Maximally entangled bipartite unitary operators or gates find various applications from quantum information to many-body physics wherein they are building blocks of minimal models of quantum chaos. In the latter case, they are referred to as "dual unitaries." Dual unitary operators that can create the maximum average entanglement when acting on product states have to satisfy additional constraints. These have been called "2-unitaries" and are examples of perfect tensors that can be used to construct absolutely maximally entangled states of four parties. Hitherto, no systematic method exists in any local dimension, which results in the formation of such special classes of unitary operators. We outline an iterative protocol, a nonlinear map on the space of unitary operators, that creates ensembles whose members are arbitrarily close to being dual unitaries. For qutrits and ququads we find that a slightly modified protocol yields a plethora of 2-unitaries.
Maximally entangled bipartite unitary operators or gates find various applications from quantum information to many-body physics wherein they are building blocks of minimal models of quantum chaos. In the latter case, they are referred to as "dual unitaries." Dual unitary operators that can create the maximum average entanglement when acting on product states have to satisfy additional constraints. These have been called "2-unitaries" and are examples of perfect tensors that can be used to construct absolutely maximally entangled states of four parties. Hitherto, no systematic method exists in any local dimension, which results in the formation of such special classes of unitary operators. We outline an iterative protocol, a nonlinear map on the space of unitary operators, that creates ensembles whose members are arbitrarily close to being dual unitaries. For qutrits and ququads we find that a slightly modified protocol yields a plethora of 2-unitaries.
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 has been proposed in Phys.Rev. Lett.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 these are equivalent to four-party absolutely maximally entangled states.It is known that they only exist if the local dimension is larger than d = 2.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 d = 3.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 d = 4.
Dual unitary gates are highly non-local two-qudit unitary gates that have been studied extensively in quantum many-body physics and quantum information in the recent past. A special class of dual unitary gates consists of rank-four perfect tensors that are equivalent to highly entangled multipartite pure states called absolutely maximally entangled (AME) states. In this work, numerical and analytical constructions of dual unitary gates and perfect tensors that are diagonal in a special maximally entangled basis are presented. The main ingredient in our construction is a phase-valued (unimodular) two-dimensional array whose discrete Fourier transform is also unimodular. We obtain perfect tensors for several local Hilbert space dimensions, particularly, in dimension six. A perfect tensor in local dimension six is equivalent to an AME state of four qudits, denoted as AME(4,6). Such a state cannot be constructed from existing constructions of AME states based on error-correcting codes and graph states. An explicit construction of AME(4,6) states is provided in this work using two-qudit controlled and single-qudit gates making it feasible to generate such states experimentally.
Deterministic classical dynamical systems have an ergodic hierarchy, from ergodic through mixing, to Bernoulli systems that are ``as random as a coin toss.'' Dual-unitary circuits have been recently introduced as solvable models of many-body nonintegrable quantum chaotic systems having a hierarchy of ergodic properties. We extend this to include the apex of a putative quantum ergodic hierarchy which is Bernoulli, in the sense that correlations of single and two-particle observables vanish at space-time separated points. We derive a condition based on the entangling power ep(U) of the basic two-particle unitary building block, U, of the circuit that guarantees mixing, and when maximized, corresponds to Bernoulli circuits. Additionally, we show, both analytically and numerically, how local averaging over random realizations of the single-particle unitaries ui and vi such that the building block is ${U}^{\ensuremath{'}}=({u}_{1}\ensuremath{\bigotimes}{u}_{2})U({v}_{1}\ensuremath{\bigotimes}{v}_{2})$ leads to an identification of the average mixing rate as being determined predominantly by the entangling power ep(U). Finally, we provide several, both analytical and numerical, ways to construct dual-unitary operators covering the entire possible range of entangling power. We construct a coupled quantum cat map, which is dual-unitary for all local dimensions and a 2-unitary or perfect tensor for odd local dimensions, and can be used to build Bernoulli circuits.
The dynamics of quantum many-body systems in the chaotic regime are of particular interest due to the associated phenomena of information scrambling and entanglement generation within the system. While these systems are typically intractable using traditional numerical methods, an effective framework can be implemented based on dual-unitary circuits which have emerged as a minimal model for maximally chaotic dynamics. In this work, we investigate how individual two-body operators influence the global dynamics of circuits composed of dual-unitaries. We study their effect on entanglement generation while examining it from both bipartite and multipartite perspectives. Here we also highlight the significant role of local unitaries in the dynamics when paired with operators from the dual-unitary class, showing that systems with identical entangling power can exhibit a range of differing entanglement growth rates. Furthermore, we present calculations establishing time-step-dependent lower bounds, which depend on both the initial state and the entangling power of the constituent operators. Finally, we find that time-evolving an initial state composed of pair products generates a state with nearly maximal multipartite entanglement content, approaching the bounds established by Absolutely Maximally Entangled (AME) states.
The problem of finding the resource-free, closest local unitary, to any bipartite unitary gate U is addressed. Previously discussed as a measure of nonlocality, the distance KD(U) to the nearest product unitary has implications for circuit complexity and related quantities. Dual unitaries, currently of great interest in models of complex quantum many-body systems, are shown to have a preferred role as these are maximally and equally away from the set of local unitaries. This is proved here for the case of qubits, and we present strong numerical and analytical evidence that it is true in general. An analytical evaluation of KD(U) is presented for general two-qubit gates. For arbitrary local dimensions, that KD(U) is largest for dual unitaries is substantiated by its analytical evaluations for an important family of dual-unitary and for certain nondual gates. A closely allied result concerns for any bipartite unitary the existence of a pair of maximally entangled states that it connects. We give efficient numerical algorithms to find such states and to find KD(U) in general.
Many alternative approaches to construct quantum channels with large entangling capacity were proposed in the past decade, resulting in multiple isolated gates. In this work, we put forward a novel one, inspired by convolution, which provides greater freedom of nonlocal parameters. Although quantum counterparts of convolution have been shown not to exist for pure states, several attempts with various degrees of rigorousness have been proposed for mixed states. In this work, we follow the approach based on coherifications of multi-stochastic operations and demonstrate a surprising connection to gates with high entangling power. In particular, we identify conditions necessary for the convolutional channels constructed using our method to possess maximal entangling power. Furthermore, we establish new, continuous classes of bipartite 2-unitary matrices of dimension d 2 for d = 7 and d = 9 , with 2 and 4 free nonlocal parameters beyond simple phasing of matrix elements, corresponding to perfect tensors of rank 4 or 4-partite absolutely maximally entangled states.