Dissipation and gate timing errors in SWAP operations of qubits

Abstract

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 $\ensuremath{-}i\ensuremath{\gamma}\mathbf{1}$ in the Hamiltonian, and $\ensuremath{\gamma}$ 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 ${J}_{\mathrm{SWAP}}$, 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 ${J}_{\mathrm{SWAP}}$, beyond which infidelity begins to increase.