  Observing a physical quantity without disturbing it is a key capability for
the control of individual quantum systems. Such back-action-evading or
quantum-non-demolition measurements were first introduced in the 1970s in the
context of gravitational wave detection to measure weak forces on test masses
by high precision monitoring of their motion. Now, such techniques have become
an indispensable tool in quantum science for preparing, manipulating, and
detecting quantum states of light, atoms, and other quantum systems. Here we
experimentally perform rapid optical quantum-noise-limited measurements of the
position of a mechanical oscillator by using pulses of light with a duration
much shorter than a period of mechanical motion. Using this back-action evading
interaction we performed both state preparation and full state tomography of
the mechanical motional state. We have reconstructed mechanical states with a
position uncertainty reduced to 19 pm, limited by the quantum fluctuations of
the optical pulse, and we have performed `cooling-by-measurement' to reduce the
mechanical mode temperature from an initial 1100 K to 16 K. Future improvements
to this technique may allow for quantum squeezing of mechanical motion, even
from room temperature, and reconstruction of non-classical states exhibiting
negative regions in their phase-space quasi-probability distribution.
