  How fast a state of a system converges to a stationary state is one of the
fundamental questions in science. Some Markov chains and random walks on finite
groups are known to exhibit the non-asymptotic convergence to a stationary
distribution, called the cutoff phenomenon. Here, we examine how quickly a
random quantum circuit could transform a quantum state to a Haar-measure random
quantum state. We find that random quantum states, as stationary states of
random walks on a unitary group, are invariant under the quantum Fourier
transform. Thus the entropic uncertainty of random quantum states has balanced
Shannon entropies for the computational bases and the quantum Fourier transform
bases. By calculating the Shannon entropy for random quantum states and the
Wasserstein distances for the eigenvalues of random quantum circuits, we show
that the cutoff phenomenon occurs for the random quantum circuit. It is also
demonstrated that the Dyson-Brownian motion for the eigenvalues of a random
unitary matrix as a continuous random walk exhibits the cutoff phenomenon. The
results here imply that random quantum states could be generated with shallow
random circuits.
