  In this paper, we develop a new channel model, which we name the $q$-ary
partial erasure channel (QPEC). The QPEC has a $q$-ary input, and its output is
either the input symbol or a set of $M$ ($2 \le M \le q$) symbols, containing
the input symbol. This channel serves as a generalization to the binary erasure
channel, and mimics situations when a symbol output from the channel is known
only partially, that is, the output symbol contains some ambiguity, but is not
fully erased. This type of channel is motivated by non-volatile memory
multi-level read channels. In such channels the readout is obtained by a
sequence of current/voltage measurements, which may terminate with partial
knowledge of the stored level. Our investigation is concentrated on the
performance of low-density parity-check (LDPC) codes when used over this
channel, thanks to their low decoding complexity using belief propagation. We
provide the exact QPEC density-evolution equations that govern the decoding
process, and suggest a cardinality-based approximation as a proxy. We then
provide several bounds and approximations on the proxy density evolutions, and
verify their tightness through numerical experiments. Finally, we provide tools
for the practical design of LDPC codes for use over the QPEC.
