  Low complexity decoding algorithms are necessary to meet data rate
requirements in excess of 1 Tbps. In this paper, we study one and two bit
message passing algorithms for belief propagation decoding of low-density
parity-check (LDPC) codes and analyze them by density evolution. The variable
nodes (VNs) exploit soft information from the channel output. To decrease the
data flow, the messages exchanged between check nodes (CNs) and VNs are
represented by one or two bits. The newly proposed quaternary message passing
(QMP) algorithm is compared asymptotically and in finite length simulations to
binary message passing (BMP) and ternary message passing (TMP) for spectrally
efficient communication with higher-order modulation and probabilistic
amplitude shaping (PAS). To showcase the potential for high throughput forward
error correction, spatially coupled LDPC codes and a target spectral efficiency
(SE) of 3 bits/QAM symbol are considered. Gains of about 0.7 dB and 0.1 dB are
observed compared to BMP and TMP, respectively. The gap to unquantized belief
propagation (BP) decoding is reduced to about 0.75 dB. For smaller code rates,
the gain of QMP compared to TMP is more pronounced and amounts to 0.24 dB in
the considered example.
