On sampling determinantal and Pfaffian point processes on a quantum computer

Abstract

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, . . . ,N} parametrized by an N×N
Hermitian kernel matrix. Vanilla sampling consists in two steps, of respective
costs O(N3) and O(Nr2) 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
O(Nr) gates that sample a given DPP, with depth varying from O(N) to
O(r logN) 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 machines.