# Quantum Compressive Sensing: Mathematical Machinery, Quantum Algorithms, and Quantum Circuitry
## 7 Conclusions  
In this paper, we have presented a protocol for performing \"quantum\" compressive sensing, and suggested several ways in which each step of the protocol may be implemented on a quantum computer. The protocol begins by pre-preparing a set of qubits into a quantum state \(|\Psi\rangle\) which captures the statistics of all potential signals. The quantum state \(|\Psi\rangle\) is then evolved with a series of quantum operations which cause \(|\Psi\rangle\) to collapse into a lower-dimensional sub-space consistent with the compressed measurement made on the signal. Finally, the qubits are measured, producing a probabilistic estimate of the original image. Our formalism generalizes to all linear sensing matrices, and we have presented a preliminary demonstration of our method on a very small, classically-simulable model. The algorithms discussed in this paper all depend on a one-qubit-per-pixel approach, relegating the application of these results to meaningfully-sized images to far in the future, when quantum computers with hundreds or thousands of qubits are available. We have also not contended with the formidable technical challenge of mitigating noise inherent to all quantum computing systems available today. Developing alternative implementations with a more compact pixel-qubit mapping would be an excellent direction for further research. For example, in the case where the signal basis is indeed sparse, as in the classical compressive sensing paradigm, one may assign \(\log(n)\) qubits to each of \(k\) \"on\" bits, indexing at which pixel its particular \"on\" bit is found (for a total of \(k\log n\) qubits). The quantum chemistry literature, from which we have borrowed the majority of our techniques, names this the \"first-quantization\" approach, and developing efficient algorithms for navigating this qubit mapping is an area of active research. In the long term, when large, error-corrected quantum computers are readily available, the compressive sensing methods in this paper are expected to be useful for processing high-dimensional, high-resolution signals. Thus, they have wide-ranging applications to earth science, astrophysics, communications, and beyond. Moreover, the classical constraint requiring a basis in which the signal is known to be sparse is no longer necessary. Instead, we use a data-driven approach, using a representative sample of signals in the basis most convenient for representing the sensing matrix. This promises to be a very powerful perspective in the quantum implementation of the compressive sensing literature. Conceptualization, K.M.S., N.N., H.S., H.C.S. and M.S. ; methodology, K.M.S. and N.N. ; software, K.M.S. and N.N. ; validation, K.M.S., N.N., H.S., H.C.S. and M.S. ; formal analysis, K.M.S., N.N. and H.S. ; investigation, K.M.S. and N.N. ; resources, H.S., H.C.S. and M.S. ; data curation, K.M.S. ; writing--original draft preparation, K.M.S. ; writing--review and editing, N.N., H.S., H.C.S. and M.S. ; visualization, K.M.S. and N.N; supervision, H.S., H.C.S. and M.S. ; project administration, H.S., H.C.S. and M.S. ; funding acquisition, H.S., H.C.S. and M.S. All authors have read and agreed to the published version of the manuscript. Funding for this research was provided by the NASA Goddard Space Flight Center Advanced Communications Capabilities for Exploration and Science Systems (ACCESS) Project Office. This research was also partially funded by the National Reconnaissance Office and the Universities Space Research Association. **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** Not applicable. **Acknowledgments:** We thank [PERSON], [PERSON], and [PERSON] for useful discussions. **Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.