Symmetry-Adapted Eigensolver from Real-Space Sampling of Tensor-Network-Bridged Quantum Circuits

We propose a symmetry-adapted eigensolver for strongly correlated systems that separates where the wave function is sampled from where it is defined. The lattice is divided into equal blocks: parallel quantum circuits absorb the entanglement within each block, while a matrix product state (MPS) backbone bridges the blocks and restores global correlations. The resulting real-space amplitude ϕ(a real) acts only as a sample generator---configurations are drawn from ϕ(a real) squared directly, block by block, without any Markov chain. The variational state itself is defined in a symmetry-adapted basis by two separate rules: the modulus of ψ(a symm) is the square root of the total weight accumulated over the entire equivalence class, (a symm)= sum a real ing a reprϕ(a real) squared, whereas its phase is inherited from the single representative amplitude ϕ(a repr). The modulus rule is what makes the scheme both exact and cheap: the distribution (a symm) sampled in the symmetric basis is exactly the one induced by the mapping from the real-space distribution p(a real), so every generated configuration is an unbiased sample of ψ(a symm) squared; the phase rule then fixes the only freedom that remains. Because amplitudes within a class are combined in probability rather than summed with symmetry characters, ψ is not a group projection of the circuit-MPS state; the class-wide aggregation instead supplies long-range, symmetry-enforced entanglement that the MPS never has to carry itself.