Operator spreading in random unitary circuits with unitary-invariant gate distributions

Random unitary circuits have become a model system to investigate information scrambling in quantum systems. In the literature, mostly random circuits with Haar-distributed gate operations have been considered. In this work, we investigate operator spreading in random unitary circuits in which the elementary gate operations are drawn from general unitary-invariant ensembles, which include the well-studied Haar-distributed random unitary circuits as a special case. Similar to the Haar-distributed case, the long-time behavior of operator spreading with the more general unitary-invariant gate distribution is governed by drift-diffusion equations characterized by the butterfly velocity <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:msub><a:mi>v</a:mi><a:mi mathvariant="normal">B</a:mi></a:msub></a:math> and a diffusion constant <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi mathvariant="script">D</c:mi></c:math>. Differences with the Haar-random case are (i) that it takes a finite time <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:msub><e:mi>τ</e:mi><e:mi mathvariant="normal">b</e:mi></e:msub></e:math> until ensemble-averaged Pauli-string weights take a “binary” form, in which they depend only on whether Pauli operators inside the support of the Pauli strong are equal to the identity matrix, and (ii) that the operator spreading is characterized by a finite “domain-wall width” <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:msub><g:mi>n</g:mi><g:mi>DW</g:mi></g:msub></g:math> separating regions with a random-matrix-like Pauli-string distribution. To illustrate these findings, we perform explicit calculations for random unitary circuits distributed according to the Poisson kernel, which interpolates between the trivial and Haar-distributed circuits.