DPPs were introduced by Macchi as a model in quantum optics the 1970s. Since then, they have been widely used as models and subsampling tools in statistics and computer science. Most applications require sampling from a DPP, and given their quantum origin, it is natural to wonder whether sampling a DPP on a quantum computer is easier than on a classical one. We focus here on DPPs over a finite state space, which are distributions over the subsets of $\{1,\dots,N\}$ parametrized by an $N\times N$ Hermitian kernel matrix. Vanilla sampling consists in two steps, of respective costs $\mathcal{O}(N^3)$ and $\mathcal{O}(Nr^2)$ operations on a classical computer, where $r$ is the rank of the kernel matrix. A large first part of the current paper consists in explaining why the state-of-the-art in quantum simulation of fermionic systems already yields quantum DPP sampling algorithms. We then modify existing quantum circuits, and discuss their insertion in a full DPP sampling pipeline that starts from practical kernel specifications. The bottom line is that, with $P$ (classical) parallel processors, we can divide the preprocessing cost by $P$ and build a quantum circuit with $\mathcal{O}(Nr)$ gates that sample a given DPP, with depth varying from $\mathcal{O}(N)$ to $\mathcal{O}(r\log N)$ depending on qubit-communication constraints on the target machine. We also connect existing work on the simulation of superconductors to Pfaffian point processes, which generalize DPPs and would be a natural addition to the machine learner's toolbox. Finally, the circuits are empirically validated on a classical simulator and on 5-qubit machines.

翻译：DPP由Macchi于20世纪70年代作为量子光学中的一种模型提出。此后，它们被广泛应用于统计学和计算机科学中的模型与子采样工具。大多数应用需要从DPP中采样，而鉴于其量子起源，自然引发疑问：在量子计算机上采样DPP是否比经典计算机更容易？我们此处关注有限状态空间上的DPP，即由$N\times N$厄米核矩阵参数化的$\{1,\dots,N\}$子集上的分布。经典采样包含两个步骤，计算成本分别为$\mathcal{O}(N^3)$和$\mathcal{O}(Nr^2)$，其中$r$为核矩阵的秩。本文前半部分主要阐释为何费米子系统量子模拟的现有技术已能衍生出量子DPP采样算法。随后我们修改现有量子电路，并讨论如何将其嵌入从实用核规范出发的完整DPP采样流程。核心结论是：借助$P$个（经典）并行处理器，可将预处理成本降低$P$倍，并构建一个含$\mathcal{O}(Nr)$个门的量子电路来采样给定DPP，其深度在$\mathcal{O}(N)$至$\mathcal{O}(r\log N)$之间变化，具体取决于目标机器上的量子比特通信约束。我们还与超导体模拟的现有工作建立联系，将其扩展至Pfaffian点过程——此类过程推广了DPP，将成为机器学习工具箱中的自然补充。最后，我们在经典模拟器和5量子比特机器上对电路进行了实证验证。