Impact of nuclear spin dynamics on two-qubit SWAP oscillations in donors*

Phosphorus atoms in silicon offer a rich quantum computing platform where both nuclear and electron spins can be used to store and process quantum information. While the control of the individual electron spin and nuclear spin has been demonstrated, the interplay between electron and nuclear spins in multi-qubit architectures remains unexplored experimentally. Here, we investigate the role that the phosphorus nuclear spins play during the exchange-based operations between donor-bound electron spins. We perform coherent exchange oscillations between two electron spin qubits, where the left and right qubits are hosted by three and two phosphorus donors, respectively. We observe beating in exchange oscillations which arises from the dynamics of nuclear spins that effectively modulate the energy difference between the anti-parallel electron spin states |↓↑〉and |↑↓〉. Based on the observed beating, we determine the individual hyperfine couplings between electron spin and each of the qubit-hosting nuclear spins. From this result we infer the corresponding crystallographic arrangements of phosphorus atoms for each qubit, which constitutes a unique metrology technique in which probing the Hamiltonian of a multi-spin donor-based system provides an insight into the atomic composition of qubits. Furthermore, we demonstrate strategies for achieving high fidelity two-qubit √SWAP gate fidelities. The framework presented here can be used for designing donor qubits to optimise inter-qubit operations.