Mirror symmetry in the geometry of nonlocal two-qubit gates and universal two-qubit quantum circuits

We discuss the transformation of nonlocal two-qubit gates and symmetries possessed by Weyl chamber of two-qubit gates under mirror operation. We define the mirror operation as reflections about a specific set of planes of Weyl chamber in c-space which is spanned by Cartan coordinates. For the lines at the intersection of two specific reflecting planes, both the mirror operation and inverse operation cause the same transformation of Cartan coordinates. The point representing B-gate being the only common point to all reflecting planes, Cartan coordinates and all nonlocal characteristics of B-gate are invariant under mirror operation. In g-space which is spanned by local invariants, the mirror operation can be defined as parity transformation. With the point representing B-gate at the center of Weyl chamber, the mirror symmetry of Weyl chamber is apparent in g-space. In addition, we show that the gate typicality is a linear function of a local invariant and discuss its transformation under mirror operation. We also discuss the ability of a special perfect entangler (SPE) and its mirror gate to construct other nonlocal two-qubit gates and show that an SPE and its mirror together with local y-rotations can generate SWAP operation. Further, we propose universal two-qubit quantum circuits involving two applications of SPEs and local y-rotations. Finally, we propose a scheme to implement SPEs in ion-trap quantum computers which allows the usage of proposed universal two-qubit quantum circuits in quantum computation. We v/ show that B-gate can generate all two-qubit gates in four applications with suitable single-qubit gates. v/ In terms of implementation time, B-gate and one of the proposed universal two-qubit quantum circuits are advantageous over CNOT gate being used as entangling basis gate in ion-trap quantum computer.