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A quantum interior-point predictor–corrector algorithm for linear programming
Journal of Physics A: Mathematical and Theoretical ( IF 2.0 ) Pub Date : 2020-10-15 , DOI: 10.1088/1751-8121/abb439
P A M Casares , 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 [1] using a quantum linear system algorithm [2]. The (worst case) work complexity of our method is, up to polylogarithmic factors, $O\left(L\sqrt{n}\left(n+m\right)\overline{{\Vert}M{{\Vert}}_{\mathrm{F}}}\overline{\kappa }{{\epsilon}}^{-2}\right)$ for n the number of variables in the cost function, m the number of constraints, ϵ −1 the target precision, L the bit length of the input data, $\overline{{\Vert}M{{\Vert}}_{\mathrm{F}}}$ an upper bound to the Frobenius norm of the linear systems of equations that appear, ‖MF, and $\overline{\kappa }$ an upper bound to the condition number κ 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 $O\left(L\sqrt{n}{\left(n+m\right)}^{2}\overline{\kappa }\enspace \mathrm{log}\left({{\epsilon}}^{-1}\right)\right)$, or with exact methods, at least $O\left(L\sqrt{n}{\left(n+m\right)}^{2.373}\right)$. 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,可以看作是使用量子线性系统算法 [2] 对内点预测-校正算法 [1] 进行量化。我们方法的(最坏情况)工作复杂度是多对数因子,$O\left(L\sqrt{n}\left(n+m\right)\overline{{\Vert}M{{\Vert}}_{\mathrm{F}}}\overline{\kappa }{ {\epsilon}}^{-2}\right)$n成本函数中的变量数,m是约束数,ε -1是目标精度,L是输入数据的位长,$\overline{{\Vert}M{{\Vert}}_{\mathrm{F}}}$出现的线性方程组的 Frobenius 范数的上限 ‖ MF,以及条件数κ$\overline{\kappa}$的上限那些方程组。这表示当问题A的初始矩阵密集时,成本函数中变量的数量n相对于可比较的经典内点算法的量子加速:如果我们用经典算法替换算法的量子部分,例如作为共轭梯度下降,这意味着整个算法具有复杂性,或者至少具有精确的方法。此外,与任何量子线性系统算法相比,本文描述的算法输出解向量的经典描述,以及最优解的值。$O\left(L\sqrt{n}{\left(n+m\right)}^{2}\overline{\kappa }\enspace \mathrm{log}\left({{\epsilon}}^{ -1}\右)\右)$$O\left(L\sqrt{n}{\left(n+m\right)}^{2.373}\right)$

更新日期:2020-10-15
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