  Extracting classical information from quantum systems is of fundamental
importance, and classical shadows allow us to extract a large amount of
information using relatively few measurements. Conventional shadow estimators
are unbiased and thus approach the true mean in the infinite-sample limit. In
this work, we consider a biased scheme, intentionally introducing a bias by
rescaling the conventional classical shadows estimators can reduce the error in
the finite-sample regime. The approach is straightforward to implement and
requires no quantum resources. We analytically prove average case as well as
worst- and best-case scenarios, and rigorously prove that it is, in principle,
always worth biasing the estimators. We illustrate our approach in a quantum
simulation task of a $12$-qubit spin-ring problem and demonstrate how
estimating expected values of non-local perturbations can be significantly more
efficient using our biased scheme.
