  We theoretically describe the weak measurement of a two-level system (qubit)
and quantify the degree to which such a qubit measurement has a quantum
non-demolition (QND) character. The qubit is coupled to a harmonic oscillator
which undergoes a projective measurement. Information on the qubit state is
extracted from the oscillator measurement outcomes, and the QND character of
the measurement is inferred by the result of subsequent measurements of the
oscillator. We use the positive operator valued measure (POVM) formalism to
describe the qubit measurement. Two mechanisms lead to deviations from a
perfect QND measurement: (i) the quantum fluctuations of the oscillator, and
(ii) quantum tunneling between the qubit states $|0>$ and $|1>$ during
measurements. Our theory can be applied to QND measurements performed on
superconducting qubits coupled to a circuit oscillator.
