Quantum Speedup for Network Coordination via Fourier Sparsity

TL;DR

The Fourier Network Coordination problem (Fourier-NC) is introduced, unifying eight application domains and formalising the abelian index \alpha(G) = [G : Amax] as the structural invariant governing the quantum-classical gap.

Abstract

Network coordination - synchronising traffic signals, scheduling trains, assigning communication slots requires minimising pairwise costs across coupled systems. These problems are NP-hard yet share a common Fourier-sparse structure exploitable by quantum algorithms. We introduce the Fourier Network Coordination problem (Fourier-NC),unifying eight application domains. For abelian and dihedral groups, classical sparse Fourier transforms match quantum in the same oracle model, limiting the advantage to at most polynomial. The genuine separation emerges for the symmetric group Sk: a conditional super-exponential speedup of k! -> poly(k) for class-function costs with non-trivial minimisers. When the minimising conjugacy class is structurally determined, the problem lies in NP (int) BQP and is conditionally outside P (Corollary 6.5), placing it in the intermediate complexity regime alongside integer factorisation and graph isomorphism. We formalise the abelian index α(G) = [G : Amax] as the structural invariant governing the quantum-classical gap and identify a three-regime complexity trichotomy: abelian ({α= 1, classical sFFT suffices), nearly abelian (α= dmax, polynomial advantage), and strongly non-abelian (α>>dmax, super-exponential advantage).

Figures (8)

• Figure 1: Fourier spectrum $|\hat{H}(k)|$ for a single edge ($C=10$). Left: 2-sparse ($r=2$)---only 2 non-zero entries on the anti-diagonal $k_j=-k_i\bmod C$. Centre: dense but cyclic ($r=C$) with $O(C)$ modes, Grover quadratic speedup. Right: unstructured---$O(C^2)$ modes, no quantum shortcut. The Fourier-NC algorithm exploits left-panel structure.
• Figure 1: Irreducible representation landscape for $S_k$. (a) Young diagrams and hook-length dimensions for $S_4$. (b) Number of irreps and maximum dimension vs. $k$. (c) The super-exponential abelian index gap.
• Figure 2: Quantum circuit for Algorithm 1. (a) High-level circuit: each of the $n$ vertex registers is prepared in a uniform superposition via Hadamard gates, the phase oracle $U_H$ encodes all edge constraints as a product of pairwise unitaries $U_{ij}$, and an inverse QFT extracts the Fourier-mode label $\mathbf{k}^*$. (b) Phase oracle detail for a single edge $(i,j)$: a modular subtraction gate computes $d=\mu_i-\mu_j\bmod C$ on an ancilla register, $r$ controlled-phase rotations imprint the Fourier coefficients of the edge potential, and the subtraction is uncomputed. Gate complexity per edge is $O(r\log^2 C)$.
• Figure 3: Convergence: fraction of distinct Fourier modes recovered versus raw measurement count $T$ for the $|V|=4$ instance ($mr=12$, complete graph, $r=2$). Blue line: expected convergence; shaded band: $\pm1$ standard deviation over 50 Monte Carlo trials; grey traces: individual trials. Dashed line: theoretical threshold $T^*=mr\cdot\ln(mr)\approx29.8$.
• Figure 4: NP-hardness reduction (Theorem \ref{['thm:nphard']}). A MAX-CUT partition on a 5-node graph (left) maps directly to offset assignments $\mu\in\{0,1\}^n$ with cost $f_{ij}(\Delta\mu)=\cos(\pi\Delta\mu)$. Minimising $H(\mu)=m-2\cdot\mathrm{cut}(\mu)$ is equivalent to MAX-CUT, confirming NP-hardness at $r=1$.

Theorems & Definitions (33)

• Definition 3.1: $r$-Sparse Delay Function
• Theorem 3.2: Fourier Factorization
• proof : Proof sketch (full proof in Appendix \ref{['app:fourier_fact_proof']})
• Definition 3.3: Frustration-free and frustrated graph
• Theorem 4.1
• proof : Proof sketch (full proof in Appendix \ref{['app:correctness_proof']})
• Theorem 5.1
• proof : Proof sketch (full proof in Appendix \ref{['app:complexity_proofs']})
• Theorem 5.2: Classical query lower bound
• proof : Proof sketch (full proof in Appendix \ref{['app:complexity_proofs']})