Multi-qubit DC gates over an inhomogeneous array of quantum dots

The prospect of large-scale quantum computation with an integrated chip of spin qubits is imminent as technology improves. This invites us to think beyond the traditional 2-qubit-gate framework and consider a naturally supported ``instruction set'' of multi-qubit gates. In this work, we systematically study such a family of multi-qubit gates implementable over an array of quantum dots by DC evolution. A useful representation of the computational Hamiltonian is proposed for a dot-array with strong spin-orbit coupling effects, distinctive $g$-factor tensors and varying interdot couplings. Adopting a perturbative treatment, we model a multi-qubit DC gate by the first-order dynamics in the qubit frame and develop a detailed formalism for decomposing the resulting gate, estimating and optimizing the coherent gate errors with appropriate local phase shifts for arbitrary array connectivity. Examples of such multi-qubit gates and their applications in quantum error correction and quantum algorithms are also explored, demonstrating their potential advantage in accelerating complex tasks and reducing overall errors.