We describe a quantum algorithm based on an interior point method for solving a linear program with $n$ inequality constraints on $d$ variables. The algorithm explicitly returns a feasible solution that is $\varepsilon$-close to optimal, and runs in time $\sqrt{n} \cdot \mathrm{poly}(d,\log(n),\log(1/\varepsilon))$ which is sublinear for tall linear programs (i.e., $n \gg d$). Our algorithm speeds up the Newton step in the state-of-the-art interior point method of Lee and Sidford [FOCS '14]. This requires us to efficiently approximate the Hessian and gradient of the barrier function, and these are our main contributions. To approximate the Hessian, we describe a quantum algorithm for the \emph{spectral approximation} of $A^T A$ for a tall matrix $A \in \mathbb R^{n \times d}$. The algorithm uses leverage score sampling in combination with Grover search, and returns a $δ$-approximation by making $O(\sqrt{nd}/δ)$ row queries to $A$. This generalizes an earlier quantum speedup for graph sparsification by Apers and de Wolf [FOCS '20]. To approximate the gradient, we use a recent quantum algorithm for multivariate mean estimation by Cornelissen, Hamoudi and Jerbi [STOC '22]. While a naive implementation introduces a dependence on the condition number of the Hessian, we avoid this by pre-conditioning our random variable using our quantum algorithm for spectral approximation.

翻译：我们提出一种基于内点法的量子算法，用于求解具有$d$个变量和$n$个不等式约束的线性规划问题。该算法显式返回一个可行解，其目标值与最优解的误差在$\varepsilon$范围内，运行时间为$\sqrt{n} \cdot \mathrm{poly}(d,\log(n),\log(1/\varepsilon))$，对于高维线性规划问题（即$n \gg d$）具有亚线性复杂度。我们的算法加速了Lee和Sidford [FOCS '14]提出的最先进内点法中的牛顿步骤。这要求我们高效逼近障碍函数的Hessian矩阵和梯度，这些是我们的核心贡献。为逼近Hessian矩阵，我们提出一种用于高矩阵$A \in \mathbb R^{n \times d}$的$A^T A$谱近似的量子算法。该算法结合杠杆值采样与Grover搜索，通过对$A$进行$O(\sqrt{nd}/δ)$次行查询，返回$δ$近似结果。这推广了Apers和de Wolf [FOCS '20]早期在图稀疏化方面的量子加速工作。为逼近梯度，我们采用Cornelissen、Hamoudi和Jerbi [STOC '22]最近提出的多元均值估计量子算法。虽然朴素实现会引入对Hessian条件数的依赖，但我们通过使用谱近似量子算法对随机变量进行预处理来规避此问题。