Convex optimization is the powerhouse behind the theory and practice of optimization. We introduce a quantum analogue of unconstrained convex optimization: computing the minimum eigenvalue of a Schr\"odinger operator $h = -\Delta + V $ with convex potential $V:\mathbb R^n \rightarrow \mathbb R_{\ge 0}$ such that $V(x)\rightarrow\infty $ as $\|x\|\rightarrow\infty$. For this problem, we present an efficient quantum algorithm, called the Fundamental Gap Algorithm (FGA), that computes the minimum eigenvalue of $h$ up to error $\epsilon$ in polynomial time in $n$, $1/\epsilon$, and parameters that depend on $V$. Adiabatic evolution of the ground state is used as a key subroutine, which we analyze with novel techniques that allow us to focus on the low-energy space. We apply the FGA to give the first known polynomial-time algorithm for finding the lowest frequency of an $n$-dimensional convex drum, or mathematically, the minimum eigenvalue of the Dirichlet Laplacian on an $n$-dimensional region that is defined by $m$ linear constraints in polynomial time in $n$, $m$, $1/\epsilon$ and the radius $R$ of a ball encompassing the region.

翻译：凸优化是优化理论与实践的强大基础。我们提出了一种无约束凸优化的量子类比：计算具有凸势函数 $V:\\mathbb R^n \\rightarrow \\mathbb R_{\\ge 0}$ 的薛定谔算子 $h = -\\Delta + V$ 的最小本征值，其中 $V(x)\\rightarrow\\infty$ 当 $\\|x\\|\\rightarrow\\infty$。针对该问题，我们提出了一种高效的量子算法——基本能隙算法（FGA），该算法能够在 $n$、$1/\\epsilon$ 以及依赖于 $V$ 的参数的多项式时间内，以误差 $\\epsilon$ 计算 $h$ 的最小本征值。基态的绝热演化被用作关键子程序，我们采用新颖的分析技术将关注点聚焦于低能空间。应用 FGA 算法，我们首次给出了在 $n$、$m$、$1/\\epsilon$ 以及包含该区域的球半径 $R$ 的多项式时间内，求解 $n$ 维凸鼓面最低频率（数学上即由 $m$ 个线性约束定义的 $n$ 维区域上狄利克雷拉普拉斯算子的最小本征值）的多项式时间算法。