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The revolutionary technology of \emph{Stacked Intelligent Metasurfaces (SIM)}
has been recently shown to be capable of carrying out advanced signal
processing directly in the native electromagnetic (EM) wave domain. An SIM is
fabricated by a sophisticated amalgam of multiple stacked metasurface layers,
which may outperform its single-layer metasurface counterparts, such as
reconfigurable intelligent surfaces (RISd) and metasurface lenses. We harness
this new SIM concept for implementing efficient holographic multiple-input
multiple-output (HMIMO) communications that dot require excessive
radio-frequency (RF) chains, which constitutes a substantial benefit compared
to existing implementations. We first present an HMIMO communication system
based on a pair of SIMs at the transmitter (TX) and receiver (RX),
respectively. In sharp contrast to the conventional MIMO designs, the
considered SIMs are capable of automatically accomplishing transmit precoding
and receiver combining, as the EM waves propagate through them. As such, each
information data stream can be directly radiated and recovered from the
corresponding transmit and receive ports. Secondly, we formulate the problem of
minimizing the error between the actual end-to-end SIMs'parametrized channel
matrix and the target diagonal one, with the latter representing a flawless
interference-free system of parallel subchannels. This is achieved by jointly
optimizing the phase shifts associated with all the metasurface layers of both
the TX-SIM and RX-SIM. We then design a gradient descent algorithm to solve the
resultant non-convex problem. Furthermore, we theoretically analyze the HMIMO
channel capacity bound and provide some useful fundamental insights. Extensive
simulation results are provided for characterizing our SIM-based HMIMO system,
quantifying its substantial performance benefits.