  Quantum error correction (QEC) is required for a practical quantum computer
because of the fragile nature of quantum information. In QEC, information is
redundantly stored in a large Hilbert space and one or more observables must be
monitored to reveal the occurrence of an error, without disturbing the
information encoded in an unknown quantum state. Such observables, typically
multi-qubit parities such as <XXXX>, must correspond to a special symmetry
property inherent to the encoding scheme. Measurements of these observables, or
error syndromes, must also be performed in a quantum non-demolition (QND) way
and faster than the rate at which errors occur. Previously, QND measurements of
quantum jumps between energy eigenstates have been performed in systems such as
trapped ions, electrons, cavity quantum electrodynamics (QED), nitrogen-vacancy
(NV) centers, and superconducting qubits. So far, however, no fast and repeated
monitoring of an error syndrome has been realized. Here, we track the quantum
jumps of a possible error syndrome, the photon number parity of a microwave
cavity, by mapping this property onto an ancilla qubit. This quantity is just
the error syndrome required in a recently proposed scheme for a
hardware-efficient protected quantum memory using Schr\"{o}dinger cat states in
a harmonic oscillator. We demonstrate the projective nature of this measurement
onto a parity eigenspace by observing the collapse of a coherent state onto
even or odd cat states. The measurement is fast compared to the cavity
lifetime, has a high single-shot fidelity, and has a 99.8% probability per
single measurement of leaving the parity unchanged. In combination with the
deterministic encoding of quantum information in cat states realized earlier,
our demonstrated QND parity tracking represents a significant step towards
implementing an active system that extends the lifetime of a quantum bit.
