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

TL;DR

This paper introduces a quantum interior-point method for second-order cone programming that demonstrates polynomial speedup over classical algorithms, with promising applications to support vector machines and numerical simulations supporting its efficiency.

Contribution

The paper develops a quantum algorithm for SOCP that achieves a theoretical polynomial speedup and provides numerical evidence of practical advantages over classical methods.

Findings

Quantum IPM runs faster than classical algorithms for SOCP.

Numerical simulations show polynomial speedup in solving SOCPs.

Quantum SVM scaling is better than classical SVM on random instances.

Abstract

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 $3.11$.