Random unitary circuits with constant spectral gap

What the math gives to ML

The paper gives a representation-uniform mixing guarantee for random walks generated by local random unitaries: the moment operator contracts every non-Haar component by a constant factor independent of the number of qubits. This is potentially transferable to unitary neural layers, recurrent networks, and randomized feature maps, where dense Haar-random matrices are expensive but local brickwork gates are cheap. The practical asset is not merely randomness, but a provable constant-rate approach toward Haar-like moments using depth proportional to the desired mixing accuracy rather than system size. A useful first adaptation is to replace expensive dense unitary initialization or fixed random projections with shallow brickwork SU(4) circuits and test whether their feature covariance and trainability match Haar initialization at much lower cost.