithm. The input state $|\mathsf{\Psi}\rangle $ is a trained quantum state such as the one prepared by the circuit in Figure 3. The quantum gate H is the Hadamard gate. The quantum gate ${R}_{z}\left(\theta \right)$ is defined as $exp(-i\frac{\theta}{2}\widehat{Z})$, where $\widehat{Z}$ is the Pauli Z operator. The quantum gate $P\left(\theta \right)$ is the phase gate, identical to ${R}_{z}$ except by a global phase such that the |0〉 component is unchanged. Meters at the end of the circuit represent measurement.

**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.

**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.

**Figure 10.**Square amplitude of each basis vector for an ideal state (bars) and each trained state (squares). (

**a**) uses a “midpoint” value of $p=0.5$ while (

**b**) optimizes p for each state to maximize the information entropy, as visualized in Figure 9.

**Figure 11.**Relative log likelihood of each basis vector for an ideal state. (

**a**) uses a “midpoint” value of $p=0.5$ while (

**b**) optimizes p for each state to maximize the information entropy, as visualized in Figure 9.

**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.

**Figure 13.**A comparison of the Rodeo protocol with different choices of $\sigma $. (

**a**) shows the number of syndrome failures out of 1024 attempts, and (

**b**) shows the median relative log likelihood for successful attempts. (

**a**) also includes failures for the Decomposition protocol, for comparison. We clearly observe in (

**a**) that the chance of failure increases with the size of $\sigma $ used in the Rodeo protocol and can see in (

**b**) that we can expect an optimal choice of $\sigma $ which maximizes the increase in median RLL.

**Figure 14.**Comparison of each Quantum Compressive Sensing protocol, for different classes of sensing matrix. (

**a**)–(

**c**) utilize a binary A with ${A}_{ij}$ being 0 or 1 with equal probability, a binary A with ${A}_{ij}$ being 1 with 20% probability, and a binary A with each row of A having only a single 1 respectively. (

**d**) utilizes a uniform A where only m columns of A are nonzero. QITE uses $\sigma =\frac{1}{2}$ and Rodeo uses $\sigma =\pi $. In (

**a**,

**b**) we see (as anticipated) that the Decomposition protocol lowers the accuracy of reconstruction when measurements are introduced, since these classes of sensing matrix do not satisfy the condition that A is non-zero on only m columns. However, (

**b**) and especially (

**c**) demonstrate that sparse matrices can approximate this condition, increasing accuracy of the Decomposition protocol for larger m. In (

**d**), the performance of QITE and Rodeo fall off due to the restrictions of the sensing matrix, but the Decomposition performance is impressive.

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**MDPI and ACS Style**

Sherbert, K.M.; Naimipour, N.; Safavi, H.; Shaw, H.C.; Soltanalian, M.
Quantum Compressive Sensing: Mathematical Machinery, Quantum Algorithms, and Quantum Circuitry. *Appl. Sci.* **2022**, *12*, 7525.
https://doi.org/10.3390/app12157525

**AMA Style**

Sherbert KM, Naimipour N, Safavi H, Shaw HC, Soltanalian M.
Quantum Compressive Sensing: Mathematical Machinery, Quantum Algorithms, and Quantum Circuitry. *Applied Sciences*. 2022; 12(15):7525.
https://doi.org/10.3390/app12157525

**Chicago/Turabian Style**

Sherbert, Kyle M., Naveed Naimipour, Haleh Safavi, Harry C. Shaw, and Mojtaba Soltanalian.
2022. "Quantum Compressive Sensing: Mathematical Machinery, Quantum Algorithms, and Quantum Circuitry" *Applied Sciences* 12, no. 15: 7525.
https://doi.org/10.3390/app12157525