Finding Gaussian Structure in Bosonic States

A fundamental question in physics is: When does classical behavior emerge from quantum systems? Bosonic Gaussian states provide a natural setting to explore this quantum-classical boundary, as they capture both the classical field behavior and the intrinsic quantum nature of light. Here, we address this problem from a learning-theoretic perspective by asking: When are bosonic Gaussian states classical to learn? That is, under what conditions (if any) can an n-mode bosonic Gaussian state be learned with as few samples, and with operations as simple, as are needed to learn a classical 2n-variate Gaussian distribution? We establish a smooth crossover in learnability governed by the state's thermal fluctuations: - Cold Gaussian states are non-classical to learn: When the covariance matrix satisfies $Σ\le(\frac12+O(\frac1n))I$, i.e. close to the vacuum covariance, tomography under single-copy (i.e., non-entangled) measurements fundamentally requires $Ω(n^3)$ copies, strictly exceeding the sample complexity $Θ(n^2)$ of learning classical Gaussian distributions. We show that this hardness persists even when few-copy entangled measurements are allowed. - Warm Gaussian states are classical to learn: When thermal fluctuations exceed the vacuum noise, parameterized by $Σ\ge(\frac12+ν)I$ for any parameter $ν>0$, we prove that single-copy tomography requires $N=Θ\left(n^2\min(n,1+ν^{-1})\right)$ copies. This bound is tight and is achieved by simple, non-adaptive, unentangled heterodyne measurements. Crucially, for $ν=Ω(1)$, the sample complexity drops to $Θ(n^2)$, matching the classical case. Our results tightly characterize a quantum-to-classical crossover in the learnability of bosonic Gaussian states, reveal a novel connection between fundamental physics and statistical learning theory, and have implications for real-world sensing experiments.

Bosonic Gaussian unitaries are fundamental building blocks of central continuous-variable quantum technologies such as quantum-optic interferometry and bosonic error-correction schemes. In this work, we present the first time-efficient algorithm for learning bosonic Gaussian unitaries with a rigorous analysis. Our algorithm produces an estimate of the unknown unitary that is accurate to small worst-case error, measured by the physically motivated energy-constrained diamond distance. Its runtime and query complexity scale polynomially with the number of modes, the inverse target accuracy, and natural energy parameters quantifying the allowed input energy and the unitary's output-energy growth. The protocol uses only experimentally friendly photonic resources: coherent and squeezed probes, passive linear optics, and heterodyne/homodyne detection. We then employ an efficient classical post-processing routine that leverages a symplectic regularization step to project matrix estimates onto the symplectic group. In the limit of unbounded input energy, our procedure attains arbitrarily high precision using only $2m+2$ queries, where $m$ is the number of modes. To our knowledge, this is the first provably efficient learning algorithm for a multiparameter family of continuous-variable unitaries.

Quantum state tomography is sample-starved, and the states one prepares live on a narrow, learnable manifold. A $k{=}0$ prior-only control shows that on concentrated families a prior estimate is already near-optimal, so ``high fidelity at few measurements'' can be family memorization rather than tomography; genuine measurement-efficiency needs a model that conditions on the measurements and demonstrably uses them. On a shared matrix-product-state (MPS) core parameterization we study two routes. ApproachA learns a generative prior over MPS cores with measurement-guided posterior inference (gold-standard-validated, but whose few-measurement accuracy the control shows is largely the prior). ApproachB, our main proposal, is a \emph{fixed-protocol amortized} MPS estimator trained once with a gauge-invariant fidelity loss; we deliberately do not rest it on a permutation-invariant set encoder (a plain MLP matches it). The decisive lever is the measurement design: motivated by the fact that local reduced density matrices determine a $χ$-MPS, conditioning on an \emph{informative local} Pauli set rather than random strings turns a modest, memorization-prone estimator into a high-fidelity one ($\approx\!0.95$, up to $+0.59$ over prior-only, decisively passing a shuffled-measurement control). A dropout ensemble, conformally recalibrated, gives $\approx\!90\%$-coverage intervals -- including for observables never measured, where a shot-based interval does not exist. Quality holds as the system grows (fidelity $0.90$ at $n{=}10$, gain \emph{growing} in $n$; $0.88$ at bond dimension $χ{=}4$), the parameterization is polynomial (native contraction to $20$ qubits), and we close the loop on IBM hardware ($5$ states at $0.97$ from hardware-measured Paulis).