  We propose a simple scheme to reduce readout errors in experiments on quantum
systems with finite number of measurement outcomes. Our method relies on
performing classical post-processing which is preceded by Quantum Detector
Tomography, i.e., the reconstruction of a Positive-Operator Valued Measure
(POVM) describing the given quantum measurement device. If the measurement
device is affected only by an invertible classical noise, it is possible to
correct the outcome statistics of future experiments performed on the same
device. To support the practical applicability of this scheme for near-term
quantum devices, we characterize measurements implemented in IBM's and
Rigetti's quantum processors. We find that for these devices, based on
superconducting transmon qubits, classical noise is indeed the dominant source
of readout errors. Moreover, we analyze the influence of the presence of
coherent errors and finite statistics on the performance of our
error-mitigation procedure. Applying our scheme on the IBM's 5-qubit device, we
observe a significant improvement of the results of a number of single- and
two-qubit tasks including Quantum State Tomography (QST), Quantum Process
Tomography (QPT), the implementation of non-projective measurements, and
certain quantum algorithms (Grover's search and the Bernstein-Vazirani
algorithm). Finally, we present results showing improvement for the
implementation of certain probability distributions in the case of five qubits.
