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：行列式点过程（DPP）由Macchi于1970年代作为量子光学模型引入。此后，它们被广泛用作统计学和计算机科学中的模型及子采样工具。大多数应用需要从DPP中采样，且鉴于其量子起源，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易实现。本文聚焦于有限状态空间上的DPP，即由$N\times N$厄密核矩阵参数化的$\{1,\dots,N\}$子集上的分布。经典计算机上的朴素采样包含两步，分别需要$\mathcal{O}(N^3)$和$\mathcal{O}(Nr^2)$次操作，其中$r$为核矩阵的秩。本文第一部分的主要内容在于解释为何费米子系统量子模拟的现有技术已能实现量子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具体核规范出发的完整DPP采样流程。关键结论是：使用$P$个（经典）并行处理器，可降共享预处理成本$P$倍，并构建包含$\mathcal{O}(Nr)$个门、深度取决于目标机器量子比特通信约束（从$\mathcal{O}(N)$到$\mathcal{O}(r\log N)$）的量子电路，从而实现对给定DPP的采样。此外，我们将超导体模拟的现有工作与普法夫点过程联系起来——这类过程推广了DPP，或将成为机器学习工具集的自然补充。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经验验证。