| name | quantum-gaussian-state-learning |
| description | Sample-optimal learning of bosonic Gaussian quantum states. Provides sharp bounds on sample complexity for characterizing unknown n-mode Gaussian states: Omega(n^3/epsilon^2) for Gaussian measurements, Omega(n^2/epsilon^2) for arbitrary measurements. Proves non-Gaussian measurements required for optimal learning of passive Gaussian states. Use when: quantum state tomography, bosonic Gaussian states, quantum learning theory, sample complexity bounds, quantum sensing benchmarking, Wigner distribution learning, continuous-variable quantum systems. Source: arXiv:2603.18136
|
Quantum Gaussian State Learning
Description
Sample-optimal algorithms for learning bosonic Gaussian quantum states from
minimal copies. Establishes fundamental limits on the number of samples needed
to characterize unknown n-mode Gaussian states to epsilon trace distance.
Sample Complexity Bounds
General Case
- Gaussian measurements: Lower bound Ω(n³/ε²), matching best known upper
bound up to doubly-log energy dependence
- Arbitrary measurements: Lower bound Ω(n²/ε²)
Special Cases
- Pure or passive states: Upper bound Õ(n²/ε²)
- Single-mode, non-entangling Gaussian measurements: Õ(E/ε²) for
non-adaptive schemes; adaptivity is indispensable for energy-independent
scaling
Key Theoretical Results
1. Measurement Type Matters
| State Type | Optimal Measurement | Sample Complexity |
|---|
| Pure Gaussian | Gaussian measurements suffice | Õ(n²/ε²) |
| Passive Gaussian | Non-Gaussian required | Õ(n²/ε²) |
| General Gaussian | Arbitrary measurements | Ω(n²/ε²) |
2. Trace Distance vs Wigner Distribution
Sharp bounds established relating trace distance between Gaussian states to
total variation distance between their Wigner distributions:
d_TV(W_ρ, W_σ) ≤ d_trace(ρ, σ) ≤ C · d_TV(W_ρ, W_σ)
This enables learning via Wigner distribution sampling.
3. Adaptivity is Essential
For single-mode Gaussian states with non-entangling Gaussian measurements:
- Non-adaptive schemes: Ω(E/ε²) — energy-dependent
- Adaptive schemes: nearly energy-independent scaling
- Conclusion: adaptivity is indispensable
Practical Algorithm Design
Step 1: Determine State Type
def choose_measurement_strategy(state_type, n_modes, energy_bound):
if state_type == "pure":
return "gaussian_measurements"
elif state_type == "passive":
return "non_gaussian_measurements"
else:
return "arbitrary_measurements"
Step 2: Compute Required Samples
def required_samples(n_modes, epsilon, measurement_type="arbitrary"):
if measurement_type == "gaussian":
return Omega(n_modes**3 / epsilon**2)
elif measurement_type == "arbitrary":
return Omega(n_modes**2 / epsilon**2)
Step 3: Wigner Distribution Learning
For learning the Wigner distribution to ε total variation distance:
def learn_wigner_distribution(samples, n_modes):
"""
Nearly tight sample complexity bound for learning Wigner distribution
of any Gaussian state to epsilon TV distance.
"""
pass
Applications
- Quantum sensing: Gravitational-wave detection, dark-matter detection
- Quantum communication: Characterizing continuous-variable channels
- Quantum computing: Benchmarking Gaussian state preparation
- Quantum metrology: Optimal parameter estimation strategies
Key Insights
- Non-Gaussian measurements are provably required for optimal learning of
passive Gaussian states — this is a fundamental theoretical result
- Adaptivity matters — non-adaptive schemes cannot achieve
energy-independent sample complexity
- Pure states are easier — Gaussian measurements suffice for nearly
optimal learning of pure Gaussian states
- Wigner-TV connection provides a practical path to learning via phase
space sampling
References
- arXiv:2603.18136 — "Towards sample-optimal learning of bosonic Gaussian
quantum states" (Senrui Chen, Francesco Anna Mele, Marco Fanizza, Alfred Li,
Zachary Mann, Hsin-Yuan Huang, Yanbei Chen, John Preskill, 2026)
