Title: Streamlined Learning of Hamiltonians, Structure, and Trace Distance for Gaussian States

This study pioneers the investigation of Hamiltonian learning within the context of bosonic Gaussian states at positive temperatures, representing the quantum counterpart to the extensively researched problem of learning Gaussian graphical models. We present efficient protocols for deducing the parameters of the underlying quadratic Hamiltonian, demonstrating favorable scaling in both sample and computational complexity. These protocols operate under the assumption that the system’s temperature, squeezing, displacement, and the maximum degree of the interaction graph remain bounded.

The proposed method relies exclusively on heterodyne measurements, a type of detection that is frequently achievable in experimental settings. Notably, the sample complexity exhibits a logarithmic dependence on the number of modes. Additionally, we demonstrate that the underlying interaction graph can be reconstructed with similar sample efficiency.

Beyond Hamiltonian inference, we apply our methodologies to achieve the first known results for learning Gaussian states in trace distance. This approach achieves a quadratic scaling with respect to precision and polynomial scaling with the number of modes, although it requires specific constraints on the Gaussian states being studied.

The core technical advancements of this work include novel continuity bounds for the covariance and Hamiltonian matrices of Gaussian states, which hold independent significance, alongside a method we term "local inversion." Fundamentally, the local inversion technique enables the accurate deduction of a Gaussian state’s Hamiltonian by estimating only parallel submatrices of the covariance matrix. The size of these submatrices scales with the desired precision rather than the total number of modes. Consequently, this strategy circumvents the necessity for precise global estimates of the covariance matrix, thereby effectively managing sample complexity.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC