Nonlinear equation solvers in SPICE simulators
How spice managed to solve such a huge system of nonlinear equations quickly and without running out of memory ? 


Google sparse matrix solver. 

A good SPICE very likely defaults (or knows when to switch to) a sparse matrix solver, since large circuits typically produce sparse matrices (each node connects to only a small fraction of the branches) it would be an obvious optimization to use (or have available) a sparse matrix solver in a SPICE. Even Nagel's original 1975 report (Thesis?) on SPICE discusses using sparse matrix methods.

Matlab certainly has a sparse matrix solver available, but you probably have to invoke it explicitly.

Qucs may not have this capability, or it might not be implemented particularly well, because it's a relatively raw open source project and its developers might not have got to the point of testing it on anything bigger than a toy problem.

(Hat tip to @jonk for the link to the Nagel report)


  Is this example fair enough ? i mean do we need to consider more practical circuits ? 


I would think you want to demonstrate your solver on a wide variety of different types of circuits. You probably want to consider circuits that are known to produce ill-conditioned matrices. Positive feedback circuits also often produce difficulties for non-linear solvers.


  and if so can any one give an example for a circuit(s) where DC operating point simulation might be the bottleneck of the simulation time?


I would expec this to be common in any ill-conditioned circuit when setting up an AC simulation.


  Is the DC operating point the simulation type we should be targeting ? i mean should we focus to interface our solver to other types of simulations e.g. the transient analysis ?


The other main simulation types (ac and transient) only require linear solvers. The AC simulation explicitly is about small variations about the operating point, so that the circuit can can be considered linear by perturbation theory. The transient solver linearizes the circuit at each time step, but re-calculates the local linear equivalent circuit for each time step. So if you're trying to demonstrate a non-linear solver, the DC solver is the one to demonstrate.