  We describe a novel index modulation (IM) scheme exploiting a unique feature
of the recently proposed affine frequency division multiplexing (AFDM) in
doubly-dispersive (DD) channels. Dubbed AFDM chirp-permutation-index modulation
(CPIM), the proposed method encodes additional information via the permutation
of the discrete affine Fourier Transform (DAFT) chirp sequence, without any
sacrifice of the various beneficial properties of the AFDM waveform in DD
channels. The effectiveness of the proposed method is validated via simulation
results leveraging a novel reduced-complexity minimum mean-squared-error
(MMSE)-based maximum-likelihood (ML) detector, highlighting the gains over the
classical AFDM. As part of the work two interesting problems related to
optimizing AFDM-CPIM are identified: the optimal codebook design problem, over
a discrete solution space of dimension $\binom{N!}{K}$, where $N$ is the number
of subcarriers and $K$ is the number of codewords; and the ML detection problem
whose solution space is of dimension $KM^N$, where $M$ is the constellation
size. In order to alleviate the computational complexity of these problems and
enable large-scale variations of AFDM-CPIM, the two problems are reformulated
as a higher-order binary optimization problem and mapped to the well-known
quantum Grover adaptive search (GAS) algorithm for their solution.
