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 $\delta$-approximation by making $O(\sqrt{nd}/\delta)$ 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：我们描述了一种基于内点法的量子算法，用于求解具有$n$个不等式约束和$d$个变量的线性规划问题。该算法显式返回一个$\varepsilon$-接近最优的可行解，运行时间为$\sqrt{n} \cdot \mathrm{poly}(d,\log(n),\log(1/\varepsilon))$，这对于高瘦型线性规划问题（即$n \gg d$）是次线性的。我们的算法加速了Lee和Sidford [FOCS~'14] 最新内点法中的牛顿步。这要求我们高效逼近障碍函数的Hessian矩阵和梯度，这也是我们的主要贡献。为了逼近Hessian矩阵，我们描述了一种用于高瘦型矩阵$A \in \mathbb R^{n \times d}$的$A^T A$的\emph{谱逼近}量子算法。该算法结合杠杆值采样与Grover搜索，通过向$A$进行$O(\sqrt{nd}/\delta)$次行查询，返回一个$\delta$-逼近。这推广了Apers和de Wolf [FOCS~'20] 早期在图稀疏化上的量子加速。为了逼近梯度，我们使用了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矩阵条件数的依赖，但我们通过使用谱逼近量子算法对随机变量进行预处理来避免这一问题。