 Solution

The essential requirement is an efficient quantum state preparation circuit $B$ satisfyingA standard sufficient promise is that $b$ has only $\mathrm{poly}(\log N)$ nonzero components, with their positions and values classically computable to the required precision in $\mathrm{poly}(\log N)$ time, and with efficiently computable normalization. More structured dense vectors are also allowed whenever cumulative weights or an equivalent data-access oracle permit amplitude encoding in polylogarithmic time. Without such a promise, merely loading $N$ arbitrary classical entries already costs $\Omega (N)$ and removes the claimed exponential dependence on dimension.