  With the advent of gravitational wave detectors employing squeezed light,
quantum waveform estimation---estimating a time-dependent signal by means of a
quantum-mechanical probe---is of increasing importance. As is well known,
backaction of quantum measurement limits the precision with which the waveform
can be estimated, though these limits can in principle be overcome by "quantum
nondemolition" (QND) measurement setups found in the literature. Strictly
speaking, however, their implementation would require infinite energy, as their
mathematical description involves Hamiltonians unbounded from below. This
raises the question of how well one may approximate nondemolition setups with
finite energy or finite-dimensional realizations. Here we consider a
finite-dimensional waveform estimation setup based on the "quasi-ideal clock"
and show that the estimation errors due to approximating the QND condition
decrease slowly, as a power law, with increasing dimension. As a result, we
find that good QND approximations require large energy or dimensionality. We
argue that this result can be expected to also hold for setups based on
truncated oscillators or spin systems.
