  We theoretically consider the temporal dynamics of two coupled spin qubits
(e.g., semiconductor quantum dots) driven by the inter-qubit spin-spin
coupling. The presence of environmental noise (e.g., charge traps, nuclear
spins, random magnetic impurities) is accounted for by including random
magnetic field and random inter-qubit coupling terms in the Hamiltonian. Both
Heisenberg coupling and Ising coupling between the spin qubits are considered,
corresponding respectively to exchange and capacitive gates as appropriate for
single spin and singlet-triplet semiconductor qubit systems, respectively. Both
exchange (Heisenberg) and capacitive (Ising) coupling situations can be solved
numerically exactly even in the presence of noise, leading to the key findings
that (i) the steady-state return probability to the initial state remains close
to unity in the presence of strong noise for many, but not all, starting spin
configurations, and (ii) the return probability as a function of time is
oscillatory with a characteristic noise-controlled decay toward the
steady-state value. We also provide results for the magnetization dynamics of
the coupled two-qubit system. Our predicted dynamics can be directly tested in
the already existing semiconductor spin qubit setups providing insight into
their coherent interaction dynamics. Retention of the initial state spin memory
even in the presence of strong environmental noise has important implications
for quantum computation using spin qubits.
