fermionic-bell-sampling-non-gaussianity

# Fermionic Non-Gaussianity via Bell Sampling ## Context Based on arXiv:2606.05066 (Tarabunga, Jun 2026). Fermionic non-Gaussianity is an essential resource for unlocking the full computational power of fermionic quantum platforms. ## Core Methodology 1. **Define the operator** Λ = Σ_{j=1}^{2n} γ_j ⊗ γ_j on two copies of an n-mode fermionic state, accessible via Bell sampling 2. **Bridge degree**: Introduce the bridge degree as the largest eigenvalue sector of Λ populated by two copies of the state — a novel non-Gaussianity monotone 3. **Prove monotonicity**: Show bridge degree is non-increasing under post-selected Gaussian protocols 4. **Derive no-go theorems**: Use monotonicity to prove stronger Gaussian conversion impossibility results than previously known monotones 5. **Irreversibility**: Show the resource theory of fermionic non-Gaussianity is irreversible in the exact-conversion setting 6. **Efficient witnessing**: The bridge degree is efficiently witnessed through Bell sampling with polynomial sample complexity 7. **Two algorithmic primitives**: - Two-copy Gaussianity test with perfect completeness (optimal among two-copy tests) - Test for state 2-design property of matchgate-invariant ensembles ## Implementation Steps 1. Prepare two copies of the target fermionic state 2. Perform Bell sampling to access the eigenvalue structure of Λ 3. Compute the bridge degree from the Bell sampling outcomes 4. Compare against known Gaussian states to certify non-Gaussianity 5. Use the approximate variant with efficiently measurable lower bound for experimental certification ## Key Results - Bridge degree is easy to compute and efficiently witnessed through Bell sampling - Lower-bounds the non-Gaussian gate complexity of state preparation - Controls the non-Gaussian gate complexity of producing quantum state designs - Extends naturally to mixed states via Choi–Jamiołkowski isomorphism - Provides experimentally certifiable lower bound on non-Gaussian cost of approximately preparing any state ## Pitfalls - Post-selected Gaussian protocols only: monotonicity holds under post-selection, not unconditional Gaussian operations - The bridge degree is defined for even pure states; extension to mixed states requires Choi isomorphism - Two-copy tests share the perfect completeness property; single-copy tests may have different completeness bounds ## Verification - Verify bridge degree computation against known Gaussian states (should yield zero for Gaussian states) - Test monotonicity: apply post-selected Gaussian protocol, verify bridge degree does not increase - Compare with existing non-Gaussianity monotones to confirm stronger no-go theorems ## Activation - fermionic non-gaussianity, bell sampling, bridge degree, gaussian conversion, fermionic quantum computing - 费米子非高斯性, 贝尔采样, 桥接度, 高斯转换