Quantum Approximate Optimization Algorithm for Maximum Likelihood Detection in Massive MIMO

TL;DR

This work tackles the NP-hard maximum likelihood (ML) detection problem in Massive MIMO by deploying the quantum approximate optimization algorithm (QAOA) on NISQ devices. It derives a universal, compact analytical expression for the $1$-level QAOA cost $C_1(oldsymbol{eta})$ by transforming ML detection into an Ising Hamiltonian $H_C= frac{1}{2}ig(\sum_{i<j} 2A_{ij}\sigma^z_i \sigma^z_j - oexpand{\sum_k} 2b_k \sigma^z_kig)$ with $A=H^T H$ and $b=y^T H$, and provides a causal-cone-based closed-form for the expectation terms. To address the nonconvex optimization landscape, the paper proposes a Bayesian optimization-based initialization that learns $2p$ parameters from small-scale instances to produce high-quality starting points for larger $p$-level runs, accelerating convergence and increasing the chance of obtaining the exact ML solution. Numerical experiments on synthetic data show that the proposed initialization leads to convergence to lower local minima and higher probability of measuring the exact bitstring (e.g., probability improvements like from $0.067$ to $0.099$ for a particular instance), demonstrating practical gains on ML-detection tasks. Overall, the approach offers a more universal $C_1$ expression and a data-driven initialization strategy that enhance QAOA-based ML detection for Massive MIMO in the NISQ era.

Abstract

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 the transmitting antennas and the 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 the maximum likelihood detection solver of binary symbols. In 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 have 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.

Figures (5)

• Figure 1: Schematic of a $p$-level quantum approximation optimization algorithm.
• Figure 2: An undirected graph corresponding to the problem Hamiltonian with $N_{t}=4$.
• Figure 3: Schematic of the bayesian optimization based parameters initialization.
• Figure 4: The convergence curve of the cost function using COBYLA optimizer with and without parameters initialization.
• Figure 5: The probability measuring the states using COBYLA optimizer with and without parameters initialization.