There are many different computational methods used in matter modelling. For example: solving multi-electron Schroedinger equation, DFT with different approximations, Quantum Monte Carlo, etc.
How does each method scale with system size (and if possible, can you give a brief reason why)?
From my experience with Stochastic Series Expansion (SSE) QMC (a type of discrete-time QMC) the computational cost scales like $\beta L^d$.
In practice, it's often important to account for the finite-size gap $\Delta \propto 1/L$, so to stay consistently above or below that finite-size gap, $\beta$ is typically set to scale as $\beta = cL$, where $c$ is some constant, so the full scaling to see a desired effect is usually $L^{d+1}$.
I don't have experience with other forms of QMC, like continuous time QMC, but I believe they are similar.
Conventional implementations of Kohn-Sham DFT scale cubically with system size. This is principally because at some point they:
The number of basis states in the basis, $M$, also scales with simulation size. Precisely how it scales with simulation size depends on the nature of the basis set; for example, in plane-wave bases the number of states is proportional to the simulation volume, whereas in local basis sets it is proportional to the number of electrons.
The problem can be recast in terms of the density matrix, rather than the trial states directly, and for systems with a band-gap the off-diagonal terms in the real-space density-matrix $D({\bf r},{\bf r^\prime})$ decay exponentially with distance $\vert{\bf r}-{\bf r^\prime}\vert$. This allows the density matrix to be truncated safely beyond some chosen cut-off distance which, in turn, allows linear scaling methods to be used.
