Juq470 [ 2027 ]

are ubiquitous in scientific and engineering domains. Classical approaches rely on either direct factorisations (LU, Cholesky) – infeasible for massive sparse matrices due to fill‑in – or iterative Krylov‑subspace methods (CG, GMRES, BiCGSTAB) that depend critically on matrix conditioning and preconditioning strategies.

The QSG stage leverages a Hardware‑Efficient Ansatz (HEA) comprising alternating layers of single‑qubit rotations (R_Y(\theta)) and nearest‑neighbour CNOTs. The number of layers (L) is chosen such that circuit depth (d\approx 2L) stays within the device’s coherence budget (typically (d\le 40) for 127‑qubit IBM Eagle). To capture the dominant eigen‑vectors, we perform a with only 3–4 bits of phase, sufficient to discriminate eigenvalues larger than a threshold (\lambda_\textcut). The eigenvectors associated with (\lambda > \lambda_\textcut) are retained as candidates for the subspace. juq470