General Hamiltonian Representation of ML Detection Relying on the Quantum Approximate Optimization Algorithm

TL;DR

This paper introduces a quantum algorithm-based approach for maximum likelihood detection in communication systems, transforming the problem into a form suitable for the quantum approximate optimization algorithm (QAOA) and analyzing its effectiveness.

Contribution

It develops a novel Hamiltonian construction for ML detection using QAOA, connecting constellation patterns to the objective function's degree, and demonstrates separate detection of in-phase and quadrature components.

Findings

QAOA can be adapted for ML detection in communication systems.

The problem is transformed into a weighted N-satisfiability problem with a pseudo Boolean function.

The approach achieves promising approximation ratios with different QAOA circuit depths.

Abstract

The quantum approximate optimization algorithm (QAOA) conceived for solving combinatorial optimization problems has attracted significant interest since it can be run on the existing noisy intermediate-scale quantum (NISQ) devices. A primary step of using the QAOA is the efficient Hamiltonian construction based on different problem instances. Hence, we solve the maximum likelihood (ML) detection problem for general constellations by appropriately adapting the QAOA, which gives rise to a new paradigm in communication systems. We first transform the ML detection problem into a weighted minimum $N$-satisfiability (WMIN-$N$-SAT) problem, where we formulate the objective function of the WMIN-$N$-SAT as a pseudo Boolean function. Furthermore, we formalize the connection between the degree of the objective function and the Gray-labelled modulation constellations. Explicitly, we show a series of results exploring the connection between the coefficients of the monomials and the patterns of the associated constellation points, which substantially simplifies the objective function with respect to the problem Hamiltonian of the QAOA. In particular, for an M-ary Gray-mapped quadrature amplitude modulation (MQAM) constellation, we show that the specific qubits encoding the in-phase components and those encoding the quadrature components are independent in the quantum system of interest, which allows the in-phase and quadrature components to be detected separately using the QAOA. Furthermore, we characterize the degree of the objective function in the WMIN-$N$-SAT problem corresponding to the ML detection of multiple-input and multiple-output (MIMO) channels. Finally, we evaluate the approximation ratio of the QAOA for the ML detection problem of quadrature phase shift keying (QPSK) relying on QAOA circuits of different depths.