Comment on the paper "Random Quantum Circuits are Approximate 2-designs"

In [A.W. Harrow and R.A. Low, Commun. Math. Phys. 291, 257-302 (2009)], it was shown that a quantum circuit composed of random 2-qubit gates converges to an approximate quantum 2-design in polynomial time. We point out and correct a flaw in one of the paper’s main arguments. Our alternative argument highlights the role played by transpositions induced by the random gates in achieving convergence.