This number can be reduced if we look at specific symmetries only. However, the dimensionality explodes quickly!

• For Hamiltonian matrices of dimensionalities which are smaller than \( d\sim 10^5 \), we would use so-called direct methods for diagonalizing the Hamiltonian matrix
• For larger dimensionalities iterative eigenvalue solvers like Lanczos' method are used. The most efficient codes at present can handle matrices of \( d\sim 10^{10} \).