sample-optimal-gaussian-state-learning

# Sample-Optimal Learning of Bosonic Gaussian States Implement efficient learning protocols for bosonic Gaussian quantum states with provable sample complexity bounds. ## Core Insight For n-mode Gaussian states, the sample complexity depends critically on the measurement strategy: - Gaussian measurements: Ω(n³/ε²) lower bound - Arbitrary measurements: Ω(n²/ε²) lower bound - Pure or passive Gaussian states: Õ(n²/ε²) achievable Non-Gaussian measurements are provably required for optimal learning of passive Gaussian states. ## When to Use - Quantum state tomography for continuous-variable systems - Gravitational-wave detector characterization - Dark-matter detection state estimation - Any Gaussian state learning task with limited samples ## Sample Complexity Bounds ### General n-mode Gaussian States | Measurement Type | Lower Bound | Upper Bound | Notes | |-----------------|-------------|-------------|-------| | Gaussian only | Ω(n³/ε²) | Õ(n³/ε²) | Nearly tight | | Arbitrary | Ω(n²/ε²) | ? | Gap unknown | | Pure Gaussian | Ω(n²/ε²) | Õ(n²/ε²) | Gaussian measurements suffice | | Passive Gaussian | Ω(n²/ε²) | Õ(n²/ε²) | Requires non-Gaussian measurements | ### Single-Mode Case For learning single-mode Gaussian states with non-entangling Gaussian measurements: - Non-adaptive: Θ̃(E/ε²) where E is energy - Adaptive: Energy-independent scaling achievable - Adaptivity is indispensable for nearly energy-independent scaling ## Implementation Pattern ### 1. Gaussian State Parameterization An n-mode Gaussian state is fully characterized by: - First moments (displacement vector): 2n real parameters - Second moments (covariance matrix): 2n² + n real parameters ```python def gaussian_state_params(cov_matrix, displacement): """Parameterize Gaussian state by covariance and displacement.""" # Covariance matrix: 2n × 2n, symmetric, positive definite # Displacement: 2n-dimensional real vector return { 'cov': cov_matrix, 'displacement': displacement, 'n_modes': len(displacement) // 2 } ``` ### 2. Measurement Strategy Selection ```python def select_measurement_strategy(state_type, n_modes, target_eps): """Choose optimal measurement strategy based on state properties.""" if state_type == 'pure_gaussian': return 'gaussian_measurements' # Õ(n²/ε²) achievable elif state_type == 'passive_gaussian': return 'non_gaussian_measurements' # Required for optimality elif state_type == 'general_gaussian': if adaptivity_available: return 'adaptive_non_gaussian' # Best scaling else: return 'gaussian_measurements' # Õ(n³/ε²) ``` ### 3. Adaptive Learning Protocol ```python def adaptive_gaussian_learning(n_modes, n_samples, measurement_fn): """Adaptive protocol for Gaussian state learning.""" estimates = [] for round in range(n_rounds): # Choose measurement based on previous estimates measurement = adapt_measurement(estimates, round) # Perform measurements results = measure_states(measurement, n_samples // n_rounds) # Update estimate estimate = update_estimate(results, measurement) estimates.append(estimate) return combine_estimates(estimates) ``` ## Key Theorems 1. **Gaussian measurement lower bound**: Ω(n³/ε²) samples needed with Gaussian measurements only 2. **Arbitrary measurement lower bound**: Ω(n²/ε²) is fundamental limit 3. **Pure state sufficiency**: Gaussian measurements suffice for pure Gaussian states 4. **Passive state necessity**: Non-Gaussian measurements provably required for passive states 5. **Adaptivity necessity**: Adaptive schemes needed for energy-independent scaling in single-mode case ## Verification Steps 1. Verify sample complexity matches bounds for known cases 2. Check that pure state learning converges with Gaussian measurements 3. Validate that passive state learning requires non-Gaussian measurements 4. Compare adaptive vs non-adaptive performance for single-mode case ## References - Chen, Mele, Fanizza, Li, Mann, Huang, Chen, Preskill (2026): "Towards sample-optimal learning of bosonic Gaussian quantum states" (arXiv:2603.xxxxx, quant-ph, cs.IT, cs.LG, math-ph)