  Measuring properties of quantum systems is a fundamental problem in quantum
mechanics. We provide a simple method for estimating the expectation value of
observables with an unknown quantum state. The idea is to use a data structure
to sample the terms of observables based on the Pauli decomposition
proportionally to their importance. We call this technique operator shadow as a
shorthand for the procedure of preparing a sketch of an operator to estimate
properties. Only when the numbers of observables are small for multiple local
observables, the sample complexity of this method is better than the classical
shadow technique. However, if we want to estimate the expectation value of a
linear combination of local observables, for example the energy of a local
Hamiltonian, the sample complexity is better on all parameters. The time
complexity to construct the data structure is $2^{O(k)}$ for $k$-local
observables, similar to the post-processing time of classical shadows.
