  We examine how dissipation and gate timing errors affect the fidelity of a
sequence of SWAP gates on a chain of interacting qubits in comparison to noise
in the interqubit interaction. Although interqubit interaction noise and gate
timing errors are always present in any qubit platform, dissipation is a
special case that can arise in multivalley semiconductor spin qubit systems,
such as Si-based qubits, where dissipation may be used as a general model for
valley leakage. In our Hamiltonian, each qubit is coupled via Heisenberg
exchange to every other qubit in the chain, with the strength of the exchange
interaction decreasing exponentially with distance between the qubits.
Dissipation is modeled through the term $-i\gamma\mathbf{1}$ in the
Hamiltonian, and $\gamma$ is chosen so as to be consistent with the
experimentally observed intervalley tunneling in Si. We show that randomness in
the dissipation parameter should have little to no effect on the SWAP gate
fidelity in the currently fabricated Si circuits. We introduce quasistatic
noise in the interqubit interaction and random gate timing error and average
the fidelities over 10,000 realizations for each set of parameters. The
fidelities are then plotted against $J_\text{SWAP}$, the strength of the
exchange coupling corresponding to the SWAP gate. We find that dissipation
decreases the fidelity of the SWAP operation -- though the effect is small
compared to that of the known noise in the interqubit interaction -- and that
gate timing error creates an effective optimal value of $J_\text{SWAP}$, beyond
which infidelity begins to increase.
