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 $\epsilon$-close to optimal, and runs in time $\sqrt{n}\, \mathrm{poly}(d,\log(n),\log(1/\varepsilon))$ 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 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与最优解$\epsilon$-接近的可行解，运行时间为$\sqrt{n}\, \mathrm{poly}(d,\log(n),\log(1/\varepsilon))$，对于"高瘦型"线性规划问题（即$n \gg d$）呈亚线性复杂度。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。这需要高效近似障碍函数的Hessian矩阵和梯度，而这正是我们的主要贡献。为近似Hessian矩阵，我们描述了一种针对高瘦型矩阵$A \in \mathbb R^{n \times d}$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，通过向$A$进行$O(\sqrt{nd}/\delta)$次行查询，返回一个$\delta$-近似解。这推广了Apers和de Wolf [FOCS '20]关于图稀疏化的早期量子加速方案。为近似梯度，我们使用Cornelissen、Hamoudi和Jerbi [STOC '22]提出的近期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，但我们利用谱近似的量子算法对随机变量进行预处理，从而避免了这一依赖。