Quantum Legendre-Fenchel Transform

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.