Random Quantum Circuits are Approximate 2-designs
Aram W. Harrow, Richard A. Low
TL;DR
The paper investigates how random quantum circuits formed from universal two-qubit gate sets rapidly emulate low-order Haar statistics. By mapping the evolution of moments to a Pauli-expansion framework and, for k=2, to a classical Markov chain, the authors prove that polynomial-length circuits yield ε-approximate unitary 1- and 2-designs. They establish tight convergence bounds, especially for the U(4) gate set, and show that universal gate sets on U(4) induce a 2-copy gap that guarantees rapid mixing to Haar-like second moments. The results unify and extend prior constructions (e.g., Clifford-based designs) and provide a scalable route to efficient pseudo-random unitaries with broad applicability in decoupling and quantum information processing. The work also sets the stage for extending the approach to higher k-designs and other circuit models, including stabiliser and Clifford circuits, with practical implications for quantum simulations and information-theoretic tasks.
Abstract
Given a universal gate set on two qubits, it is well known that applying random gates from the set to random pairs of qubits will eventually yield an approximately Haar-distributed unitary. However, this requires exponential time. We show that random circuits of only polynomial length will approximate the first and second moments of the Haar distribution, thus forming approximate 1- and 2-designs. Previous constructions required longer circuits and worked only for specific gate sets. As a corollary of our main result, we also improve previous bounds on the convergence rate of random walks on the Clifford group.