  A large obstacle for realizing quantum photonic logic is the weak optical
nonlinearity of available materials, which results in large power consumption.
In this paper, we present the theoretical design of all-optical logic with
second order ($\chi^{(2)}$) nonlinear bimodal cavities and their networks.
Using semiclassical models derived from the Wigner quasi-probability
distribution function, we analyze the power consumption and signal-to-noise
ratio (SNR) of networks implementing an optical AND gate and an optical latch.
Comparison between the second and third order $(\chi^{(3)})$ optical logic
reveals that while the $\chi^{(3)}$ design outperforms the $\chi^{(2)}$ design
in terms of the SNR for the same input power, employing the $\chi^{(3)}$
nonlinearity necessitates the use of cavities with ultra high quality factors
($Q\sim 10^6$) to achieve gate power consumption comparable to that of the
$\chi^{(2)}$ design at significantly smaller quality factors ($Q \sim 10^4$).
Using realistic estimates of the $\chi^{(2)}$ and $\chi^{(3)}$ nonlinear
susceptibilities of available materials, we show that at achievable quality
factors ($Q \sim 10^4$), the $\chi^{(2)}$ design is an order of magnitude more
energy efficient than the corresponding $\chi^{(3)}$ design.
