Construction and local equivalence of dual-unitary operators: from dynamical maps to quantum combinatorial designs

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 $d=2$. 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 $d=4$.

26+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

References in corpus (7)

• Supplementary information for "Quantum supremacy using a programmable superconducting processor"
• Postselection-free entanglement dynamics via spacetime duality
• Exact thermalization dynamics in the "Rule 54" Quantum Cellular Automaton
• Entangling Power of Permutations
• Many Body Quantum Chaos and Dual Unitarity Round-a-Face
• Lower bounds on the complexity of simulating quantum gates
• Existence of Universal Entangler

Cited by in corpus (15)

• Growth of entanglement of generic states under dual-unitary dynamics
• Crystalline Quantum Circuits
• Krylov complexity and Trotter transitions in unitary circuit dynamics
• Absolutely maximally entangled state equivalence and the construction of infinite quantum solutions to the problem of 36 officers of Euler
• From dual-unitary to biunitary: a 2-categorical model for exactly-solvable many-body quantum dynamics
• Exactly solvable many-body dynamics from space-time duality
• Operator dynamics and entanglement in space-time dual Hadamard lattices
• Operator Space Entangling Power of Quantum Dynamics and Local Operator Entanglement Growth in Dual-Unitary Circuits
• Fundamental charges for dual-unitary circuits
• Quantum convolutional channels and multiparameter families of 2-unitary matrices
• The Foliage Partition: An Easy-to-Compute LC-Invariant for Graph States
• Tensor Product Structure Geometry under Unitary Channels
• Quantum Circuits for High-Dimensional Absolutely Maximally Entangled States
• Absolutely maximally entangled pure states of multipartite quantum systems
• Nonlocal characteristics and argand diagram of two-qubit gates