Construction and Local Equivalence of Dual-Unitary Operators: From Dynamical Maps to Quantum Combinatorial Designs

While quantum circuits built from two-particle dual-unitary (maximally entangled) operators serve as minimal models of typically nonintegrable many-body systems, the construction and characterization themselves are only partially understood. A nonlinear map on space unitary was proposed in PRL.~125, 070501 (2020) that results being arbitrarily close to dual unitaries. Here we study analytically for two-qubit case describing basins attraction, fixed points, rates approach subset having maximum entangling power 2-unitary or perfect tensors, equivalent four-party absolutely maximally entangled states. It is known they exist if local dimension larger than $d=2$. We use map, introduce stochastic variants it, construct explicit examples new operators. necessary criterion their equivalence distinguish classes also introduced used display various concrete a conjecture $d=3$. orthogonal Latin squares provide ``classical combinatorial design" constructing permutations 2-unitary. extend underlying design classical genuine ones general give an example what might be smallest sized genuinely $d=4$.