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 discretizaton 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 certain scenarios, such as computing an expectation value of an efficiently-computable observable associated with a Legendre-Fenchel-transformed convex function, the quantum algorithm provides an exponential speedup compared to any classical algorithm.

中文翻译：

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量子勒让德-芬切尔变换

我们提出了一种计算离散勒让德-芬歇尔变换的量子算法。如果访问在 $N$ 点处评估的凸函数，该算法会输出在变换空间中的 $K$ 点处评估的相应离散勒让德-芬歇尔变换的量子力学表示。对于对偶空间的固定正则离散化，预期运行时间缩放为 $O(\sqrt{\kappa}\,\mathrm{polylog}(N,K))$，其中 $\kappa$ 是条件数功能。如果自适应地选择对偶空间的离散化，其中 $K$ 等于 $N$，运行时间减少到 $O(\mathrm{polylog}(N))$。我们解释了如何将所提出的算法扩展到多变量设置并证明查询复杂性的下限，表明我们的量子算法在多对数因子上是最优的。对于某些场景，