A simultaneous feedback and feed-forward control and its application to realize a random walk on the Bloch sphere in a superconducting Xmon-qubit system

Measurement-based feedback control is central in quantum computing and precise quantum control. Here we realize a fast and flexible field-programmable-gate-array-based feedback control in a superconducting Xmon qubit system. The latency of room-temperature electronics is custom optimized to be as short as 140 ns. Projective measurement of a signal qubit produces a feedback tag to actuate a conditional pulse gate to the qubit. In a feed-forward process, the measurement-based feedback tag is brought to a different target qubit for a conditional control. In a two-qubit experiment, the feedback and feed-forward controls are simultaneously actuated in consecutive steps. A quantum number is then generated by the signal qubit, and a random walk of the target qubit is correspondingly triggered and realized on the Bloch sphere. Our experiment provides a conceptually simple and intuitive benchmark for the feedback control in a multi-qubit system. The feedback system can be further scaled up for more complex feedback control experiments.