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Article Dans Une Revue Journal of Physics A: Mathematical and Theoretical Année : 2024

On sampling determinantal and Pfaffian point processes on a quantum computer

Résumé

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(N^3) and O(N r^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 O(N r) gates that sample a given DPP, with depth varying from O(N) to 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.
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hal-04107251 , version 1 (26-05-2023)

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Rémi Bardenet, Michaël Fanuel, Alexandre Feller. On sampling determinantal and Pfaffian point processes on a quantum computer. Journal of Physics A: Mathematical and Theoretical, 2024, 57 (5), pp.055202. ⟨10.1088/1751-8121/ad1b75⟩. ⟨hal-04107251⟩
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