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

arXiv:2603.18136 · quant-ph, cs.IT, cs.LG, math-ph, math.IT, math.MP · Submitted 2026-03-18 · Read on arXiv

quant-ph, cs.IT, cs.LG, math-ph, math.IT, math.MP

Submitted: 2026-03-18

Updated: 2026-09-13

Comments: v2: Results on passive Gaussian states improved. Minor errors and typos corrected. Notes added on subsequent works

Code: https://github.com/ccanonne/probabilitydistributiontoolbox

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Terminology

• Trace — 93× in this paper · explained in 2 episodes
• Trace Distance — 73× in this paper · explained in 3 episodes
• Sample Complexity — 50× in this paper · explained in 15 episodes
• Wigner function — 22× in this paper · explained in 2 episodes
• Adaptivity — 19× in this paper · explained in 1 episode
• Fidelity — 14× in this paper · explained in 2 episodes
• Entanglement — 6× in this paper · explained in 3 episodes
• Hilbert Space — 4× in this paper · explained in 2 episodes
• Benchmarking — 3× in this paper · explained in 3 episodes
• Adam — 2× in this paper · explained in 2 episodes
• Hamiltonian Learning — 2× in this paper · explained in 2 episodes
• Sample Complexity Scaling — 2× in this paper · explained in 2 episodes
• Accuracy — 1× in this paper · explained in 1 episode
• Additivity — 1× in this paper · explained in 1 episode
• Fisher Information — 1× in this paper · explained in 3 episodes
• Gibbs State — 1× in this paper · explained in 2 episodes
• Reconstruction Error — 1× in this paper · explained in 2 episodes
• Shadow Tomography — 1× in this paper · explained in 4 episodes

Sources

• Optimal convergence rates in trace distance and relative entropy for the quantum central limit theorem
• Energy-independent tomography of Gaussian states
• Adaptivity can help exponentially for shadow tomography
• Instance-optimal high-precision shadow tomography with few-copy measurements: A metrological approach
• Exponential advantage in continuous-variable quantum state learning
• The total variation distance between high-dimensional Gaussians with the same mean
• Efficient learning of bosonic Gaussian unitaries
• Optimal Fidelity Estimation from Binary Measurements for Discrete and Continuous Variable Systems
• Efficient Hamiltonian, structure and trace distance learning of Gaussian states
• Precision Bounds on Continuous-Variable State Tomography using Classical Shadows
• Encoding a qubit in an oscillator
• Random purification channel made simple
• Is it Gaussian? Testing bosonic quantum states
• On estimates of trace-norm distance between quantum Gaussian states
• Higher moment theory and learnability of bosonic states
• Learning and simulating bosonic systems via finite-energy locality
• Achievable rates in non-asymptotic bosonic quantum communication
• Calculating trace distances of bosonic states in Krylov subspace
• Random purification channel for passive Gaussian bosons
• Nonparametric Learning Non-Gaussian Quantum States of Continuous Variable Systems

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