  We formulate a convergent sequence for the energy gap estimation in the
worldline quantum Monte Carlo method. The ambiguity left in the conventional
gap calculation for quantum systems is eliminated. Our estimation will be
unbiased in the low-temperature limit and also the error bar is reliably
estimated. The level spectroscopy from quantum Monte Carlo data is developed as
an application of the unbiased gap estimation. From the spectral analysis, we
precisely determine the Kosterlitz-Thouless quantum phase-transition point of
the spin-Peierls model. It is stablished that the quantum phonon with a finite
frequency is essential to the critical theory governed by the antiadiabatic
limit, i.e., the $k=1$ SU(2) Wess-Zumino-Witten model.
