An understanding of materials’ electronic properties requires calculations that involve excited

states of electrons. These calculations can be completed in linear time by a stochastic

approach, which involves sampling random vectors in the electrons’ Hilbert space. However, in

practice, it has been observed that the stochastic fluctuations are higher for strongly localized

electronic states than for other states, which increases the computational cost in those cases.

In order to gain a better understanding of why the stochastic fluctuations may be higher for

certain states, we present several toy models that exhibit similar behavior and can be more

easily analyzed than many-body electronic systems . Specifically, we perform simple calculations

on energy eigenstates of a modified quantum harmonic oscillator, using a Python

implementation of the stochastic approach. In these toy models, we find that strongly localized

states exhibit lower stochastic fluctuations. However, by selecting a finely spaced representation

in position space, the stochastic fluctuations can be made uniform across all states. Our

analysis provides a simple case study of how the representation of quantum states can lead to

an uneven distribution of fluctuations in stochastic calculations.