A quantum central path algorithm for linear optimization

TL;DR

This paper introduces a quantum algorithm for linear optimization that directly simulates the central path, achieving improved query complexity and efficiency over classical methods by leveraging quantum simulation of nonlinear equations.

Contribution

It presents a novel quantum approach that directly models the nonlinear complementarity equations of the central path, reducing complexity compared to iterative classical interior point methods.

Findings

Achieves $ ilde{O}(\sqrt{m+n} R_1/\varepsilon)$ query complexity.

Can obtain highly-precise solutions with $ ilde{O}(\sqrt{m+n} extsf{nnz}(A) R_1/\varepsilon)$ elementary gates.

Provides a quantum algorithm that potentially outperforms classical interior point methods in efficiency.

Abstract

We propose a novel quantum algorithm for solving linear optimization problems by quantum-mechanical simulation of the central path. While interior point methods follow the central path with an iterative algorithm that works with successive linearizations of the perturbed KKT conditions, we perform a single simulation working directly with the nonlinear complementarity equations. This approach yields an algorithm for solving linear optimization problems involving $m$ constraints and $n$ variables to $\varepsilon$-optimality using $\mathcal{O} \left( \sqrt{m + n} \frac{R_{1}}{\varepsilon}\right)$ queries to an oracle that evaluates a potential function, where $R_{1}$ is an $\ell_{1}$-norm upper bound on the size of the optimal solution. In the standard gate model (i.e., without access to quantum RAM) our algorithm can obtain highly-precise solutions to LO problems using at most $$\mathcal{O} \left( \sqrt{m + n} \textsf{nnz} (A) \frac{R_1}{\varepsilon}\right)$$ elementary gates, where $\textsf{nnz} (A)$ is the total number of non-zero elements found in the constraint matrix.