Efficient simulation of logical magic state preparation protocols by removing the non-Cliffordness

Developing space- and time-efficient logical magic state preparation protocols will likely be an essential step towards building a large-scale fault-tolerant quantum computer. However, understanding the performance of these protocols has been difficult because of the simulation cost; the complexity of existing methods scale exponentially with the number of non-Clifford gates, making large-scale simulation challenging. We introduce a scalable method for simulating these protocols under a standard circuit-level noise model. Applied to the protocols based on code switching and magic state cultivation, our method yields a complexity polynomial in (i) the number of qubits and (ii) the stabilizer rank of the final state in the protocol. Because the stabilizer rank of the final state is $\mathcal{O}(1)$ in many practical protocols of interest, our method can simulate those protocols efficiently. We also provide a method applicable to stabilizer simulation tools that do not track the global phase, where the complexity of the simulation depends on a more convenient magic monotone known as Pauli rank.