Quantum Metric-Driven Functional Redundancy Graphs

• QM-FRGs are quantum-inspired graphs that encode functional similarity between neural operators using Fubini–Study and Wasserstein-2 distances.
• They enable one-shot structured pruning by clustering redundant operators with provable guarantees on functional output consistency.
• QM-FRGs are applied to optimize neural architectures and analyze neural data, supported by rigorous spectral geometric analysis and empirical validation.

Quantum Metric-Driven Functional Redundancy Graphs (QM-FRG) encode operator-level functional similarity in neural systems using quantum-inspired metrics. They provide a structurally principled framework for detecting redundancy and guiding pruning through spectral geometry, with guarantees deriving from the Fubini–Study and Wasserstein-2 distances on operator spectra. QM-FRGs underpin modern approaches to understanding and optimizing complex neural networks and have proven applications in both neuroscience and large-scale machine learning models (Shao et al., 30 Nov 2025, Chan et al., 23 Aug 2025).

1. Conceptual Definition and Motivation

Quantum Metric-Driven Functional Redundancy Graphs are undirected, weighted graphs constructed such that each vertex represents a neural operator—be it a convolutional filter, linear transformation, attention head, or other parametric computation—and each edge encodes a quantum-inspired functional distance. The definition (Shao et al., 30 Nov 2025):

• Vertices: $V = \{\Phi_1, \dots, \Phi_N\}$, with each $\Phi_i$ an operator in the network.
• Edge Weights: $w_{ij} = d_{\mathrm{FS}}(\Phi_i, \Phi_j)$, where $d_{\mathrm{FS}}$ is the Fubini–Study distance between quantum-encoded operator spectra.

Clusters of tightly connected vertices identify groups of mutually redundant operators. This structural representation underpins efficient one-shot structured pruning and cross-modal/network analysis. The overarching aim is to transcend classical similarity metrics by capturing functional equivalence via high-dimensional spectral geometry (Shao et al., 30 Nov 2025).

2. Mathematical Foundations: Quantum-Inspired Metrics

2.1. Spectral Embedding

Each operator $\Phi$ is augmented to a weight matrix $$\widehat{W} \in \mathbb{R}^{(d_{\text{out}}+1)\times(d_{\text{in}}+1)}$$—concatenating weights and biases. Its singular value vector $s = (\sigma_1, ..., \sigma_m)$, normalized as $|\psi_\Phi\rangle = s/\|s\|_2$, defines a quantum-inspired pure state on the Bloch hypersphere $\mathbb{S}^{m-1}$.

2.2. Fubini–Study Distance

For a pair of operators, the metric is

$$d_{\mathrm{FS}}(\Phi_1,\Phi_2) = \arccos|\langle \psi_1 | \psi_2 \rangle|$$

where $|\psi_{1,2}\rangle$ are their spectral states.

2.3. Wasserstein-2 Distance

The cumulative singular-value distributions $F_i$ for each operator yield an alternative metric: $$\mathcal{W}_2(F_1, F_2) = \left( \inf_{\pi\in\Pi(F_1,F_2)} \int_{[0,1]^2} |x-y|^2\,d\pi(x,y) \right)^{1/2}$$ with $\Pi(F_1,F_2)$ the set of couplings.

2.4. Spectral–to–Functional Equivalence

A tight theorem establishes that small spectral distances (particularly small Fubini–Study or Wasserstein-2 distances) guarantee bounded output differences between operators: $$\|\Phi_1(x) - \Phi_2(x)\|_2 \le L (R+1)(\|\widehat{W}_1\|_F + \|\widehat{W}_2\|_F) \, \mathcal{W}_2(F_1,F_2) + 2M\mathbf{1}_{\|\widehat{W}_1\|_F\neq\|\widehat{W}_2\|_F}$$ When $\|\widehat{W}_1\|_F = \|\widehat{W}_2\|_F$,

$$\mathcal{W}_2(F_1,F_2) \le \sqrt{2}d_{\mathrm{FS}}(\Phi_1,\Phi_2) \quad\text{and}\quad d_{\mathrm{FS}}=0 \Rightarrow \|\Phi_1 - \Phi_2\|_\infty=0$$

This underpins the use of QM-FRGs for functional redundancy detection and pruning (Shao et al., 30 Nov 2025).

3. Construction Algorithm

The QM-FRG is constructed as follows:

1. For each operator $\Phi_i$, compute singular values $s_i$ of $\widehat{W}_i$, then normalize $|\psi_i\rangle = s_i/\|s_i\|_2$.
2. For every unordered pair $(i,j)$, compute $w_{ij} = d_{\mathrm{FS}}(\Phi_i, \Phi_j)$.
3. The graph $G=(V,E)$ is formed with edge weights $w_{ij}$.
4. Optionally, prune edges above a threshold $\tau$ for sparsification.

Pseudocode:

Input: {Φ1,…,ΦN}, threshold τ for i in 1…N: s_i ← SVD_singular_values(Ŵ_i) ψ_i ← s_i / ‖s_i‖₂ for i in 1…N: for j in i+1…N: w_ij ← arccos(|ψ_i · ψ_j|) if w_ij ≤ τ: add edge (i, j) with weight w_ij return G = (V, E)

Computational cost is $O(N^2 m)$ for spectral overlap computations and $O(m^3)$ per operator for SVD, which is practical for $m \le 4096$ and $N \sim 10^3$ on NPUs (Shao et al., 30 Nov 2025).

4. Applications: Structural Redundancy Detection and Pruning

Redundant operators are identified as clusters with small intra-cluster $d_{\mathrm{FS}}$ values. The canonical criterion for pruning is to retain a minimal subset $S \subset C$, which spectrally covers a low-diameter cluster $C$: $$\rho_i = \sum_{j \in C} d_{\mathrm{FS}}(\Phi_i, \Phi_j)$$ Operators with lowest $\rho_i$ are retained. For a sparsity ratio $p$, select the top $(1-p)N$ operators by ascending $\rho_i$ (Shao et al., 30 Nov 2025). This yields a "one-shot" procedure—no gradient-based or iterative fine-tuning required.

Theoretical Guarantee

If every pruned operator is within spectral distance $\le \tau$ of some retained operator, the maximum possible output discrepancy is $O(\tau)$, as quantified by the spectral–to–functional equivalence theorem.

Empirical Validation

On ResNet-18, QM-FRG pruning outperformed both $L_1$-norm and random channel pruning at extreme sparsity with lower accuracy loss. All pairwise operator SVDs and distance computations took $<0.1$ ms per layer on edge NPUs (Shao et al., 30 Nov 2025).

5. Quantum State Fidelity FRG: Data-Driven Graph Inference

In the neuroscience domain, QM-FRGs are constructed by encoding neural tuning curves as quantum states and quantifying similarity via the Uhlmann fidelity: $$F(|\psi\rangle, |\phi\rangle) = |\langle \psi | \phi \rangle|^2 \in [0,1]$$ The distances $D_{ij} = 1 - F_{ij}$ are used for edge weights. Graph extraction uses strategies like minimum spanning tree (MST), top-$k\%$ edge selection, or thresholding.

Quantum state fidelities capture nonlinear, higher-order relationships among neuron encodings that classical measures (Pearson correlation, Euclidean, or Bhattacharyya overlaps) miss. Empirically, such graphs exhibited distinct modular structure and robustness, with quantum-simulator FRGs yielding higher granularity and cluster modularity than classical metrics (Chan et al., 23 Aug 2025).

6. Scalability, Complexity, and Extensions

Computational Efficiency

• Operator-Level QM-FRG: SVD and Fubini–Study distance computations are practical for standard NPUs ($<5$ ms per operator for $m \le 4096$) (Shao et al., 30 Nov 2025).
• Fidelity-Based QM-FRG: Quantum circuit executions scale as $O(N^2)$; MST or percentage-edge pruning post-processing is $O(N^2 \log N)$ (Chan et al., 23 Aug 2025).

Extensions

• Pruning and redundancy detection in transformer architectures and heterogeneous hardware (Huawei Ascend, Cambricon MLU, Kunlunxin).
• Functional motifs in neural data, complex system graphs (genomics, finance), and directed graphs via asymmetric quantum metrics.

A plausible implication is that with improved quantum hardware, the high-dimensional quantum state overlaps may enable even richer graph inference in neuroscience and broader domains, surpassing the reach of classical similarity measures (Chan et al., 23 Aug 2025).

7. Comparative Perspective and Theoretical Significance

QM-FRGs provide a rigorous and practical bridge between spectral-geometric analysis and operator-level functional similarity:

Method	Metric	Functional Guarantee
Quantum State FRG	Uhlmann fidelity	Overlap in Hilbert space; neural network structure (Chan et al., 23 Aug 2025)
Spectral QM-FRG	Fubini–Study distance	Tight bound on functional output deviation (Shao et al., 30 Nov 2025)
Classical Baseline	Pearson/Euclidean/Bhattacharyya	No provable functional closeness

By abstracting over operator modality and dimension, QM-FRG enables provable cross-modal and cross-architecture pruning—two operators from vision or language subsystems with vanishing $d_{\mathrm{FS}}$ are thus theoretically and empirically interchangeable (Shao et al., 30 Nov 2025). This suggests that QM-FRGs establish a new standard for network compression, pruning, and operator sharing, supported by both tight theoretical results and practical benchmarks.