  Flag-style fault-tolerance has become a linchpin in the realization of small
fault-tolerant quantum-error correction experiments. The flag protocol's
utility hinges on low qubit overhead, which is typically much smaller than in
other approaches. However, as in most fault-tolerance protocols, the advantages
of flag-style error correction come with a tradeoff: fault tolerance can be
guaranteed, but such protocols involve high-depth circuits, due to the need for
repeated stabilizer measurements. Here, we demonstrate that a dynamic choice of
stabilizer measurements, based on past syndromes, and the utilization of
elements from the full stabilizer group, leads to flag protocols with
lower-depth syndrome-extraction circuits for the [[5,1,3]] code, as well as for
the Steane code when compared to the standard methods in flag fault tolerance.
We methodically prove that our new protocols yield fault-tolerant lookup
tables, and demonstrate them with a pseudothreshold simulation, showcasing
large improvements for all protocols when compared to previously-established
methods. This work opens the dialogue on exploiting the properties of the full
stabilizer group for reducing circuit overhead in fault-tolerant quantum-error
correction.
