Approximating observables on eigenstates of large many-body localized systems

Eigenstates of fully many-body localized (FMBL) systems can be organized into spin algebras based on quasilocal operators called l-bits. These spin algebras define quasilocal l-bit measurement ($tau^z_i$) and l-bit flip ($tau^x_i$) operators. For a disordered Heisenberg spin chain in the MBL regime we approximate l-bit flip operators by finding them exactly on small windows of systems and extending them onto the whole system by exploiting their quasilocal nature. We subsequently use these operators to represent approximate eigenstates. We then describe a method to calculate products of local observables on these eigenstates for systems of size $L$ in $O(L^2)$ time. This algorithm is used to compute the error of the approximate eigenstates.