Local random quantum circuits are approximate polynomial-designs: numerical results
Piotr Ćwikliński, Michał Horodecki, Marek Mozrzymas, Łukasz Pańkowski, Michał Studziński · Journal of Physics A Mathematical and Theoretical · 2013
We numerically investigate the statement that local random quantum circuits acting on n qubits composed of polynomially many nearest-neighbor two-qubit gates form an approximate unitary poly( n )-design (Brandão et al 2012 arXiv:1208.0692). Using a group theory formalism, spectral gaps that give a ratio of convergence to a given t -design are evaluated for a different number of qubits n (up to 20) and degrees t ( t = 2, 3, 4 and 5), improving previously known results for n = 2 in the case of t = 2 and 3. Their values lead to the conclusion that the previously used lower bound that bounds spectral gaps values may give very little information about the real situation and in most cases, only tells us that a gap is closed. We compare our results to another lower bounding technique, again showing that its results may not be tight.