In the massive multiple-input and multiple-output (Massive MIMO) systems, the maximum likelihood (ML) detection problem is NP-hard and becoming classically intricate with the number of transmitting antennas and symbols increasing. The quantum approximate optimization algorithm (QAOA), a leading candidate algorithm running in the noisy intermediate-scale quantum (NISQ) devices, can show quantum advantage for approximately solving combinatorial optimization problems. In this paper, we propose the QAOA based on the maximum likelihood detection solver of binary symbols. In the proposed scheme, we first conduct a universal and compact analytical expression for the expectation value of the 1-level QAOA. Second, a Bayesian optimization based parameters initialization is presented, which can speedup the convergence of the QAOA to a lower local minimum and improve the probability of measuring the exact solution. Compared to the state-of-the-art QAOA based ML detection algorithm, our scheme has the more universal and compact expectation value expression of the 1-level QAOA, and requires few quantum resources and has the higher probability to obtain the exact solution.
Beyond Binary Encoding: Qudit-Based QAOA Framework for ML Detection in M-QAM MIMO Systems
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 log2M 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=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 O(Ntlog2M) to O(Nt) and the Hamiltonian interaction complexity from O((Ntlog2M)2) to O(Nt2), while preserving the geometric structure of the constellation. Bit-error-rate (BER) simulations for 2×2 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.
Quantum-approximate optimization algorithm (QAOA) is promising in Noisy Intermediate-Scale Quantum (NISQ) computers with applications for NP-hard combinatorial optimization problems. It is recently utilized for NP-hard maximum-likelihood (ML) detection problem with fundamental challenges of optimization, simulation and performance analysis for n x n multiple-input multiple output (MIMO) systems with large n. QAOA is recently applied by Farhi et al. on infinite size limit of Sherrington-Kirkpatrick (SK) model with a cost model including only quadratic terms. In this article, we extend application of QAOA on SK model by including also linear terms and then realize SK modeling of massive MIMO ML detection by ensuring independence from specific problem instance and size n while preserving computational complexity of O(16p) designed by Farhi et al. We provide both optimized and extrapolated angles for p in the set [1, 14] and signal-to-noise (SNR) < 12 dB achieving near-optimum ML performance for 25 x 25 MIMO system with p >= 4 in extensive simulations where 236500 different QAOA circuits are simulated. We present two conjectures about the concentration properties of QAOA and its near-optimum performance for massive MIMO systems with large sizes covering n < 300 promising significant performance for next generation massive MIMO systems.