Geometric theory of nonlocal two-qubit operations

Abstract

We study non-local two-qubit operations from a geometric perspective. By applying a Cartan decomposition to su(4), we find that the geometric structure of non-local gates is a 3-Torus. We derive the invariants for local transformations, and connect these local invariants to the coordinates of the 3-Torus. Since different points on the 3-Torus may correspond to the same local equivalence class, we use the Weyl group theory to reduce the symmetry. We show that the local equivalence classes of two-qubit gates are in one-to-one correspondence with the points in a tetrahedron except on the base. We then study the properties of perfect entanglers, that is, the two-qubit operations that can generate maximally entangled states from some initially separable states. We provide criteria to determine whether a given two-qubit gate is a perfect entangler and establish a geometric description of perfect entanglers by making use of the tetrahedral representation of non-local gates. We find that exactly half the non-local gates are perfect entanglers. We also investigate the nonlocal operations generated by a given Hamiltonian. We first study the gates that can be directly generated by a Hamiltonian. Then we explicitly construct a quantum circuit that contains at most three non-local gates generated by a two-body interaction Hamiltonian, together with at most four local gates generated by single qubit terms. We prove that such a quantum circuit can simulate any arbitrary two-qubit gate exactly, and hence it provides an efficient implementation of universal quantum computation and simulation.

Non-Adiabatic Universal Holonomic Quantum Gates Based on Abelian Holonomies

“…A twoqubit quantum gate is local if it belongs to the subgroup SU(2) ⊗ SU(2) generated by span(⊔ i {1 1 ⊗ σ i }) ⊕ span(⊔ i {σ i ⊗ 1 1}), which typically corresponds to the physical spin or polarization eigenstates. A non-local gate is called a perfect entangler if it can produce a maximally entangled state from a tensor product state [30][31][32], such as CNOT gate. In contrast, SWAP gate is a typical non-perfect entangler [30,31].…”

“…A non-local gate is called a perfect entangler if it can produce a maximally entangled state from a tensor product state [30][31][32], such as CNOT gate. In contrast, SWAP gate is a typical non-perfect entangler [30,31]. The Hamiltonian H ′ is capable of creating a GQG that is as well a perfect entangler for a suitable choice of parameters which can be determined by operator Schmidt decomposition [33].…”

Characterization of two-qubit perfect entanglers

“…More discussion on this matter can be found in the appendices and references therein. It is known that [5,17,20] any two-qubit unitary operator U ∈ SU(4) can be written in the following form…”

“…In this paper we investigate the problem of characterization of perfect entanglers [17] from a new perspective. Perfect entanglers are defined as unitary operators that can generate maximally entangled states from some suitably chosen separable states.…”

“…A fundamental relevant question here is how to characterize entangling capabilities of quantum operations. In fact, in this regard a lot of investigation have been done from many different aspects [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18].…”