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The HHL algorithm requires coherent, efficient and repeatable preparation of the normalized state $|b⟩$, normally through a known preparation circuit and its inverse; possession of a single unknown physical specimen does not supply that access. The component of $b$ on any discarded or unresolved small-eigenvalue subspace must also be negligible. Here $U$ is a unitary operator, so it is invertible and all its singular values equal one, giving condition number $\kappa =1$.

Standard HHL is stated for a Hermitian matrix with an efficient sparse-access or block encoding oracle. A non-Hermitian $U$ can be embedded in the Hermitian block matrixpart (a) supplies efficient access to $U^{†}$. With inverse-polynomial target precision, $m=O(\log n)$ phase bits, and an efficient preparation oracle for $|b⟩$, the runtime is $\mathrm{poly}(n)$. The output is the normalized quantum state proportional to the solution $x$, rather than a classical list of all its amplitudes.