Towards sample-optimal learning of bosonic Gaussian quantum states

Abstract

Continuous-variable systems enable key quantum technologies in computation, communication, and sensing. Bosonic Gaussian states emerge naturally in various such applications, including gravitational-wave and dark-matter detection. A fundamental question is how to characterize an unknown bosonic Gaussian state from as few samples as possible. Despite decades-long exploration, the ultimate efficiency limit remains unclear. In this work, we study the necessary and sufficient number of copies to learn an $n$-mode Gaussian state, with energy less than $E$, to $\varepsilon$ trace distance with high probability. We prove a lower bound of $Ω(n^3/\varepsilon^2)$ for Gaussian measurements, matching the best known upper bound up to doubly-log energy dependence, and $Ω(n^2/\varepsilon^2)$ for arbitrary measurements. We further show an upper bound of $\widetilde{O}(n^2/\varepsilon^2)$ given that the Gaussian state is promised to be either pure or passive. Interestingly, while Gaussian measurements suffice for nearly optimal learning of pure Gaussian states, non-Gaussian measurements are provably required for optimal learning of passive Gaussian states. Finally, focusing on learning single-mode Gaussian states via non-entangling Gaussian measurements, we provide a nearly tight bound of $\widetildeΘ(E/\varepsilon^2)$ for any non-adaptive schemes, showing adaptivity is indispensable for nearly energy-independent scaling. As a byproduct, we establish sharp bounds on the trace distance between Gaussian states in terms of the total variation distance between their Wigner distributions, and obtain a nearly tight sample complexity bound for learning the Wigner distribution of any Gaussian state to $\varepsilon$ total variation distance. Our results greatly advance quantum learning theory in the bosonic regimes and have practical impact in quantum sensing and benchmarking applications.

Figures (3)

• Figure 1: An illustration of the Gaussian state ensembles used in our lower bound proof. The left two boxes depict a total of $n$ initial single-mode Gaussian states. In this specific example, they consist of $n/9$ pure squeezed states and $8n/9$ vacuum states, which is used for the proof of Theorem \ref{['th:main_lo_any']}. For the other lower bound proofs, we will make different choices of the initial states. The right box depicts a Gaussian unitary consisting solely of beam splitters. Each choice of the unitary defines an output Gaussian state of the ensemble, and we have $2^{\Omega(n^2)}$ different choices in total. This second part is the same for all ensembles that we use in different lower bound proofs.
• Figure 2: Comparison of three different single-copy Gaussian measurement schemes. The left box shows the Wigner function of an unknown single-mode Gaussian state $\rho$. On the right show the Wigner functions of the Gaussian measurement seeds for the first few copies of $\rho$ of three different protocols. (a) Heterodyne measurements, with sample complexity $\Theta(\bar{E}^2/\varepsilon^2)$, whose seeds are always the vacuum state. (b) Angle-randomized homodyne, which is a nearly optimal non-adaptive scheme proposed in Theorem \ref{['th:main_nonada']} with sample complexity $\widetilde{\Theta}(\bar{E}/\varepsilon^2)$, whose seeds are angle-randomized squeezed states (which approximate the ideal homodyne). (c) Adaptive general-dyne measurements as proposed in bittel2025energy, which achieves nearly energy-independent sample complexity of $\widetilde{\Theta}(1/\varepsilon^2)$. The seeds are recursively and adaptively refined to be aligned with $\rho$ so as to maximize sensitivity.
• Figure S1: Example of Algorithm \ref{['alg:non-ada']} on learning a one-mode zero-mean Gaussian state. The state is shown on the left hand side with parameter $E=60$, $b=1/2E$, $a=E/2$, and $\theta=\pi/4$. Here $\phi$ represents the angle of homodyne measurements (with added variance $1/E$). On the right hand side, the solid line is the true quadrature-projected variance $\Sigma_\phi$ scanned over $\phi\in[0,\pi)$. We sample $K=1000$ uniformly random homodyne angles and measure each with $500$ shots to obtain the dots (using Lemma \ref{['le:homodyne-concen']}). The inset illustrates how Algorithm \ref{['alg:non-ada']} will pick three sampled angles $\phi_\mathrm{min}$, $\phi_-$ and $\phi_+$, which have sufficiently small shot noise and are sufficiently separated, to solve for the model parameters.

Theorems & Definitions (67)

• Theorem 1: Nearly-optimal learning of Gaussian Wigner distributions
• proof : Proof sketch
• Lemma 1: Sharp bounds between trace distance and Wigner TV distance for Gaussian states
• Theorem 2: Lower bound with classical measurements
• proof : Proof Sketch for Theorem \ref{['th:main_lo_gaussian']}
• Theorem 3: Lower bound with arbitrary measurements
• proof : Proof Sketch for Theorem \ref{['th:main_lo_any']}
• Theorem 4: Lower bound with heterodyne measurements
• Theorem 5: Nearly-optimal learning of pure Gaussian states
• proof : Proof Sketch for Theorem \ref{['th:main_up_pure']}