  Classical shadows enable us to learn many properties of a quantum state
$\rho$ with very few measurements. However, near-term and early fault-tolerant
quantum computers will only be able to prepare noisy quantum states $\rho$ and
it is thus a considerable challenge to efficiently learn properties of an
ideal, noise free state $\rho_{id}$. We consider error mitigation techniques,
such as Probabilistic Error Cancellation (PEC), Zero Noise Extrapolation (ZNE)
and Symmetry Verification (SV) which have been developed for mitigating errors
in single expected value measurements and generalise them for mitigating errors
in classical shadows. We find that PEC is the most natural candidate and thus
develop a thorough theoretical framework for PEC shadows with the following
rigorous theoretical guarantees: PEC shadows are an unbiased estimator for the
ideal quantum state $\rho_{id}$; the sample complexity for simultaneously
predicting many linear properties of $\rho_{id}$ is identical to that of the
conventional shadows approach up to a multiplicative factor which is the sample
overhead due to error mitigation. Due to efficient post-processing of shadows,
this overhead does not depend directly on the number of qubits but rather grows
exponentially with the number of noisy gates. The broad set of tools introduced
in this work may be instrumental in exploiting near-term and early
fault-tolerant quantum computers: We demonstrate in detailed numerical
simulations a range of practical applications of quantum computers that will
significantly benefit from our techniques.
