利用内点法的线性规划量子加速
Quantum speedups for linear programming via interior point methods
AI总结：
本文提出了一种基于内点法的量子算法，用于加速线性规划问题，通过高效近似Hessian和梯度实现亚线性时间复杂度。
摘要：
我们描述了一种基于内点法的量子算法，用于求解具有n个不等约束的线性规划问题。该算法明确返回一个可行解，该解在最优解附近ε范围内，并且运行时间为√n·poly(d,log(n),log(1/ε))，对于高矮线性规划（即n>>d）来说是亚线性的。我们的算法加速了Lee和Sidford[FOCS '14]最先进的内点法中的牛顿步。这要求我们高效地近似障碍函数的Hessian和梯度，而这是我们主要的贡献。为了近似Hessian，我们描述了一种量子算法，用于近似 tall 矩阵A∈R^{n×d}的谱近似。该算法结合了杠杆得分采样与Grover搜索，并通过O(√(nd)/δ)次A的行查询返回一个δ近似。这扩展了Apers和de Wolf[FOCS '20]之前关于图稀疏化的量子加速。为了近似梯度，我们使用了Cornelissen、Hamoudi和Jerbi[STOC '22]最近的多变量均值估计量子算法。虽然直接实现会引入对Hessian条件数的依赖，但我们通过使用我们的谱近似量子算法预处理随机变量来避免这种情况。
英文摘要：
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.
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