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We examine how dissipation and gate timing errors affect the fidelity of asequence of SWAP gates on a chain of interacting qubits in comparison to noisein the interqubit interaction. Although interqubit interaction noise and gatetiming errors are always present in any qubit platform, dissipation is aspecial case that can arise in multivalley semiconductor spin qubit systems,such as Si-based qubits, where dissipation may be used as a general model forvalley leakage. In our Hamiltonian, each qubit is coupled via Heisenbergexchange to every other qubit in the chain, with the strength of the exchangeinteraction decreasing exponentially with distance between the qubits.Dissipation is modeled through the term $-i\gamma\mathbf{1}$ in theHamiltonian, and $\gamma$ is chosen so as to be consistent with theexperimentally observed intervalley tunneling in Si. We show that randomness inthe dissipation parameter should have little to no effect on the SWAP gatefidelity in the currently fabricated Si circuits. We introduce quasistaticnoise in the interqubit interaction and random gate timing error and averagethe fidelities over 10,000 realizations for each set of parameters. Thefidelities are then plotted against $J_\text{SWAP}$, the strength of theexchange coupling corresponding to the SWAP gate. We find that dissipationdecreases the fidelity of the SWAP operation -- though the effect is smallcompared to that of the known noise in the interqubit interaction -- and thatgate timing error creates an effective optimal value of $J_\text{SWAP}$, beyondwhich infidelity begins to increase.

Dissipation and gate timing errors in SWAP operations of qubits