Beyond Binary Encoding: Qudit-Based QAOA Framework for ML Detection in $M$-QAM MIMO Systems

Abstract

Maximum likelihood (ML) detection in massive multiple-input multiple-output (MIMO) systems is an NP-hard combinatorial problem whose complexity grows exponentially with the number of transmit antennas and the modulation order. The Quantum Approximate Optimization Algorithm (QAOA) has recently emerged as a promising near-term quantum heuristic for such problems; however, existing qubit-based formulations require $\log_2 M$ qubits per $M$-QAM symbol, inflating both circuit width and depth. In this paper, we propose a qudit-based QAOA framework that encodes each $M$-QAM symbol directly into a pair of $d$-dimensional quantum states, where $d=\sqrt{M}$. We derive a Hermitian cost Hamiltonian via a real-imaginary decomposition of the constellation, and introduce a generalized cyclic-shift mixer that ensures ergodic exploration of the search space without binary encoding. The proposed formulation reduces the number of quantum carriers from $\mathcal{O}(N_t \log_2 M)$ to $\mathcal{O}(N_t)$ and the Hamiltonian interaction complexity from $\mathcal{O}((N_t \log_2 M)^2)$ to $\mathcal{O}(N_t^2)$, while preserving the geometric structure of the constellation. Bit-error-rate (BER) simulations for $2\times 2$ MIMO systems employing 4-QAM and 16-QAM over Rayleigh fading channels, with circuit depth $p=3$, demonstrate that the proposed detector closely approximates the performance of the optimal classical maximum-likelihood (ML) detector, while consistently outperforming the conventional minimum mean square error (MMSE) detector.