Quantum algorithms for Second-Order Cone Programming and Support Vector Machines

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