What resources are needed for optimal learning of bosonic Gaussian states?

Senrui Chen; Francesco Anna Mele; Marco Fanizza; Alfred Li; Zachary Mann; Hsin-Yuan Huang; Yanbei Chen; John Preskill

quant-phMar 18, 2026

What resources are needed for optimal learning of bosonic Gaussian states?

Authors: Senrui Chen, Francesco Anna Mele, Marco Fanizza, Alfred Li, Zachary Mann, Hsin-Yuan Huang, Yanbei Chen, John Preskill

Organizations: Institute for Quantum Information and Matter, Caltech, Pasadena, CA 91125, USA · Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy · Inria, Télécom Paris – LTCI, Institut Polytechnique de Paris, France

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 significantly advance quantum learning theory in the bosonic regimes and have practical impact in quantum sensing and benchmarking applications.

Figures & tables

Class of states	Class of measurements	Bounds
All Gaussian	Classical (including Gaussian)	$\widetilde{O}(n^{3}/\varepsilon^{2})$ [ BMEM25 ] , ${\Omega}(n^{3}/\varepsilon^{2})$ Thm. 2
	Any	$\widetilde{O}(n^{3}/\varepsilon^{2})$ [ BMEM25 ] , ${\Omega}(n^{2}/\varepsilon^{2})$ Thm. 3
	Heterodyne	${O}(\bar{E}^{2}n^{3}/\varepsilon^{2})$ [ BMEM25 ] , ${\Omega}(\bar{E}^{2}n^{3}/\varepsilon^{2})$ Thm. 4
Pure Gaussian	Any or Gaussian	$\widetilde{\Theta}(n^{2}/\varepsilon^{2})$ Thm. 3 , 5
Passive Gaussian	Any	$\widetilde{\Theta}(n^{2}/\varepsilon^{2})$ Thm. 6
Passive Gaussian	Classical (including Gaussian)	${\Theta}(n^{3}/\varepsilon^{2})$ Thm. 2 , 6

Table 1: Summary of results. The task is learning an

n

-mode Gaussian states

\rho(\mu,\Sigma)

\varepsilon

trace distance closeness with probability at least

2/3

, given the promise that

\|\Sigma\|_{\mathrm{op}}\leq\bar{E}

. Here

\widetilde{O}

\widetilde{\Omega}

\widetilde{\Theta}

, suppress

\log n

\log\varepsilon^{-1}

, and

\mathrm{poly}\log\log{\bar{E}}

factors.

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 3 . 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 7 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 [ BMEM25 ] , 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.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Figure S1: Example of Algorithm 1 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 E.2 ). The inset illustrates how Algorithm 1 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.

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Quantum state tomography reconstructs unknown quantum states from measurement data, enabling characterization and verification of quantum systems.

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Sep 22, 2026

When are bosonic Gaussian states classical to learn?

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.

Senrui Chen, Antonio Anna Mele, Francesco Anna Mele +1

Institute for Quantum Information and Matter, Caltech, Pasadena, CA 91125, USA · Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany

Quantum State Tomography Statistical Learning Theory Information-Theoretic Lower Bounds

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Oct 5, 2026

Finding Gaussian Structure in Bosonic States

We study agnostic tomography of pure bosonic Gaussian states: given copies of an arbitrary

n

-mode bosonic state

ρ

, the goal is to output a pure Gaussian state whose infidelity with

ρ

is at most

\mathrm{opt} + ε

, where

\mathrm{opt}

is the minimum infidelity achievable by any pure Gaussian state. We give efficient protocols achieving this in both the high and low fidelity regimes. When

\mathrm{opt}

is below some universal constant, our protocol has runtime and copy complexity which is strongly polynomial in

n, 1/ε

and

\log \log E

, where

E

is the energy of the closest pure Gaussian state. For arbitrary

\mathrm{opt}

, our protocol uses

(n+1)^{\mathrm{poly}(1/ε)} \mathrm{poly}\left(1+\log\log(E)\right)

copies and runtime. As a corollary, we obtain the first truly tolerant Gaussianity testing protocol for distinguishing whether

\mathrm{opt} > c + ε

\mathrm{opt} < c - ε

, for any threshold

c\in(0,1)

. We also prove

\mathrm{poly}(n,1/ε)

runtime is impossible, unless

\mathrm{NP}\subseteq\mathrm{BQP}

. Our protocols follow a shared paradigm: first, we iteratively use general Gaussian measurements combined with techniques from classical robust statistics to obtain a good warm start estimate, then we leverage non-Gaussian measurements to refine this warm start using convex and non-convex optimization methods. Interestingly, we prove that non-Gaussian measurements are necessary to match the strong agnostic guarantees we obtain, and in fact these guarantees are provably superior to what is possible for robustly estimating classical Gaussians.

Alvan Arulandu, Sitan Chen, Ziyun Chen +2

Harvard University · University of Washington · Stanford University

Quantum State Tomography Statistical Hypothesis Testing

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Oct 7, 2025

Efficient learning of bosonic Gaussian unitaries

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.

Marco Fanizza, Vishnu Iyer, Junseo Lee +2

Inria, Télécom Paris - LTCI, Institut Polytechnique de Paris · Department of Computer Science, The University of Texas at Austin · Institute of Computer Technology, Seoul National University +2

Quantum Machine Learning

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