  The direction of arrival (DOA) estimation in array signal processing is an
important research area. The effectiveness of the direction of arrival greatly
determines the performance of multi-input multi-output (MIMO) antenna systems.
The multiple signal classification (MUSIC) algorithm, which is the most
canonical and widely used subspace-based method, has a moderate estimation
performance of DOA. However, in hybrid massive MIMO systems, the received
signals at the antennas are not sent to the receiver directly, and spatial
covariance matrix, which is essential in MUSIC algorithm, is thus unavailable.
Therefore, the spatial covariance matrix reconstruction is required for the
application of MUSIC in hybrid massive MIMO systems. In this article, we
present a quantum algorithm for MUSIC-based DOA estimation in hybrid massive
MIMO systems. Compared with the best-known classical algorithm, our quantum
algorithm can achieve an exponential speedup on some parameters and a
polynomial speedup on others under some mild conditions. In our scheme, we
first present the quantum subroutine for the beam sweeping based spatial
covariance matrix reconstruction, where we implement a quantum singular vector
transition process to avoid extending the steering vectors matrix into the
Hermitian form. Second, a variational quantum density matrix eigensolver
(VQDME) is proposed for obtaining signal and noise subspaces, where we design a
novel objective function in the form of the trace of density matrices product.
Finally, a quantum labeling operation is proposed for the direction of arrival
estimation of the signal.
