The concept of projecting a large Hilbert space onto a low‑dimensional subspace spanned by quantum‑generated basis vectors has been employed in quantum chemistry (e.g., QSE [6]) and in quantum singular‑value transformation (QSVT) [7]. By selecting a set of K orthonormal quantum states (_k=1^K), one can construct the effective matrix
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Input: Sparse matrix A (N×N), RHS vector b, tolerance ε, max. quantum subspace size K_max Output: Approximate solution x̃ such that ||A x̃ – b|| / ||b|| < ε