Leveraging Symmetry Merging in Pauli Propagation

We introduce a symmetry-adapted framework for simulating quantum dynamics based on Pauli propagation. When a quantum circuit possesses a symmetry, many Pauli strings evolve redundantly under actions of the symmetry group. We exploit this by merging Pauli strings related through symmetry transformations. This procedure, formalized as the symmetry-merging Pauli propagation algorithm, propagates only a minimal set of orbit representatives. Analytically, we show that symmetry merging reduces space complexity by a factor set by orbit sizes, with explicit gains for translation and permutation symmetries. Numerical benchmarks of all-to-all Heisenberg dynamics confirm improved stability, particularly under truncation and noise. Our results establish a group-theoretic framework for enhancing Pauli propagation, supported by open-source code demonstrating its practical relevance for classical quantum-dynamics simulations.