  The classical simulation of highly-entangling quantum dynamics is conjectured
to be generically hard. Thus, recently discovered measurement-induced
transitions between highly entangling and low-entanglement dynamics are phase
transitions in classical simulability. Here, we study simulability transitions
beyond entanglement: noting that some highly-entangling dynamics (e.g.,
integrable systems or Clifford circuits) are easy to classically simulate, thus
requiring "magic"--a subtle form of quantum resource--to achieve computational
hardness, we ask how the dynamics of magic competes with measurements. We study
the resulting "dynamical magic transitions" focusing on random monitored
Clifford circuits doped by T gates (injecting magic). We identify dynamical
"stabilizer-purification"--the collapse of a superposition of stabilizer states
by measurements--as the mechanism driving this transition. We find cases where
transitions in magic and entanglement coincide, but also others with a magic
and simulability transition in a highly (volume-law) entangled phase. In
establishing our results, we use Pauli-based computation, a scheme distilling
the quantum essence of the dynamics to a magic state register subject to
mutually commuting measurements. We link stabilizer-purification to "magic
fragmentation" wherein these measurements separate into disjoint, O(1)-weight
blocks, and relate this to the spread of magic in the original circuit becoming
arrested.
