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The classical simulation of highly-entangling quantum dynamics is conjecturedto be generically hard. Thus, recently discovered measurement-inducedtransitions between highly entangling and low-entanglement dynamics are phasetransitions in classical simulability. Here, we study simulability transitionsbeyond entanglement: noting that some highly-entangling dynamics (e.g.,integrable systems or Clifford circuits) are easy to classically simulate, thusrequiring "magic"--a subtle form of quantum resource--to achieve computationalhardness, we ask how the dynamics of magic competes with measurements. We studythe resulting "dynamical magic transitions" focusing on random monitoredClifford circuits doped by T gates (injecting magic). We identify dynamical"stabilizer-purification"--the collapse of a superposition of stabilizer statesby measurements--as the mechanism driving this transition. We find cases wheretransitions in magic and entanglement coincide, but also others with a magicand simulability transition in a highly (volume-law) entangled phase. Inestablishing our results, we use Pauli-based computation, a scheme distillingthe quantum essence of the dynamics to a magic state register subject tomutually commuting measurements. We link stabilizer-purification to "magicfragmentation" wherein these measurements separate into disjoint, O(1)-weightblocks, and relate this to the spread of magic in the original circuit becomingarrested.

Dynamical Magic Transitions in Monitored Clifford+T Circuits