Practical Framework to classically simulate Permutation Equivariant unitaries

Symmetry lies at the heart of many physics-related problems, and harnessing this structure through equivariant quantum circuits has emerged as a powerful approach to enhance performance in symmetry-constrained quantum models. Among symmetry groups, the permutation group, S_n , is especially significant due to its wide applicability in areas such as quantum many-body systems, combinatorial optimization, and particle physics. In particular, S_n-equivariant quantum circuits are known to avoid barren plateaus, enabling efficient classical simulation with complexity scaling at most polynomially with the system size n.

This work presents a practical end-to-end classical framework, integrated with permutation-invariant shadows, to efficiently simulate S_n-equivariant quantum unitaries assuming that are generated by at most two-local Pauli operators. Our method achieves a classical simulation time complexity of O(n⁴), improving upon previous approaches with complexity O(n⁷). To validate our approach, we perform numerical experiments on a representative use-case involving a permutation-invariant Hamiltonian, and we compare the real-world execution times of classical and quantum simulations across different hardware platforms.