Simulating decoherence of two-spin-qubit gates using cluster correlation expansion

One of the main challenges to the realization of high fidelity spin-qubit gates is the decoherence due to magnetic noise from nuclear spins. In the context of single spin qubits, the same kind of decoherence has been well understood and simulated with the cluster correlation expansion (CCE) method. Here we apply CCE to study the decoherence and fidelity of two-spin-qubit gates. In our model, two electron spin qubits with various kinds of interactions between them are coupled to random nuclear spin baths. The reduced density matrix (RDM) of the two qubits is simulated, and the decoherence of off-diagonal elements is understood with effective two-level Hamiltonians. We discuss the effects on decoherence from factors such as the external magnetic field, the quantum number of the spins, the interactions between electron spins etc. The applicability and limitations of CCE in simulating RDM elements are also explored.