The log log jam in Gaussian state tomography

• URL: http://arxiv.org/abs/2607.12983v1
• Date: Tue, 14 Jul 2026 17:25:43 GMT
• Title: The log log jam in Gaussian state tomography
• Abstract summary: We show that any protocol that uses Gaussian measurements, even entangled or adaptively chosen ones, must incur a $log log E$ dependence.<n>With highly entangled, non-Gaussian measurements, one can learn $n$-mode pure Gaussian states with $O(n2 / 2)$ samples, independent of $E$.<n>Our results shed new light on the intriguing interplay between adaptivity, entanglement, and magic in quantum learning.
• Score: 30.395638404311537
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Unlike in finite dimensions, quantum information in continuous-variable systems has the peculiar feature that without imposing physical constraints, the sample complexity of state tomography can be unbounded. Remarkably, this is even the case for state-of-the-art protocols for learning Gaussian states, which have finite-dimensional descriptions: the best known rates scale with $\log \log E$, where $E$ is the energy of the system. We prove this is not an artifact of existing analyses, but a fundamental limitation of the measurements used. We show: (1) Any protocol that uses Gaussian measurements, even entangled or adaptively chosen ones, must incur a $\log \log E$ dependence. This answers an open question posed by a number of previous works. (2) There is a smooth tradeoff between the number of rounds of adaptivity and the energy dependence, and we give a matching protocol achieving this interpolated rate. (3) With highly entangled, non-Gaussian measurements, one can learn $n$-mode pure Gaussian states with $O(n^2 / ε^2)$ samples, independent of $E$. This answers an open question posed by Chen et al. (4) A simple protocol based on the single-copy canonical phase POVM of Holevo and Helstrom learns single-mode pure Gaussian states with $O(1/ε^2)$ samples, again independent of $E$. Our results clarify the role of energy in bosonic state tomography and shed new light on the intriguing interplay between adaptivity, entanglement, and magic in quantum learning.

Related papers

• Dimension-Free Polylogarithmic Quantum Shadow Tomography from Sequential Pretty-Good Measurements [42.01011095596403]
  We give a quantum protocol for shadow tomography with sample complexity.<n>Our approach first reduces the general shadow-tomography problem to a finite-ensemble estimation problem.<n>A refined tail analysis of the resulting estimation error yields simultaneous accuracy guarantees for all observables.
  arXiv  Detail & Related papers  (2026-08-06T17:51:13Z)
• Non-Gaussian Entanglement Hierarchy Based on the Schmidt Number [2.999888908665658]
  Non-Gaussian entanglement is a promising resource in various quantum tasks.<n>We introduce a quantitative witness $E_rm NG$ in bipartite bosonic systems.<n>We benchmark the framework with some paradigmatic non-Gaussian states, such as NOON states and squeezed Kerr states.
  arXiv  Detail & Related papers  (2026-05-18T16:15:17Z)
• Towards sample-optimal learning of bosonic Gaussian quantum states [7.657727267684591]
  A fundamental question is how to characterize an unknown bosonic Gaussian state from as few samples as possible.<n>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.<n>Our results greatly advance quantum learning theory in the bosonic regimes and have practical impact in quantum sensing and benchmarking applications.
  arXiv  Detail & Related papers  (2026-03-18T18:00:00Z)
• FFT-Accelerated Auxiliary Variable MCMC for Fermionic Lattice Models: A Determinant-Free Approach with $O(N\log N)$ Complexity [52.3171766248012]
  We introduce a Markov Chain Monte Carlo (MCMC) algorithm that dramatically accelerates the simulation of quantum many-body systems.<n>We validate our algorithm on benchmark quantum physics problems, accurately reproducing known theoretical results.<n>Our work provides a powerful tool for large-scale probabilistic inference and opens avenues for physics-inspired generative models.
  arXiv  Detail & Related papers  (2025-10-13T07:57:21Z)
• Fast quantum measurement tomography with dimension-optimal error bounds [0.6144680854063935]
  We present a two-step protocol for quantum measurement tomography that is light on classical co-processing cost.<n>We show that the protocol requires $mathcalO(d3 L ln(d)/epsilon2)$ samples to achieve error $epsilon$ in worst-case distance.<n>We also complement our findings with empirical performance studies carried out on a noisy superconducting quantum computer with flux-tunable transmon qubits.
  arXiv  Detail & Related papers  (2025-07-06T18:35:07Z)
• Learning quantum states of continuous variable systems [4.05533173496439]
  Quantum state tomography is aimed at deriving a classical description of an unknown state from measurement data. We prove that tomography of continuous-variable systems is extremely inefficient in terms of time resources. We also prove that tomography of Gaussian states is efficient.
  arXiv  Detail & Related papers  (2024-05-02T16:19:55Z)
• Gaussian Entanglement Measure: Applications to Multipartite Entanglement of Graph States and Bosonic Field Theory [50.24983453990065]
  An entanglement measure based on the Fubini-Study metric has been recently introduced by Cocchiarella and co-workers. We present the Gaussian Entanglement Measure (GEM), a generalization of geometric entanglement measure for multimode Gaussian states. By providing a computable multipartite entanglement measure for systems with a large number of degrees of freedom, we show that our definition can be used to obtain insights into a free bosonic field theory.
  arXiv  Detail & Related papers  (2024-01-31T15:50:50Z)
• Theory of free fermions under random projective measurements [43.04146484262759]
  We develop an analytical approach to the study of one-dimensional free fermions subject to random projective measurements of local site occupation numbers. We derive a non-linear sigma model (NLSM) as an effective field theory of the problem.
  arXiv  Detail & Related papers  (2023-04-06T15:19:33Z)
• Classical shadow tomography for continuous variables quantum systems [13.286165491120467]
  We introduce two experimentally realisable schemes for obtaining classical shadows of CV quantum states. We are able to overcome new mathematical challenges due to the infinite-dimensionality of CV systems. We provide a scheme to learn nonlinear functionals of the state, such as entropies over any small number of modes.
  arXiv  Detail & Related papers  (2022-11-14T17:56:29Z)
• Fermionic tomography and learning [0.5482532589225553]
  Shadow tomography via classical shadows is a state-of-the-art approach for estimating properties of a quantum state. We show how this approach leads to efficient estimation protocols for the fidelity with a pure fermionic Gaussian state. We use these tools to show that an $n$-electron, $m$-mode Slater can be learned to within $epsilon$ fidelity given $O(n2 m7 log(m / delta) / epsilon2)$ samples of the determinant.
  arXiv  Detail & Related papers  (2022-07-29T17:12:53Z)
• Deterministic Gaussian conversion protocols for non-Gaussian single-mode resources [58.720142291102135]
  We show that cat and binomial states are approximately equivalent for finite energy, while this equivalence was previously known only in the infinite-energy limit. We also consider the generation of cat states from photon-added and photon-subtracted squeezed states, improving over known schemes by introducing additional squeezing operations.
  arXiv  Detail & Related papers  (2022-04-07T11:49:54Z)
• Tightening the Dependence on Horizon in the Sample Complexity of Q-Learning [59.71676469100807]
  This work sharpens the sample complexity of synchronous Q-learning to an order of $frac|mathcalS|| (1-gamma)4varepsilon2$ for any $0varepsilon 1$. Our finding unveils the effectiveness of vanilla Q-learning, which matches that of speedy Q-learning without requiring extra computation and storage.
  arXiv  Detail & Related papers  (2021-02-12T14:22:05Z)
• Sample Complexity of Asynchronous Q-Learning: Sharper Analysis and Variance Reduction [63.41789556777387]
  Asynchronous Q-learning aims to learn the optimal action-value function (or Q-function) of a Markov decision process (MDP) We show that the number of samples needed to yield an entrywise $varepsilon$-accurate estimate of the Q-function is at most on the order of $frac1mu_min (1-gamma)5varepsilon2+ fract_mixmu_min (1-gamma)$ up to some logarithmic factor.
  arXiv  Detail & Related papers  (2020-06-04T17:51:00Z)
• Quantum Algorithms for Simulating the Lattice Schwinger Model [63.18141027763459]
  We give scalable, explicit digital quantum algorithms to simulate the lattice Schwinger model in both NISQ and fault-tolerant settings. In lattice units, we find a Schwinger model on $N/2$ physical sites with coupling constant $x-1/2$ and electric field cutoff $x-1/2Lambda$. We estimate observables which we cost in both the NISQ and fault-tolerant settings by assuming a simple target observable---the mean pair density.
  arXiv  Detail & Related papers  (2020-02-25T19:18:36Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.