 Solution

For the HHL algorithm to have runtime polynomial in $\log N$, the Hermitian matrix $A$ must be invertible, have a condition number $\kappa$ bounded by $\mathrm{poly}(\log N)$, and be a sparse matrix with its nonzero entries efficiently accessible by an oracle. The normalized state $|b⟩$ must also be preparable in $\mathrm{poly}(\log N)$ time. With precision costs suppressed, these assumptions let HHL prepare, with high probability,