  Stochastic orbital techniques offer reduced computational scaling and memory
requirements to describe ground and excited states at the cost of introducing
controlled statistical errors. Such techniques often rely on two basic
operations, stochastic trace estimation and stochastic resolution of identity,
both of which lead to statistical errors that scale with the number of
stochastic realizations ($N_{\xi}$) as $\sqrt{N_{\xi}^{-1}}$. Reducing the
statistical errors without significantly increasing $N_{\xi}$ has been
challenging and is central to the development of efficient and accurate
stochastic algorithms. In this work, we build upon recent progress made to
improve stochastic trace estimation based on the ubiquitous Hutchinson's
algorithm and propose a two-step approach for the stochastic resolution of
identity, in the spirit of the Hutch++ method. Our approach is based on
employing a randomized low-rank approximation followed by a residual
calculation, resulting in statistical errors that scale much better than
$\sqrt{N_{\xi}^{-1}}$. We implement the approach within the second-order Born
approximation for the self-energy in the computation of neutral excitations and
discuss three different low-rank approximations for the two-body Coulomb
integrals. Tests on a series of hydrogen dimer chains with varying lengths
demonstrate that the Hutch++-like approximations are computationally more
efficient than both deterministic and purely stochastic (Hutchinson) approaches
for low error thresholds and intermediate system sizes. Notably, for
arbitrarily large systems, the Hutchinson-like approximation outperforms both
deterministic and Hutch++-like methods.
