Convergence conditions for random quantum circuits

Efficient methods for generating pseudorandomly distributed unitary operators are needed for the practical application of Haar-distributed random operators in quantum communication and noise estimation protocols. We develop a theoretical framework for analyzing pseudorandom ensembles generated through a random circuit composition. We prove that the measure over random circuits converges exponentially (with increasing circuit length) to the uniform (Haar) measure on the unitary group, though the rate for uniform convergence must decrease exponentially with the number of qubits. We describe how the rate of convergence for test functions associated with specific randomization tasks leads to weaker convergence conditions that may allow efficient random circuit constructions.