Characterizing the geometrical edges of nonlocal two-qubit gates

TL;DR

This paper geometrically characterizes nonlocal two-qubit gates within the Weyl chamber, identifying key edges formed by parametric gates and analyzing their entangling properties, including optimal CNOT constructions.

Contribution

It reveals that all edges of the Weyl chamber are formed by single parametric gates and characterizes their nonlocal attributes using entangling power and invariants.

Findings

SWAP-alpha family forms an edge of the Weyl chamber

SWAP-1/2 is the only perfect entangler

Optimal CNOT constructions using specific edge gates

Abstract

Nonlocal two-qubit gates are geometrically represented by tetrahedron known as Weyl chamber within which perfect entanglers form a polyhedron. We identify that all edges of the Weyl chamber and polyhedron are formed by single parametric gates. Nonlocal attributes of these edges are characterized using entangling power and local invariants. In particular, SWAP (power)alpha family of gates constitutes one edge of the Weyl chamber with SWAP-1/2 being the only perfect entangler. Finally, optimal constructions of controlled-NOT using SWAP-1/2 gate and gates belong to three edges of the polyhedron are presented.