  We introduce a post-processing technique for classical shadow measurement
data that enhances the precision of ground state estimation through
high-dimensional subspace expansion; the dimensionality is only limited by the
amount of classical post-processing resources rather than by quantum resources.
Crucial steps of our approach are the efficient identification of useful
observables from shadow data, followed by our regularised subspace expansion
that is designed to be numerically stable even when using noisy data. We
analytically investigate noise propagation within our method, and upper bound
the statistical fluctuations due to the limited number of snapshots in
classical shadows. In numerical simulations, our method can achieve a reduction
in the energy estimation errors in many cases, sometimes by more than an order
of magnitude. We also demonstrate that our performance improvements are robust
against both coherent errors (bad initial state) and gate noise in the
state-preparation circuits. Furthermore, performance is guaranteed to be at
least as good - and in many cases better - than direct energy estimation
without using additional quantum resources and the approach is thus a very
natural alternative for estimating ground state energies directly from
classical shadow data.
