We propose a symmetry-adapted eigensolver for strongly correlated systems that separates where the wave function is from where it is . 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 $\phi(a_{\rm real})$ acts only as a sample generator—configurations are drawn from $|\phi(a_{\rm real})|^2$ 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 $\psi(a_{\rm symm})$ is the square root of the total weight accumulated over the entire equivalence class, $\widetilde{p}(a_{\rm symm})=\sum_{a_{\rm real}\in\{g a_{\rm repr}\}}|\phi(a_{\rm real})|^2$, whereas its phase is inherited from the single representative amplitude $\phi(a_{\rm repr})$. The modulus rule is what makes the scheme both exact and cheap: the distribution $\widetilde{p}(a_{\rm symm})$ sampled in the symmetric basis is exactly the one induced by the mapping from the real-space distribution $p(a_{\rm real})$, so every generated configuration is an unbiased sample of $|\psi(a_{\rm symm})|^2$; 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, $\psi$ is 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. For the periodic $J_1-J_2$ antiferromagnetic Heisenberg model with translation, reflection, inversion, and particle-number symmetries, MPS bond dimensions up to 6 yield absolute errors in the energy per site of ${10}^{-5}$ for a 64-site periodic chain and a $6\times 6$ torus after bond-dimension extrapolation.