quantum-fermionic-shadows-statistics

name	quantum-fermionic-shadows-statistics
description	Mode-independent sample complexity for fermionic classical shadows. Improves worst-case bound from O(√n log n) to O(η log η) using harmonic analysis on AIII symmetric space.

Quantum Fermionic Shadows with Mode-Independent Sample Complexity

Description

Methodology for classical shadow estimation of particle-preserving fermionic states that achieves mode-independent sample complexity O(η log η), reducing from the previous worst-case O(√n log n) bound. Uses harmonic analysis on the AIII symmetric space U(n)/(U(η) × U(n-η)) and Jacobi ensemble techniques.

Activation Keywords

• fermionic classical shadows
• mode-independent sample complexity
• Slater determinant overlap estimation
• AIII symmetric space harmonic analysis
• Jacobi ensemble quantum
• particle-preserving operators
• fermionic state learning
• 费米子经典阴影
• 粒子数守恒算符

Core Concepts

Fermionic Shadow Estimation

• Problem: Learn expectation values of particle-preserving operators from unknown η-particle n-mode fermionic states
• Key insight: Particle-preserving structure enables mode-independent sample complexity
• Randomization: First-quantized encoding with approximate unitary designs achieves polylogarithmic circuit depth

Sample Complexity Bounds

• Previous worst case: O(√n log n) samples
• New bound: O(η log η) samples (independent of total mode count n)
• Quadratic observables: O(η ||h₀||₂²) where h₀ is traceless component
• Classical post-processing: O(n η²) for generic dense orbital, O(n² η) for quadratic observables

AIII Symmetric Space Analysis

• Reduction: Extremal shadow variance → harmonic analysis on U(n)/(U(η) × U(n-η))
• Techniques: Jacobi ensembles and orthogonal polynomials for integral evaluation
• Significance: Mathematical technique of independent interest beyond quantum shadows

Usage Patterns

Pattern 1: Fermionic State Learning

1. Identify particle number η and mode count n
2. Select randomization scheme (first-quantized for polylog depth, second-quantized for nearest-neighbor)
3. Apply classical shadow protocol with O(η log η) samples
4. Post-process classically in O(n η²) time

Pattern 2: Slater Determinant Overlap Estimation

1. Target state: unknown η-particle n-mode fermionic state
2. Reference state: arbitrary Slater determinant
3. Sample complexity: O(η log η) for fixed additive precision
4. Advantage: exponential improvement when η ≪ n

Mathematical Framework

Symmetric Space Decomposition

Shadow variance extremal problem
    ↓ reduction
Harmonic analysis on AIII symmetric space U(n)/(U(η) × U(n-η))
    ↓ evaluation
Jacobi ensemble integrals + orthogonal polynomials
    ↓ result
O(η log η) sample complexity bound

Circuit Depth Comparison

• First-quantized: Polylogarithmic depth (approximate unitary designs)
• Second-quantized matchgate: Linear depth (nearest-neighbor)

Error Handling

Sample Complexity Bounds

• If η ≈ n: The O(η log η) bound approaches O(n log n), still better than O(√n log n) for large n
• If observables not particle-preserving: Different shadow protocol required
• If classical post-processing too slow: Consider sparse orbital structure optimization

References

• arXiv:2606.27254 - Particle-preserving fermionic shadows (West, Cerezo, Larocca 2026)
• Classical shadow tomography literature
• Harmonic analysis on symmetric spaces