Measurement-Free Ultrafast Quantum Error Correction by Using Multi-Controlled Gates in Higher-Dimensional State Space

Quantum error correction is a crucial step beyond the current noisy-intermediate-scale quantum device towards fault-tolerant quantum computing. However, most of the error corrections ever demonstrated rely on post-selection of events or post-correction of states, based on measurement results repeatedly recorded during circuit execution. On the other hand, real-time error correction is supposed to be performed through classical feedforward of the measurement results to data qubits. It provides unavoidable latency from conditional electronics that would limit the scalability of the next-generation quantum processors. Here we propose a new approach to real-time error correction that is free from measurement and realized by using multi-controlled gates based on higher-dimensional state space. Specifically, we provide a series of novel decompositions of a Toffoli gate by using the lowest three energy levels of a transmon that significantly reduce the number of two-qubit gates and discuss their essential features, such as extendability to an arbitrary number of control qubits, the necessity of exclusively controlled NOT gates, and usefulness of their incomplete variants. Combined with the recently demonstrated schemes of fast two-qubit gates and all-microwave qubit reset, it would substantially shorten the time required for error correction and resetting ancilla qubits compared to a measurement-based approach and provide an error correction rate of $\gtrsim1$~MHz with high accuracy for three-qubit bit- and phase-flip errors.