Subspace-Confined QAOA with Generalized Dicke States for Multi-Channel Allocation in 5G CBRS Networks

TL;DR

This work addresses CBRS multi-channel allocation by formulating it as a Graph Multi-Coloring Problem with node-dependent channel demands. It introduces a subspace-confined QAOA that initializes each node register in a Generalized Dicke state and evolves under an intra-register XY mixer, confining dynamics to a tensor-product of Johnson graphs. For an 8-node, 3-channel instance with $N_{ ext{qubits}}=24$, the feasible state count is $|\\mathcal{F}|=\prod_{i=1}^{n} \binom{m}{k_i}=2{,}916$, a reduction of about $5.8\times 10^{3}$ from the full space $2^{24}$. The method achieves near-optimal conflict levels with unit feasibility, outperforms a penalty-based QAOA and a greedy heuristic, and exhibits robustness to depolarizing noise; a dual-constraint mixer extending to per-channel capacities is also demonstrated, illustrating how problem structure can be integrated into QAOA for realistic spectrum allocation.

Abstract

Efficient spectrum sharing in the Citizens Broadband Radio Service (CBRS) band is essential for maximizing 5G network capacity, particularly when high-traffic base stations require simultaneous access to multiple channels. Standard formulations of the Quantum Approximate Optimization Algorithm (QAOA) impose such multi-channel constraints using penalty terms, so most of the explored Hilbert space corresponds to invalid assignments. We propose a subspace-confined QAOA tailored to CBRS multi-channel allocation, in which each node-wise channel register is initialized in a Generalized Dicke state and evolved under an intra-register XY mixer. This ansatz confines the dynamics to a tensor product of Johnson graphs that exactly encode per-node Hamming-weight constraints. For an 8-node CBRS interference graph with 24 qubits, the effective search space is reduced from the full Hilbert space of size $2^{24}$ to 2916 feasible configurations. Within this subspace, the algorithm converges rapidly to low-conflict assignments without large penalty coefficients. Simulations on instances with up to eight nodes show that the proposed ansatz achieves near-optimal conflict levels and consistently outperforms standard penalty-based QAOA and a greedy classical heuristic in terms of feasibility. Noise simulations with depolarizing channels further indicate that the constraint-preserving structure maintains a high feasibility ratio in NISQ-relevant error regimes.

Figures (8)

• Figure 2: (a) Optimization trajectories for the 8-node CBRS network (24 qubits) comparing standard penalty-based QAOA and the proposed subspace-confined QAOA. (b) Average deviation from the Hamming-weight constraints as a function of depolarizing error rate for a 5-node (15-qubit) instance.
• Figure 3: Average conflict landscape for the dual-constraint QAOA ansatz on an 8-node, 3-channel CBRS interference graph (24 qubits). The color scale indicates the average number of interference conflicts as a function of the problem and mixer angles $(\gamma,\beta)$ at depth $p=1$. For all parameter pairs considered, both the node-wise and channel-wise cardinality constraints remain satisfied with probability close to one.
• Figure 4: (a) Unitary evolution of the XY-Hamiltonian for a single node with four available colors. (b) Unitary evolution for a two-node, three-colorable problem. The mixer acts on intra-node qubit pairs and preserves the Hamming weight for each node, implementing a quantum walk on the corresponding Johnson graphs.
• Figure 5: Quantum circuit for preparing the initial state in a two-node, three-channel system. The gate labeled $U_W$ generates the Generalized Dicke state $|D_k^m\rangle$. X gates at the input can be used to adjust the target Hamming weight $k_i$.
• Figure 6: Overview of the QAOA circuit using the XY-Hamiltonian mixer and Dicke-state initialization for a two-node, three-channel system. The dynamics are confined to the tensor-product Johnson-graph subspace defined by the node-wise Hamming-weight constraints.

Theorems & Definitions (2)

• Definition 1: Node-wise Johnson-graph structure of the feasible subspace
• Proposition 1: Search-space reduction from Johnson-graph confinement