# Optimizing Multidimensional Pooling for Variational Quantum Algorithms
## 6 Conclusions  
In this work, we proposed efficient quantum average and Euclidean pooling methods for multidimensional data that can be used in quantum machine learning (QML). Compared to existing classical and quantum techniques of pooling, our proposed techniques are highly generalizable for any dimensionality of data or levels of decomposition. Moreover, compared to the existing classical pooling techniques on GPUs, our proposed techniques can achieve significant speedup--from \(\mathcal{O}(N)\) to \(\mathcal{O}(1)\) for a data size of \(N\) values. We experimentally validated the correctness of our proposed quantum pooling techniques against the corresponding classical pooling techniques on 1-D audio data, 2-D image data, and 3-D hyperspectral data in a noise-free quantum simulator. We also presented results illustrating the effect on fidelity due to statistical and measurement errors using noisy quantum simulation. In future work, we will explore applications of the proposed pooling layers in QML algorithms.  
Conceptualization: M.J., V.J., D.K. and E.E.-A.; Methodology: M.J., V.J., D.K. and E.E.-A.; Software: M.J., D.L., D.K. and E.E.-A.; Validation: M.J., V.J., D.L., A.N., D.K. and E.E.-A.; Formal analysis: M.J., V.J., D.L., D.K. and E.E.-A.; Investigation: [PERSON], A.N., V.J., D.L., D.K., M.C., I.I., E.B., E.V., A.F., M.S., A.A. and E.E.-A.; Resources: [PERSON], A.N., V.J., D.L., D.K., M.C., I.I., E.B., E.V., A.F., M.S., A.A. and E.E.-A.; Data curation: M.J., A.N., V.J., D.L., D.K., M.C., I.I., E.B., E.V., A.F., M.S., A.A. and E.E.-A.; Writing--original draft preparation: [PERSON], A.N., V.J., D.L., D.K., M.C., I.I. and E.E.-A.; Writing--review and editing: [PERSON], A.N., V.J., D.L., D.K., M.C., I.I., E.B., E.V., A.F., M.S., A.A. and E.E.-A.; Visualization: [PERSON], A.N., V.J., D.L., D.K., M.C., I.I., E.B., E.V., A.F., M.S., A.A. and E.E.-A.; Supervision: E.E.-A.; Project administration: E.E.-A.; Funding acquisition: E.E.-A. All authors have read and agreed to the published version of the manuscript.  
This research used resources from the Oak Ridge Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC05-00 OR22725.  
The Jayhawk images used in this work are available on request from Brand Center, University of Kansas, at [[https://brand.ku.edu/](https://brand.ku.edu/)]([https://brand.ku.edu/](https://brand.ku.edu/)) (last accessed on 12 December 2023) [33]. The audio samples used in this work are publicly available from the European Broadcasting Union at [[https://tech.ebu.ch/publications/sqamcd](https://tech.ebu.ch/publications/sqamcd)]([https://tech.ebu.ch/publications/sqamcd](https://tech.ebu.ch/publications/sqamcd)) (accessed on 19 October 2023) as file 64.flac [32]. The hyperspectral data used in this work are publicly available from the Grupo de Inteligencia Computacional (GIC) at [[https://www.ehu.eus/ccwintco/index.php/Hyperspectral_Remote_Sensing_Scenes#Kennedy_Space_Center_](https://www.ehu.eus/ccwintco/index.php/Hyperspectral_Remote_Sensing_Scenes#Kennedy_Space_Center_)]([https://www.ehu.eus/ccwintco/index.php/Hyperspectral_Remote_Sensing_Scenes#Kennedy_Space_Center_](https://www.ehu.eus/ccwintco/index.php/Hyperspectral_Remote_Sensing_Scenes#Kennedy_Space_Center_))(KSC) (accessed on 19 October 2023) under the heading Kennedy Space Center (KSC) [34].  
The authors declare no conflicts of interest.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Classical** & **PQC-Based** & **Measurement-** \\  & [21; 22; 23] & [10] & **Based**[29] & **Proposed** \\ \hline Pooling Method & Arbitrary & N/A & Euclidean & Average, Euclidean \\ \hline Time & \(\mathcal{O}(2^{n})\) & \(\mathcal{O}(1)\) & \(\mathcal{O}(1)\) & \(\mathcal{O}(1)\) \\ Complexity & 2-D, 3-D & N/A & 2-D & Arbitrary \\ \hline Window Size & Arbitrary & N/A & \((2\times 2)\) & Arbitrary \\ \hline \hline \end{tabular}
\end{table}
Table 5: Comparison of Related Work to Proposed Methods.