Efficient sampling from shallow Gaussian quantum-optical circuits with local interactions

Understand

We prove that a classical computer can efficiently sample from the photon-number probability distribution of a Gaussian state prepared by using an optical circuit that is shallow and local.

• Our work generalizes previous known results for qubits to the continuous-variable domain.
• The key to our proof is the observation that the adjacency matrices characterizing the Gaussian states generated by shallow and local circuits have small bandwidth.
• To exploit this structure, we devise fast algorithms to calculate loop hafnians of banded matrices.