Fig. 1: Evolution of weight-1 Pauli strings.
From: Fast pseudorandom quantum state generators via inflationary quantum gates
An effective 2-qubit reversible gate obtained by averaging uniformly over 2-qubit gates in U(4) leads to equal transition amplitudes among the 15 (=42 − 1) non-trivial weight-1 and weight-2 string states, including a finite stay probability, pw=1 < 1, for weight-1 strings (i.e., a finite amplitude for the transition from a weight-1 to another weight-1 string state). Applying \(\log n\) layers of 2-qubit gates leads to a polynomial tail \({({p}_{w = 1})}^{\log n}={n}^{-\log (1/{p}_{w = 1})}\) in the stay probability of weight-1 strings that, in turn, translates into polynomial tails in OTOCs. Exponential decay of OTOCs in \(\log n\) depth quantum circuits requires the use of special “inflationary” gates that map all weight-1 strings into weight-2 strings, thus eliminating the stay probability for weight-1 strings, as depicted schematically in the inset.
