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, 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, an upper bound to the Frobenius norm of the linear systems of equations that appear, ‖M‖_F, and 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 , or with exact methods, at least . 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.

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，n是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，出现的线性方程组的 Frobenius 范数的上限 ‖ M ‖ _F，以及条件数κ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。