  Although the sphere decoder (SD) is a powerful detector for multiple-input
multiple-output (MIMO) systems, it has become computationally prohibitive in
massive MIMO systems, where a large number of antennas are employed. To
overcome this challenge, we propose fast deep learning (DL)-aided SD (FDL-SD)
and fast DL-aided $K$-best SD (KSD, FDL-KSD) algorithms. Therein, the major
application of DL is to generate a highly reliable initial candidate to
accelerate the search in SD and KSD in conjunction with candidate/layer
ordering and early rejection. Compared to existing DL-aided SD schemes, our
proposed schemes are more advantageous in both offline training and online
application phases. Specifically, unlike existing DL-aided SD schemes, they do
not require performing the conventional SD in the training phase. For a $24
\times 24$ MIMO system with QPSK, the proposed FDL-SD achieves a complexity
reduction of more than $90\%$ without any performance loss compared to
conventional SD schemes. For a $32 \times 32$ MIMO system with QPSK, the
proposed FDL-KSD only requires $K = 32$ to attain the performance of the
conventional KSD with $K=256$, where $K$ is the number of survival paths in
KSD. This implies a dramatic improvement in the performance--complexity
tradeoff of the proposed FDL-KSD scheme.
