Table 1.

Comparison of depth/time complexity of proposed generalized quantum convolution technique against related work.

a Depth complexity of quantum data encoding (I/O) techniques
FRQI [14]	NEQR [15]	C2Q [17]
${\mathcal{O}}_{\mathrm{I}/\mathrm{O}}\left(4^n\right)$	${\mathcal{O}}_{\mathrm{I}/\mathrm{O}}\left(qn{2}^n\right)$	${\mathcal{O}}_{\mathrm{I}/\mathrm{O}}\left(2^n\right)$
b Depth complexity of quantum convolution algorithms for a fixed filter
Proposed	Related Work [4,5,6,7,8]
${\mathcal{O}}_{\mathrm{alg}}\left(n_{\max }^2\right)$	${\mathcal{O}}_{\mathrm{alg}}\left(n^2\right)$
c Complexity of proposed technique compared to classical convolution
Proposed	Direct (CPU) [10]	FFT (CPU/GPU) [10,11]	GEMM (GPU) [12,13]
$\begin{array}{c}\hfill {\mathcal{O}}_{\mathrm{alg}}\left(\begin{array}{c}\hfill n_{f_{\max }}{n}_{\max }^2-n_{f_{\max }}^2{n}_{\max }\\ \hfill +2^{n_f}\end{array}\right)\\ \hfill {\mathcal{O}}_{\mathrm{alg}}+\mathrm{I}/\mathrm{O}\left(\begin{array}{c}\hfill n_{f_{\max }}{n}_{\max }^2-n_{f_{\max }}^2{n}_{\max }\\ \hfill +{\mathbf{2}}^{\mathit{n}}+2^{n_f}\end{array}\right)\end{array}$	$\begin{array}{c}{\mathcal{O}}_{\mathrm{alg}}\left(N^2\right)\equiv \hfill \\ {\mathcal{O}}_{\mathrm{alg}}\left(4^n\right)\hfill \end{array}$	$\begin{array}{c}{\mathcal{O}}_{\mathrm{alg}}\left(NlogN\right)\equiv \hfill \\ {\mathcal{O}}_{\mathrm{alg}}\left(n{2}^n\right)\hfill \end{array}$	$\begin{array}{c}{\mathcal{O}}_{\mathrm{alg}}\left(N_{F}N\right)\equiv \hfill \\ {\mathcal{O}}_{\mathrm{alg}}\left(2^{(n+n_f)}\right)\hfill \end{array}$