Efficient simulation of low-entanglement bosonic Gaussian states in polynomial time

Bosonic Gaussian states appear ubiquitously in quantum optics and condensed matter physics but remain difficult to simulate classically due to the hafnian bottleneck. We present an efficient algorithm that converts pure bosonic Gaussian states into matrix product states (MPSs), with a computational cost governed solely by the entanglement and not by the number of bosonic modes. Our method combines a Gaussian singular value decomposition with a projected-creation-operator mapping that constructs local MPS tensors without computing hafnians. Benchmarking on covariance matrices from the Jiuzhang 2.0 and Jiuzhang 4.0 Gaussian boson sampling experiments demonstrates substantial speedups over previous tensor-network approaches in the low-entanglement regime relevant to lossy devices. The method provides a scalable classical simulation framework for bosonic Gaussian states with limited entanglement and extends the applicability of MPS-based methods to a broad range of bosonic systems.