Construction and local equivalence of dual-unitary operators: from dynamical maps to quantum combinatorial designs
arXiv:2205.08842 · doi:10.1103/PRXQuantum.3.040331
Abstract
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 of dual-unitary operators themselves are only partially understood. A nonlinear map on the space of unitary operators was proposed in PRL.~125, 070501 (2020) that results in operators being arbitrarily close to dual unitaries. Here we study the map analytically for the two-qubit case describing the basins of attraction, fixed points, and rates of approach to dual unitaries. A subset of dual-unitary operators having maximum entangling power are 2-unitary operators or perfect tensors, and are equivalent to four-party absolutely maximally entangled states. It is known that they only exist if the local dimension is larger than . We use the nonlinear map, and introduce stochastic variants of it, to construct explicit examples of new dual and 2-unitary operators. A necessary criterion for their local unitary equivalence to distinguish classes is also introduced and used to display various concrete results and a conjecture in . It is known that orthogonal Latin squares provide a ``classical combinatorial design" for constructing permutations that are 2-unitary. We extend the underlying design from classical to genuine quantum ones for general dual-unitary operators and give an example of what might be the smallest sized genuinely quantum design of a 2-unitary in .
arXiv comment26+4 pages, 13+2 Figures. Main changes: Proof of Theorem 1 modified, Theorem 2 concerning two-qubit gates is added, and summary of main results included. Version accepted for publication in PRX Quantum
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