  Following the first paper on quantum algorithms for SDP-solving by Brand\~ao
and Svore in 2016, rapid developments has been made on quantum optimization
algorithms. Recently Brand\~ao et al. improved the quantum SDP-solver in the
so-called quantum state input model, where the input matrices of the SDP are
given as purified mixed states. They also gave the first non-trivial
application of quantum SDP-solving by obtaining a more efficient algorithm for
the problem of shadow tomography (proposed by Aaronson in 2017). In this paper
we improve on all previous quantum SDP-solvers. Mainly we construct better
Gibbs-samplers for both input models, which directly gives better bounds for
SDP-solving. For an SDP with $m$ constraints involving $n\times n$ matrices,
our improvements yield an $\widetilde{\mathcal O}\left( \left( \sqrt{m} +
\sqrt{n}\gamma \right)s \gamma^4\right)$ upper bound on SDP-solving in the
sparse matrix input model and an $\widetilde{\mathcal O}\left(
\left(\sqrt{m}+B^{2.5}\gamma^{3.5} \right)B\gamma^4 \right)$ upper bound in the
quantum state input model. We then apply these results to the problem of shadow
tomography to simultaneously improve the best known upper bounds on sample
complexity due to Aaronson and complexity due Brandao et al. Furthermore, we
apply our quantum SDP-solvers to the problems of quantum state discrimination
and E-optimal design. In both cases we beat the classical lower bound in terms
of some parameters, at the expense of heavy dependence on some other
parameters. Finally we prove two lowers bounds for solving SDPs using quantum
algorithms: (1) $\tilde{\Omega}(\sqrt{m}B/\eps)$ in the quantum state input
model, and (2) $\tilde{\Omega}(\sqrt{m}\alpha/\eps)$ in the quantum operator
input model. These lower bounds show that the $\sqrt{m}$ factor and the
polynomial dependence on the parameters $B,\alpha$, and $1/\eps$ are necessary.
