Periodic Symmetry-Adapted Encoding: Qubit Reduction in Crystalline Electronic Structure

We extend the symmetry-adapted encoding (SAE) framework to periodic electronic structure, enabling qubit-efficient quantum simulation of crystalline materials. By constructing a Γ-point supercell Hamiltonian from a folded k-point calculation and systematically identifying all applicable space-group symmetry generators - including spin-parity, point-group, and crystal translation symmetries - we obtain qubit Hamiltonians with fewer qubits than the Jordan--Wigner starting point. We benchmark diamond, silicon, 3C-SiC, MgO, NaCl, CsCl, h-BN, wurtzite AlN, α-quartz SiO₂, and MgF₂ using active spaces chosen to preserve complete near-degenerate frontier manifolds across cubic, hexagonal, trigonal, and tetragonal space groups. Across the suite the periodic SAE removes 4--8 qubits. The B2 CsCl benchmark realises eight independent Boolean generators, i.e. a symmetry group isomorphic to mathbb{Z}₂⁸, reducing CAS(6,7) from 14 to 6 qubits. This exceeds the mathbb{Z}₂⁵ maximum of molecular SAE, where only two spin parities and at most three independent Boolean point-group generators are available, because the folded crystal supplies three additional half-translation symmetries. Noiseless UCCSD-VQE benchmarks against exact diagonalisation in the active-space sector show that the reduced encodings preserve the target energies to well below chemical accuracy while reducing variational parameter counts by 3--8times and CNOT counts by up to 309times. The largest circuit savings occur when translation and point-group generators act independently in the active space, demonstrating that periodic symmetry can be converted directly into both qubit and ansatz compression. The method is implemented in the open-source QuantumSymmetry package and requires no manual specification of symmetry generators.