  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.
