  Supermassive black hole (SMBH) coalescences are ubiquitous in the history of
the Universe and often exhibit strong accretion activities and powerful jets.
These SMBH mergers are also promising candidates for future gravitational wave
detectors such as Laser Space Inteferometric Antenna (LISA). In this work, we
consider neutrino counterpart emission originating from the jet-induced shocks.
The physical picture is that relativistic jets launched after the merger will
push forward inside the premerger disk wind material, and then they
subsequently get collimated, leading to the formation of internal shocks,
collimation shocks, forward shocks and reverse shocks. Cosmic rays can be
accelerated in these sites and neutrinos are expected via the photomeson
production process. We formulate the jet structures and relevant interactions
therein, and then evaluate neutrino emission from each shock site. We find that
month-to-year high-energy neutrino emission from the postmerger jet after the
gravitational wave event is detectable by IceCube-Gen2 within approximately
five to ten years of operation in optimistic cases where the cosmic-ray loading
is sufficiently high and a mildly super-Eddington accretion is achieved. We
also estimate the contribution of SMBH mergers to the diffuse neutrino
intensity, and find that a significant fraction of the observed very
high-energy ($E_\nu\gtrsim1$ PeV) IceCube neutrinos could originate from them
in the optimistic cases. In the future, such neutrino counterparts together
with gravitational wave observations can be used in a multimessenger approach
to elucidate in greater detail the evolution and the physical mechanism of SMBH
mergers.
