Characterizing Pauli Propagation via Operator Complexity in Quantum Spin Systems

Simulating real-time quantum dynamics in interacting spin systems is a fundamental challenge, where exact diagonalization suffers from exponential Hilbert-space growth and tensor-network methods face entanglement barriers. Recently, Pauli-propagation-based methods have emerged as a promising alternative. In this work, we bridge operator complexity and the complexity of Pauli-propagation-based methods in simulating real-time dynamics of quantum spin systems. Specifically, we derive a priori error bounds governed by the Operator Stabilizer R\'enyi entropy (OSE) $\mathcal{S}^\alpha(O)$, which explicitly links the truncation accuracy to operator complexity. For the 1D Heisenberg model with $J_z = 0$, we prove the number of non-zero Pauli coefficients scales quadratically in Trotter steps, establishing the compressibility of Heisenberg-evolved operators. We then consider a Pauli propagation instance with a Top-$K$ truncation strategy, and benchmark the method numerically on XXZ Heisenberg chain benchmarks, showing high accuracy with small truncation number $K$ in free regimes ($J_z = 0$) and competitive performance against tensor-network methods (e.g., TDVP) in interacting cases ($J_z = 0.5$). Analogous to the role of entanglement entropy in tensor networks, these results establish a new perspective on using OSE to quantify the computational complexity of Pauli-propagation-based methods.