Comment on the paper "Random Quantum Circuits are Approximate 2-designs"
Igor Tuche Diniz, Daniel Jonathan
TL;DR
The note identifies a flaw in HL08a's Corollary 5.1, which claimed that a random quantum circuit reaches an approximate 2-design in $\Theta(n\log(n/\\varepsilon))$ steps based on a SST argument. It then presents an alternative, symmetry-based strategy that reduces the analysis to random transpositions, leveraging known results from Diaconis to show that the full Markov chain $P$ mixes in the same $\Theta(n\log(n/\\varepsilon))$ bound. By connecting a projected chain $Q$ to a transposition chain and proving equivalences between their mixing times, the authors restore the desired convergence result within the original circuit model. Consequently, the polynomial-time convergence of random circuits to approximate 2-designs is upheld, with the transpositions playing a central role in achieving mixing. The work clarifies the mechanism behind convergence and reinforces the relevance of symmetry and representation theory in analyzing quantum circuit randomness.
Abstract
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.