mesoscopic-quantum-graph-statistics

name	mesoscopic-quantum-graph-statistics
description	Mesoscopic linear spectral statistics for random quantum graph ensembles - proves variance coincides with GOE/GUE in large graph limit. Trigger words: mesoscopic statistics, quantum graphs, random graph ensemble, spectral variance, GOE, GUE, Haar measure
tags	["quantum-computing","graph-theory","random-matrix-theory","spectral-analysis","mathematical-physics"]

Mesoscopic Linear Statistics for Quantum Graphs

Methodology from arXiv:2607.02356 for analyzing mesoscopic linear spectral statistics in ensembles of random quantum graphs.

Core Methodology

1. Quantum Graph Model

A quantum graph is defined by:

• A discrete graph $G$
• A unitary-matrix-valued function $U(k)$ indexed by directed edges
• Local unitary matrices $U^{(v)}$ at each vertex $v$

2. Two Ensembles

Ensemble 1 - Random Graph Sampling:

• Sample the underlying discrete graph uniformly from $d$-regular graphs
• Fixed unitary matrices at vertices
• Studies graph structure randomness

Ensemble 2 - Haar Measure Sampling:

• Sample $U^{(v)}$ uniformly from Haar measure
• Independent for each vertex
• Studies unitary randomness

3. Main Result

The variance of linear spectral statistics in the large graph limit on polynomial mesoscopic scales coincides with the Gaussian Orthogonal Ensemble (GOE) or Gaussian Unitary Ensemble (GUE).

4. Mathematical Framework

Linear Spectral Statistic: L_f = Σ_i f(λ_i)
Variance: Var(L_f) → Var_GOE/GUE(f) as N → ∞

Where the convergence holds on polynomial mesoscopic scales.

Implementation Pattern

import numpy as np
from scipy.linalg import eigvals

def build_quantum_graph(adjacency, d):
    """Build quantum graph with unitary edge matrices"""
    n = len(adjacency)
    # Construct scattering matrix from adjacency
    S = np.zeros((n*d, n*d), dtype=complex)
    
    for i in range(n):
        neighbors = np.where(adjacency[i])[0]
        for j in neighbors:
            # Random unitary for each edge
            U_ij = random_unitary(d)
            S[i*d:(i+1)*d, j*d:(j+1)*d] = U_ij
    
    return S

def spectral_statistics(scattering_matrix, test_function, scale):
    """Compute linear spectral statistics at mesoscopic scale"""
    eigenvalues = eigvals(scattering_matrix)
    phases = np.angle(eigenvalues)
    
    # Apply test function at mesoscopic scale
    return sum(test_function(p * scale) for p in phases)

def variance_analysis(graph_ensemble, test_function, scales):
    """Analyze variance convergence to GOE/GUE"""
    variances = []
    for scale in scales:
        stats = []
        for G in graph_ensemble:
            S = build_quantum_graph(G, d=3)
            stat = spectral_statistics(S, test_function, scale)
            stats.append(stat)
        variances.append(np.var(stats))
    return variances

Applications

• Quantum chaos detection: Identifying chaotic vs regular quantum systems
• Random matrix universality: Proving universality classes in quantum systems
• Network analysis: Spectral properties of complex networks
• Quantum transport: Understanding transport in disordered quantum systems

Key Theorems

1. GOE Convergence: For time-reversal symmetric ensembles, variance converges to GOE prediction
2. GUE Convergence: For broken time-reversal symmetry, variance converges to GUE prediction
3. Mesoscopic Scale: Results hold on polynomial scales between microscopic and macroscopic

Related Skills

• [[quantum-graph-neural-drug-discovery]]
• [[neural-quantum-graph-embedding]]
• [[random-matrix-quantum-statistics]]