eigensolvers

Use NumPyEigensolver from qiskit_algorithms.eigensolvers with the k parameter set to the number of eigenvalues needed. Call compute_eigenvalues(operator) to get sorted eigenvalues and eigenstates via exact NumPy/SciPy diagonalization.

Instantiate VQD with an estimator primitive, a ComputeUncompute fidelity primitive, a parameterized ansatz like RealAmplitudes, an optimizer such as SLSQP, and k for the number of states. VQD adds overlap penalties against previously found states to enforce orthogonality.

Use NumPyEigensolver for exact deterministic results on small operators up to roughly 10-12 qubits. Use VQD when a variational workflow is preferred, when testing ansatz-based pipelines, or as a scalable design pattern despite its dependence on optimizer and penalty tuning.

The dense matrix grows as 2^n by 2^n, so 10 qubits need about 8 MB and 12 qubits about 128 MB in complex128. Beyond 12 qubits the dense path may exhaust memory, making exact diagonalization impractical.

The overlap penalty weights (betas) are too weak to enforce orthogonality against previously found states. Increase betas starting around 1.0, or raise optimizer maxiter so the penalty term is properly minimized.

Yes, pass a filter_criterion callable with signature (eigenstate, eigenvalue, aux_values) returning a boolean. Only eigenpairs passing the filter are returned, so the result may contain fewer than k entries.