Leveraging Symmetries in Pauli Propagation

We present a symmetry-adapted framework for simulating quantum dynamics based on Pauli propagation (PP). When a quantum circuit exhibits a symmetry, many Pauli strings evolve redundantly under the action of the symmetry group. We exploit this redundancy by merging Paulis related through symmetry transformations, thereby reducing computational costs. We formalize this procedure as "symmetry PP", which propagates only a minimal set of orbit representatives of Pauli strings. We prove that the algorithm yields exactly the same expectation values as standard PP when the quantum circuit and initial state respect the symmetry. Analytically, we show that symmetry merging reduces the space complexity by a factor proportional to the size of the orbit representatives, with explicit results for translational and permutation symmetries. Numerical benchmarks of all-to-all interacting Heisenberg dynamics on a periodic two-dimensional lattice geometry confirm that symmetry-merging improves stability, especially under truncation and noise. To contribute to the landscape of powerful open-source software for simulating quantum dynamics, our systematically improvable algorithm is published as part of the PauliPropagation.jl library.