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A Quantum Interior-Point Predictor-Corrector Algorithm for Linear Programming
arXiv - CS - Data Structures and Algorithms Pub Date : 2019-02-18 , DOI: arxiv-1902.06749
P. A. M. Casares and M. A. Martin-Delgado

We introduce a new quantum optimization algorithm for dense Linear Programming problems, which can be seen as the quantization of the Interior Point Predictor-Corrector algorithm \cite{Predictor-Corrector} using a Quantum Linear System Algorithm \cite{DenseHHL}. The (worst case) work complexity of our method is, up to polylogarithmic factors, $O(L\sqrt{n}(n+m)\overline{||M||_F}\bar{\kappa}^2\epsilon^{-2})$ for $n$ the number of variables in the cost function, $m$ the number of constraints, $\epsilon^{-1}$ the target precision, $L$ the bit length of the input data, $\overline{||M||_F}$ an upper bound to the Frobenius norm of the linear systems of equations that appear, $||M||_F$, and $\bar{\kappa}$ an upper bound to the condition number $\kappa$ of those systems of equations. This represents a quantum speed-up in the number $n$ of variables in the cost function with respect to the comparable classical Interior Point algorithms when the initial matrix of the problem $A$ is dense: if we substitute the quantum part of the algorithm by classical algorithms such as Conjugate Gradient Descent, that would mean the whole algorithm has complexity $O(L\sqrt{n}(n+m)^2\bar{\kappa} \log(\epsilon^{-1}))$, or with exact methods, at least $O(L\sqrt{n}(n+m)^{2.373})$. Also, in contrast with any Quantum Linear System Algorithm, the algorithm described in this article outputs a classical description of the solution vector, and the value of the optimal solution.

中文翻译:

一种用于线性规划的量子内点预测校正算法

我们为密集线性规划问题引入了一种新的量子优化算法,可以将其视为使用量子线性系统算法 \cite{DenseHHL} 对内点预测-校正算法 \cite{Predictor-Corrector} 进行量化。我们方法的(最坏情况)工作复杂度是,最多为多对数因子,$O(L\sqrt{n}(n+m)\overline{||M||_F}\bar{\kappa}^2\ epsilon^{-2})$ 为 $n$ 成本函数中变量的数量,$m$ 约束的数量,$\epsilon^{-1}$ 目标精度,$L$ 的位长输入数据,$\overline{||M||_F}$ 是出现的线性方程组的 Frobenius 范数的上限,$||M||_F$ 和 $\bar{\kappa}$这些方程组的条件数 $\kappa$ 的上限。当问题的初始矩阵 $A$ 密集时,这表示成本函数中变量数量 $n$ 相对于可比较的经典内点算法的量子加速:如果我们替换算法的量子部分通过共轭梯度下降等经典算法,这意味着整个算法的复杂度为 $O(L\sqrt{n}(n+m)^2\bar{\kappa} \log(\epsilon^{-1}) )$,或使用精确方法,至少为 $O(L\sqrt{n}(n+m)^{2.373})$。此外,与任何量子线性系统算法相比,本文中描述的算法输出解向量的经典描述和最优解的值。如果我们用经典算法(如共轭梯度下降)代替算法的量子部分,则意味着整个算法的复杂度为 $O(L\sqrt{n}(n+m)^2\bar{\kappa} \log (\epsilon^{-1}))$,或使用精确方法,至少 $O(L\sqrt{n}(n+m)^{2.373})$。此外,与任何量子线性系统算法相比,本文中描述的算法输出解向量的经典描述和最优解的值。如果我们用经典算法(如共轭梯度下降)代替算法的量子部分,则意味着整个算法的复杂度为 $O(L\sqrt{n}(n+m)^2\bar{\kappa} \log (\epsilon^{-1}))$,或使用精确方法,至少 $O(L\sqrt{n}(n+m)^{2.373})$。此外,与任何量子线性系统算法相比,本文中描述的算法输出解向量的经典描述和最优解的值。
更新日期:2020-10-15
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