Measurement-Induced Phase Transition in the Classical Simulability of Random Clifford+T Circuits

Classical simulations of quantum systems are conjectured to be generically inefficient, particularly when the system is highly entangled. Recently discovered "measurement-induced phase transitions" in entanglement suggest a corresponding phase transition in the classical simulability of quantum systems. However, some entanglement-generating quantum (such as Clifford) circuits are nevertheless simple to classically simulate, suggesting a more nuanced treatment of simulability is required. Here, we find a phase transition in the simulability of random quantum circuits composed of Clifford gates, T gates and measurements. The transition is related to the amount of "magic" in the circuit—another quantum resource distinct from entanglement—that is related to the number of T gates and distribution of measurements in the circuit. We put forward "stabilizer-purification"—a process by which measurements collapse a superposition of stabilizer states to a single one—as the mechanism driving this simulability phase transition.