Cutoff phenomenon and entropic uncertainty for random quantum circuits

Understand

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.