  Shadow estimation is a recent protocol that allows estimating exponentially
many expectation values of a quantum state from ``classical shadows'', obtained
by applying random quantum circuits and computational basis measurements. In
this paper we study the statistical efficiency of this approach in light of
near-term quantum computing. We propose a more practical variant of the
protocol, thrifty shadow estimation, in which quantum circuits are reused many
times instead of having to be freshly generated for each measurement. We show
that reuse is maximally effective when sampling Haar random unitaries, and
maximally ineffective when sampling from the Clifford group, i.e., one should
not reuse circuits when performing shadow estimation with the Clifford group.
We provide an efficiently simulable family of quantum circuits that
interpolates between these extremes, which we believe should be used instead of
the Clifford group. Finally, we consider tail bounds for shadow estimation and
discuss when median-of-means estimation can be replaced with standard mean
estimation.
