We present quantum algorithms for sampling from non-logconcave probability distributions in the form of $\pi(x) \propto \exp(-\beta f(x))$. Here, $f$ can be written as a finite sum $f(x):= \frac{1}{N}\sum_{k=1}^N f_k(x)$. Our approach is based on quantum simulated annealing on slowly varying Markov chains derived from unadjusted Langevin algorithms, removing the necessity for function evaluations which can be computationally expensive for large data sets in mixture modeling and multi-stable systems. We also incorporate a stochastic gradient oracle that implements the quantum walk operators inexactly by only using mini-batch gradients. As a result, our stochastic gradient based algorithm only accesses small subsets of data points in implementing the quantum walk. One challenge of quantizing the resulting Markov chains is that they do not satisfy the detailed balance condition in general. Consequently, the mixing time of the algorithm cannot be expressed in terms of the spectral gap of the transition density, making the quantum algorithms nontrivial to analyze. To overcome these challenges, we first build a hypothetical Markov chain that is reversible, and also converges to the target distribution. Then, we quantified the distance between our algorithm's output and the target distribution by using this hypothetical chain as a bridge to establish the total complexity. Our quantum algorithms exhibit polynomial speedups in terms of both dimension and precision dependencies when compared to the best-known classical algorithms.

翻译：我们提出了量子算法，用于从形式为$\pi(x) \propto \exp(-\beta f(x))$的非对数凹概率分布中采样。其中，$f$可表示为有限和$f(x):= \frac{1}{N}\sum_{k=1}^N f_k(x)$。我们的方法基于慢变马尔可夫链上的量子模拟退火，该链源自未校正Langevin算法，从而避免了函数求值——这在混合建模和多稳态系统中，对于大规模数据集可能带来高昂的计算成本。我们还引入了一个随机梯度预言机，该预言机仅利用小批量梯度来非精确地实现量子游走算子。因此，我们的基于随机梯度的算法在执行量子游走时仅需访问数据点的小规模子集。量化此类马尔可夫链的一个挑战在于，它们通常不满足细致平衡条件。因此，算法的混合时间无法用转移密度的谱间隙表示，这使得量子算法的分析变得复杂。为克服这些困难，我们首先构建一个可逆的假设马尔可夫链，该链同时收敛于目标分布。随后，我们以该假设链为桥梁，量化了算法输出与目标分布之间的距离，从而确立了总复杂度。与已知最优经典算法相比，我们的量子算法在维度和精度依赖关系上均展现出多项式加速。