Unusual energy spectra of matrix product states

In the simulation of ground states of strongly-correlated quantum systems, the decomposition of an approximate solution into the exact eigenstates of the Hamiltonian -- the energy spectrum of the state -- determines crucial aspects of the simulation's performance. For example, in approaches based on imaginary-time evolution, the spectrum falls off exponentially with the energy, ensuring rapid convergence. Here we consider the energy spectra of approximate matrix product state ground states, such as those obtained with the density matrix renormalization group. Despite the high accuracy of these states, contributions to the spectra are roughly constant out to surprisingly high energy, with an increase in bond dimension reducing the amplitude but not the extent of these high-energy tails. The unusual spectra, which appear to be a general feature of compressed wavefunctions, have a strong effect on sampling-based methods, yielding large fluctuations. For example, estimating the energy variance using sampling performs much more poorly than one might expect. Bounding the most extreme samples makes the variance estimate much less noisy but introduces a strong bias. However, we find that this biased variance estimator is an excellent surrogate for the variance when extrapolating the ground-state energy, and this approach outperforms competing extrapolation methods in both accuracy and computational cost.