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Rigorous measurement error correction

We review an experimental technique used to correct state preparation and measurement errors on gate-based quantum computers, and discuss its rigorous justification. Within a specific biased quantum measurement model, we prove that nonideal measurement of an arbitrary $n$-qubit state is equivalent to ideal projective measurement followed by a classical Markov process $Γ$ acting on the output probability distribution. Measurement errors can be removed, with rigorous justification, if $Γ$ can be learned and inverted. We show how to obtain $Γ$ from gate set tomography (R. Blume-Kohout et al., arXiv:1310.4492) and apply the error correction technique to single IBM Q superconducting qubits.