Local random quantum circuits are approximate polynomial-designs - numerical results

Understand

We numerically investigate the statement that local random quantum circuits acting on n qubits composed of polynomially many nearest neighbour two-qubit gates form an approximate unitary poly(n)-design [F.G.S.L.

• Brandao et al., 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 a 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 that a gap is closed.