  Quantum state preparation is a fundamental and significant subroutine in
quantum computing. In this paper, we conduct a systematic investigation on the
circuit size for sparse quantum state preparation. A quantum state is said to
be $d$-sparse if it has only $d$ non-zero amplitudes. For the task of preparing
$n$-qubit $d$-sparse quantum states, we obtain the following results:
  (a) We propose the first approach that uses $o(dn)$ elementary gates without
using ancillary qubits. Specifically, it is proven that any $n$-qubit
$d$-sparse quantum state can be prepared by a quantum circuit of size
$O(\frac{dn}{\log n} + n)$ without using ancillary qubits. This is
asymptotically optimal when $d = poly(n)$, and this optimality extends to a
broader scope under some reasonable assumptions.
  (b) We show that any $n$-qubit $d$-sparse quantum state can be prepared by a
quantum circuit of size $O(\frac{dn}{\log d})$ and depth $\Theta(\log dn)$
using at most $O(\frac{n{d}}{\log d} )$ ancillary qubits, which not only
reduces the circuit size compared to the one without ancillary qubits when $d =
\omega(poly(n))$, but also achieves the same asymptotically optimal depth while
utilizing fewer ancillary qubits and applying fewer quantum gates compared to
the result given in [PRL, 129, 230504(2022)]. (ii) We establish the lower bound
$\Omega(\frac{dn}{\log(n + m) + \log d} + n)$ on the circuit size with $m$
ancillary qubits available. we also obtain a slightly stronger lower bound
under reasonable assumptions.
  (c) We prove that with arbitrary amount of ancillary qubits available, the
circuit size for preparing $n$-qubit $d$-sparse quantum states is
$\Theta({\frac{dn}{\log dn} + n})$.
