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. In particular, we describe "projective" Pfaffian point processes, the cardinality of which has constant parity, almost surely. Finally, the circuits are empirically validated on a classical simulator and on 5-qubit IBM machines.

翻译：DPPs由Macchi于20世纪70年代在量子光学中作为模型提出。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，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？本文聚焦于有限状态空间上的DPP，其定义为由$N\times N$埃尔米特核矩阵参数化的$\{1,\dots,N\}$子集上的分布。经典计算机上的朴素采样包含两个步骤，计算复杂度分别为$\mathcal{O}(N^3)$和$\mathcal{O}(Nr^2)$，其中$r$为核矩阵的秩。本文前半部分主要解释为何当前费米子系统量子模拟的前沿技术已能实现量子DPP采样算法。随后我们改进现有量子电路，并讨论如何将其嵌入从实际核规范出发的完整DPP采样流程。最终结论是：使用$P$个（经典）并行处理器时，可将预处理成本降低至原来的$1/P$，并构建包含$\mathcal{O}(Nr)$个门、深度从$\mathcal{O}(N)$到$\mathcal{O}(r\log N)$（取决于目标机器的量子比特通信约束）的量子电路，用于采样指定DPP。本文还将超导体模拟的现有工作与普法夫点过程相关联——后者是DPP的推广，有望成为机器学习工具库的自然扩展。特别地，我们描述了"投影型"普法夫点过程，其基数几乎必然具有恒定奇偶性。最后，通过经典模拟器和5量子比特IBM量子计算机对电路进行了实验验证。