We propose a novel quantum algorithm for solving linear optimization problems by quantum-mechanical simulation of the central path. While interior point methods follow the central path with an iterative algorithm that works with successive linearizations of the perturbed KKT conditions, we perform a single simulation working directly with the nonlinear complementarity equations. Combining our approach with iterative refinement techniques, we obtain an exact solution to a linear optimization problem involving $m$ constraints and $n$ variables using at most $\mathcal{O} \left( (m + n) \text{nnz} (A) \kappa (\mathcal{M}) L \cdot \text{polylog} \left(m, n, \kappa (\mathcal{M}) \right) \right)$ elementary gates and $\mathcal{O} \left( \text{nnz} (A) L \right)$ classical arithmetic operations, where $ \text{nnz} (A)$ is the total number of non-zero elements found in the constraint matrix, $L$ denotes binary input length of the problem data, and $\kappa (\mathcal{M})$ is a condition number that depends only on the problem data.

翻译：我们提出了一种新颖的量子算法，通过量子力学模拟中心路径来求解线性优化问题。内点法通过迭代算法遵循中心路径，该算法对扰动后的KKT条件进行逐次线性化处理，而我们直接对非线性互补方程进行单次模拟。将我们的方法与迭代精化技术相结合，可求解包含$m$个约束和$n$个变量的线性优化问题，并得到精确解，所需基本门数量为$\mathcal{O} \left( (m + n) \text{nnz} (A) \kappa (\mathcal{M}) L \cdot \text{polylog} \left(m, n, \kappa (\mathcal{M}) \right) \right)$，经典算术运算量为$\mathcal{O} \left( \text{nnz} (A) L \right)$。其中$\text{nnz} (A)$为约束矩阵中非零元素总数，$L$表示问题数据的二进制输入长度，而$\kappa (\mathcal{M})$为仅取决于问题数据的条件数。