  We present a theory of a two qubit gate with macroscopic singlet-triplet (ST)
qubits in synthetic spin-one chains in InAsP quantum dot nanowires. The
macroscopic topologically protected singlet-triplet qubits are built with two
spin-half Haldane quasiparticles. The Haldane quasiparticles are hosted by
synthetic spin-one chain realized in chains of InAsP quantum dots embedded in
an InP nanowire, with four electrons each. The quantum dot nanowire is
described by a Hubbard-Kanamori (HK) Hamiltonian derived from an interacting
atomistic model. Using exact diagonalization and Matrix Product States (MPS)
tools, we demonstrate that the low-energy behavior of the HK Hamiltonian is
effectively captured by an antiferromagnetic spin-one chain Hamiltonian. Next
we consider two macroscopic qubits and present a method for creating a tunable
coupling between the two macroscopic qubits by inserting an intermediate
control dot between the two chains. Finally, we propose and demonstrate two
approaches for generating highly accurate two-ST qubit gates : (1) by
controlling the length of each qubit, and (2) by employing different background
magnetic fields for the two qubits.
