  We present a quantum interior-point method (IPM) for second-order cone
programming (SOCP) that runs in time $\widetilde{O} \left( n\sqrt{r}
\frac{\zeta \kappa}{\delta^2} \log \left(1/\epsilon\right) \right)$ where $r$
is the rank and $n$ the dimension of the SOCP, $\delta$ bounds the distance of
intermediate solutions from the cone boundary, $\zeta$ is a parameter upper
bounded by $\sqrt{n}$, and $\kappa$ is an upper bound on the condition number
of matrices arising in the classical IPM for SOCP. The algorithm takes as its
input a suitable quantum description of an arbitrary SOCP and outputs a
classical description of a $\delta$-approximate $\epsilon$-optimal solution of
the given problem.
  Furthermore, we perform numerical simulations to determine the values of the
aforementioned parameters when solving the SOCP up to a fixed precision
$\epsilon$. We present experimental evidence that in this case our quantum
algorithm exhibits a polynomial speedup over the best classical algorithms for
solving general SOCPs that run in time $O(n^{\omega+0.5})$ (here, $\omega$ is
the matrix multiplication exponent, with a value of roughly $2.37$ in theory,
and up to $3$ in practice). For the case of random SVM (support vector machine)
instances of size $O(n)$, the quantum algorithm scales as $O(n^k)$, where the
exponent $k$ is estimated to be $2.59$ using a least-squares power law. On the
same family random instances, the estimated scaling exponent for an external
SOCP solver is $3.31$ while that for a state-of-the-art SVM solver is $3.11$.
