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arXiv:2210.13307 · Physical Review A

Dual unitaries as maximizers of the distance to local product gates

Shrigyan Brahmachari, Rohan Narayan Rajmohan, Suhail Ahmad Rather, Arul Lakshminarayan

Published Feb 20, 2024• 8 citations
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Abstract

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 ${K}_{D}(U)$ 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 ${K}_{D}(U)$ is presented for general two-qubit gates. For arbitrary local dimensions, that ${K}_{D}(U)$ 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 ${K}_{D}(U)$ in general.

DOI
10.1103/physreva.109.022610

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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.
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We show that any unitary operator on the ${d}_{A}\ifmmode\times\else\texttimes\fi{}{d}_{B}$ system $({d}_{A}\ensuremath{\ge}2)$ can be decomposed into the product of at most $4{d}_{A}\ensuremath{-}5$ controlled unitary operators. The number can be reduced to $2{d}_{A}\ensuremath{-}1$ when dA{d}_{A}dA​ is a power of two. We also prove that three controlled unitaries can implement a bipartite complex permutation operator, and discuss the connection to an analogous result on classical reversible circuits. We further show that any nnn-partite unitary on the space ${\mathbb{C}}^{{d}_{1}}\ensuremath{\bigotimes}\ensuremath{\cdots}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}_{n}}$ is the product of at most $[2{\ensuremath{\prod}}_{j=1}^{n\ensuremath{-}1}(2{d}_{j}\ensuremath{-}2)\ensuremath{-}1]$ controlled unitary gates, each of which is controlled from $n\ensuremath{-}1$ systems. We also decompose any bipartite unitary into the product of a simple type of bipartite gate and some local unitaries. We derive dimension-independent upper bounds for the controlled-not gate cost or entanglement cost of bipartite permutation unitaries (with the help of ancillas of fixed size) as functions of the Schmidt rank of the unitary. It is shown that such costs under a simple protocol are related to the log-rank conjecture in communication complexity theory via the link of nonnegative rank.
We consider two capacity quantities associated with bipartite unitary gates: the entangling and the disentangling power. Here, we prove that these capacities are different in general by constructing an explicit example of a qubit-qutrit unitary whose entangling power is maximal (2 ebits), but whose disentangling power is strictly less. A corollary is that there can be no unique ordering for unitary gates in terms of their ability to perform nonlocal tasks. Finally, we show that in large dimensions, almost all bipartite unitaries have entangling and disentangling capacities close to maximal.
The entanglement in operator space is a well established measure for the complexity of the quantum many-body dynamics. In particular, that of local operators has recently been proposed as dynamical chaos indicator, i.e. as a quantity able to discriminate between quantum systems with integrable and dynamics. For systems the local-operator entanglement is expected to grow linearly in time, while it is expected to grow at most logarithmically in the integrable case. Here we study local-operator entanglement in dual-unitary quantum circuits, a class of statistically solvable quantum circuits that we recently introduced. We identify a class of completely chaotic dual-unitary circuits where the local-operator entanglement grows linearly and we provide a conjecture for its asymptotic behaviour which is in excellent agreement with the numerical results. Interestingly, our conjecture also predicts a phase transition in the slope of the local-operator entanglement when varying the parameters of the circuits.