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arXiv:2607.27534

Efficient learning of bosonic unitaries beyond the Gaussian class

Xiaobin Zhao, Quntao Zhuang

quant-phSubmitted Jul 30, 2026• Announced Jul 31, 2026• 0 citations
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Abstract

Multimode quantum processes are generally difficult to learn, due to the large dimensionality and complex entanglement structure beyond the Gaussian class. Here, we show that the fundamental obstruction is not non-Gaussianity itself, but the buildup of irreducible multimode non-Gaussian correlations. We establish a tractability frontier for bosonic unitary learning: a general $m$-mode unitary with input energy at most $E$ per mode requires at least $Ω(E^{2m})$ channel uses, whereas two broad non-Gaussian families---$t$-doped Gaussian unitaries and Gaussian-entanglable unitaries---can be learned with resources polynomial in $m$. The latter can exhibit both extensive non-Gaussianity and strong multimode entanglement. Our forward-only protocols use coherent-state probes, Gaussian operations, local heterodyne detection, and classical post-processing to identify the global Gaussian mixing and reduce the remaining task to single- or few-mode learning. The analysis also yields a multimode quantum Darmois--Skitovich theorem showing that mode-spreading passive networks preserve product structure only for Gaussian input states, an almost-sure activation theorem for non-Gaussian processes showing that non-Gaussian unitaries yield non-Gaussian outputs for almost all coherent input states, and a method for learning unitaries from uncalibrated coherent probes. Our results identify that complexity of learning arises from irreducible mixing of non-Gaussianity and entanglement, rather than either resource alone.

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Multimode quantum processes are generally difficult to learn, due to the large dimensionality and complex entanglement structure beyond the Gaussian class. Here, we show that the fundamental obstruction is not non-Gaussianity itself, but the buildup of irreducible multimode non-Gaussian correlations. We establish a tractability frontier for bosonic unitary learning: a general mmm-mode unitary with input energy at most EEE per mode requires at least Ω(E2m)Ω(E^{2m})Ω(E2m) channel uses, whereas two broad non-Gaussian families---ttt-doped Gaussian unitaries and Gaussian-entanglable unitaries---can be learned with resources polynomial in mmm. The latter can exhibit both extensive non-Gaussianity and strong multimode entanglement. Our forward-only protocols use coherent-state probes, Gaussian operations, local heterodyne detection, and classical post-processing to identify the global Gaussian mixing and reduce the remaining task to single- or few-mode learning. The analysis also yields a multimode quantum Darmois--Skitovich theorem showing that mode-spreading passive networks preserve product structure only for Gaussian input states, an almost-sure activation theorem for non-Gaussian processes showing that non-Gaussian unitaries yield non-Gaussian outputs for almost all coherent input states, and a method for learning unitaries from uncalibrated coherent probes. Our results identify that complexity of learning arises from irreducible mixing of non-Gaussianity and entanglement, rather than either resource alone.
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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 nnn-mode Gaussian state, with energy less than EEE, to ε\varepsilonε trace distance with high probability. We prove a lower bound of Ω(n3/ε2)Ω(n^3/\varepsilon^2)Ω(n3/ε2) for Gaussian measurements, matching the best known upper bound up to doubly-log energy dependence, and Ω(n2/ε2)Ω(n^2/\varepsilon^2)Ω(n2/ε2) for arbitrary measurements. We further show an upper bound of O~(n2/ε2)\widetilde{O}(n^2/\varepsilon^2)O(n2/ε2) 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 Θ~(E/ε2)\widetildeΘ(E/\varepsilon^2)Θ(E/ε2) 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 ε\varepsilonε total variation distance. Our results significantly advance quantum learning theory in the bosonic regimes and have practical impact in quantum sensing and benchmarking applications.
Entanglement and nonclassicality are two fundamental features of quantum mechanics which serve as resources in quantum information processing tasks and are tightly connected by two-mode mixing. In this work, we show that the connection between entanglement and nonclassicality extends well beyond the two-mode mixing setting by considering the case of arbitrary multimode Gaussian evolutions. Using the Narcowich-Wigner spectrum as our main technical tool to quantify single-mode phase-space nonclassicality, we derive sufficient conditions for a single-mode state to generate NPT entanglement after a multimode Gaussian unitary evolution, which both tighten and generalize existing criteria. As an application, we obtain simple single-mode protocols for quantitatively testing the entanglement structure of multimode quantum states over arbitrary bipartitions. Our results identify the Narcowich-Wigner spectrum as a powerful tool for investigating fundamental properties of bosonic quantum states.
Quantum state tomography typically requires exponentially many copies of a quantum state, due to the complex correlations present in large systems. We show that, for bosonic systems, the scaling is completely determined by the nature of these correlations. Motivated by the Hong-Ou-Mandel effect and boson sampling, we define Gaussian-entanglable (GE) states, produced by generalized interference between separable bosonic modes. GE states greatly extend the Gaussian family, encompassing separable states, multi-mode Gottesman-Kitaev-Preskill codes, entangled cat states, and boson-sampling outputs-resources for error correction and quantum advantage. We prove that any pure GE state of m modes can be learned efficiently, requiring only poly(m) copies, via a protocol based on Gaussian unitaries, local tomography, and classical post-processing; for boson-sampling states, no Gaussian unitaries are needed. For states outside GE, we define an operational monotone-the minimal number of ancillary modes needed to make them GE-which exactly characterizes the exponential tomography overhead. We also show that deterministic generation of NOON states with N ≥ 3 via two-mode interference is impossible.
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≤(12+O(1n))IΣ\le(\frac12+O(\frac1n))IΣ≤(21​+O(n1​))I, i.e. close to the vacuum covariance, tomography under single-copy (i.e., non-entangled) measurements fundamentally requires Ω(n3)Ω(n^3)Ω(n3) copies, strictly exceeding the sample complexity Θ(n2)Θ(n^2)Θ(n2) 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≥(12+ν)IΣ\ge(\frac12+ν)IΣ≥(21​+ν)I for any parameter ν>0ν>0ν>0, we prove that single-copy tomography requires N=Θ(n2min⁡(n,1+ν−1))N=Θ\left(n^2\min(n,1+ν^{-1})\right)N=Θ(n2min(n,1+ν−1)) copies. This bound is tight and is achieved by simple, non-adaptive, unentangled heterodyne measurements. Crucially, for ν=Ω(1)ν=Ω(1)ν=Ω(1), the sample complexity drops to Θ(n2)Θ(n^2)Θ(n2), 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.
We present a framework for studying linear bosonic non-Gaussian channels. Our emphasis is on a class of channels that we call photon-added Gaussian channels and these are experimentally viable with current quantum-optical technologies. A strong motivation for considering these channels is the fact that it is compulsory to go beyond the Gaussian regime for numerous tasks in continuous-variable quantum information processing including universal quantum computation. These channels are obtained by extending Gaussian channels with photon addition applied to the ancilla ports giving rise to a one-parameter family of non-Gaussian channels indexed by photon number n≥1n \geq 1n≥1. We derive its operator-sum representation which is indispensable in the present context. We observe that these channels are Fock-preserving (or coherence non-generating in the Fock basis). Further, noisy Gaussian channels can be expressed as a convex mixture of these non-Gaussian channels. We then report examples of activation of nonclassicality, using this method of photon-addition, at outputs of channels that would otherwise output only classical states. We also derive many structure theorems and information theoretic properties of these channels.