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We define different classes of local random quantum circuits (L-RQC) and showthat: a) statistical properties of L-RQC are encoded into an associated familyof completely positive maps and b) average purity dynamics can be described bythe action of these maps on operator algebras of permutations (swap algebras).An exactly solvable one-dimensional case is analyzed to illustrate the power ofthe swap algebra formalism. More in general, we prove short time area-lawbounds on average purity for uncorrelated L-RQC and infinite time results forboth the uncorrelated and correlated cases.

Local random quantum circuits: ensemble CP maps and swap algebras