Efficient Pauli Truncation from Operator Stabilizer Rényi Entropy

Abstract

Operator stabilizer Rényi entropy (OSE) quantifies the magic of Heisenberg operators and serves as a diagnostic of their compressibility in the Pauli basis. We establish rigorous bounds showing when an operator can be efficiently approximated by Pauli truncation in terms of the scaling of its OSE. Specifically, we prove that volume-law scaling of OSE for $α\geq1$ prevents efficient Pauli truncation from reproducing expectation values for all states at sufficiently small fixed error. On the other hand, logarithmic growth of OSE for $0<α<1/2$ guarantees efficient Pauli truncation approximability uniformly over all states. A counterexample shows that this sufficient range cannot be extended on the basis of OSE scaling alone. If we restrict to correlations over suitable low-average state ensembles, then the sufficient range extends to $0<α<1$. We further derive cumulative error bounds for iterative Pauli propagation and approximation guarantees for fixed-order out-of-time-ordered correlators. Moreover, under an independent-sign assumption, we show that logarithmic OSE for $0<α<1$ guarantees efficient Pauli truncation with the prescribed accuracy for all input states with high probability. Our results provide a theoretical framework for understanding when operator magic constrains or enables efficient Pauli truncation and for guiding the development of classical simulation methods.