  Recent advances in quantum technologies pave the way for noisy
intermediate-scale quantum (NISQ) devices, where quantum approximation
optimization algorithms (QAOAs) constitute promising candidates for
demonstrating tangible quantum advantages based on NISQ devices. In this paper,
we consider the maximum likelihood (ML) detection problem of binary symbols
transmitted over a multiple-input and multiple-output (MIMO) channel, where
finding the optimal solution is exponentially hard using classical computers.
Here, we apply the QAOA for the ML detection by encoding the problem of
interest into a level-p QAOA circuit having 2p variational parameters, which
can be optimized by classical optimizers. This level-p QAOA circuit is
constructed by applying the prepared Hamiltonian to our problem and the initial
Hamiltonian alternately in p consecutive rounds. More explicitly, we first
encode the optimal solution of the ML detection problem into the ground state
of a problem Hamiltonian. Using the quantum adiabatic evolution technique, we
provide both analytical and numerical results for characterizing the evolution
of the eigenvalues of the quantum system used for ML detection. Then, for
level-1 QAOA circuits, we derive the analytical expressions of the expectation
values of the QAOA and discuss the complexity of the QAOA based ML detector.
Explicitly, we evaluate the computational complexity of the classical optimizer
used and the storage requirement of simulating the QAOA. Finally, we evaluate
the bit error rate (BER) of the QAOA based ML detector and compare it both to
the classical ML detector and to the classical MMSE detector, demonstrating
that the QAOA based ML detector is capable of approaching the performance of
the classical ML detector. This paves the way for a host of large-scale
classical optimization problems to be solved by NISQ computers.
