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 \emph{sampled} from where it is \emph{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 $ϕ(a_{\rm real})$ acts only as a sample generator---configurations are drawn from $|ϕ(a_{\rm real})|^2$ 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 $ψ(a_{\rm symm})$ is the square root of the total weight accumulated over the entire equivalence class, $\widetilde{p}(a_{\rm symm})=\sum_{a_{\rm real}\in\{g a_{\rm repr}\}}|ϕ(a_{\rm real})|^2$, whereas its phase is inherited from the single representative amplitude $ϕ(a_{\rm repr})$. 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 $|ψ(a_{\rm symm})|^2$; 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, $ψ$ is \emph{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.