Towards scalable multi-qubit optimal control via interaction decomposition in the diagonal frame

In this work, we introduce a general n-qubit formulation of control objectives that allows a control target to be specified in a diagonal frame, so that only the diagonal entries must be characterized, thus quadratically reducing the complexity of the cost functional in constrast to a full target matrix. We do so by representing any n-qubit unitary transformation as a diagonal phase map on the computational basis states, as they are naturally diagonalizable by unitarity. By using discrete derivative operators to analytically construct support-selective phase invariants, we enable to deterministically isolate and quantify any multi-qubit interactions encoded in the phase map. These phase invariants form a coordinate system for the formulation of specific control targets in terms of arbitrary desired multi-qubit interactions, without having to invert the diagonalization during the optimizatiion, solely relying on the experimentally accesible diagonal phases. To illustrate the framework, we synthesize two genuinely tripartite entangling gates, both, diagonal and non-diagonal. These are obtained with a single shaped microwave pulse, for a numerically simulated room-temperature nitrogen-vacancy center with a three qubit nuclear spin register, with durations of about a microsecond. These results represent a factor 10-100 reduction in operation time compared with the fastest existing NV-based entanglers that act on more than two qubits at once.