We present a quantum algorithm to compute the discrete Legendre-Fenchel transform. Given access to a convex function evaluated at $N$ points, the algorithm outputs a quantum-mechanical representation of its corresponding discrete Legendre-Fenchel transform evaluated at $K$ points in the transformed space. For a fixed regular discretization of the dual space the expected running time scales as $O(\sqrt{\kappa}\,\mathrm{polylog}(N,K))$, where $\kappa$ is the condition number of the function. If the discretization of the dual space is chosen adaptively with $K$ equal to $N$, the running time reduces to $O(\mathrm{polylog}(N))$. We explain how to extend the presented algorithm to the multivariate setting and prove lower bounds for the query complexity, showing that our quantum algorithm is optimal up to polylogarithmic factors. For multivariate functions with $\kappa=1$, the quantum algorithm computes a quantum-mechanical representation of the Legendre-Fenchel transform at $K$ points exponentially faster than any classical algorithm can compute it at a single point.
翻译:我们提出一个量子算法来计算离散的图例- Fenchel 变换。 如果访问在美元点上评估的 convex 函数, 算法输出出相应的离散图例- Fenchel 变换的量子机械表示值, 在变换空间中, 以 $K 表示, 以 $ 为单位。 对于双层空间的离散性函数, $\ kappa=1$ 为单位。 如果以 $ 等于 $ 的适应性选择双层空间, 运行时间减为 $O ( mathrm{polylog} (N)) 。 对于双层空间的固定正常分解, 我们解释如何将显示的算法扩展至多变量设置, 并证明查询复杂度的界限较低, 表明我们的量子算法最符合多元的系数。 对于以 $\ kapa=1$ 的多变量函数, 量衡算算法在任何Gligalus- fnalex- Fnchalgal orational orgal 点上快速的量- glaslegal- glaslegal- gal- gas.