  Non-Gaussian quantum gates are essential components for optical quantum
information processing. However, the efficient implementation of practically
important multi-mode higher-order non-Gaussian gates has not been
comprehensively studied. We propose a measurement-based method to directly
implement general, multi-mode, and higher-order non-Gaussian gates using only
fixed non-Gaussian ancillary states and adaptive linear optics. Compared to
existing methods, our method allows for a more resource-efficient and
experimentally feasible implementation of multi-mode gates that are important
for various applications in optical quantum technology, such as the two-mode
cubic quantum non-demolition gate or the three-mode continuous-variable Toffoli
gate, and their higher-order extensions. Our results will expedite the progress
toward fault-tolerant universal quantum computing with light.
