  We introduce a twisted Heisenberg-type interaction Hamiltonian, a
Heisenberg-type spin interaction where the coordinates of the second qubit are
twisted according to a unitary gate. We develop parallelizable adiabatic gate
teleportation (PAGT) where a sequence of unitary gates is performed in a single
step of the adiabatic process. In PAGT, numeric calculations suggest the
necessary time for the adiabatic evolution implementing a sequence of $L$
unitary gates increases at most as $O(L^5)$. However, we show that it has the
interesting property that it can map the temporal order of gates to the spatial
order of interactions specified by the final Hamiltonian. Using this property,
we present a controlled-PAGT scheme to manipulate the order of gates by a
control-qubit. In the controlled-PAGT scheme, two differently ordered
sequential unitary gates $FG$ and $GF$ are coherently performed depending on
the state of a control-qubit by simultaneously applying the twisted
Heisenberg-type interaction Hamiltonians implementing unitary gates $F$ and
$G$. We investigate why the twisted Heisenberg-type interaction Hamiltonian
allows PAGT. We show that the twisted Heisenberg-type interaction Hamiltonian
has an ability to perform a transposed unitary gate by just modifying the space
ordering of the final Hamiltonian implementing a unitary gate in adiabatic gate
teleportation. The dynamics generated by the time-reversed Hamiltonian
represented by the transposed unitary gate enables deterministic simulation of
a postselected event of parallelized gate teleportation in adiabatic
implementation.
