Geometric theory of nonlocal two-qubit operations

Journal Article · Tue Apr 01 04:00:00 EST 2003 · Physical Review. A

DOI:https://doi.org/10.1103/PhysRevA.67.042313· OSTI ID:20633958

Jun, Zhang; Sastry, Shankar ^[1]; Vala, Jiri; Whaley, K Birgitta ^[2]

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1. Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, California 94720 (United States)
2. Department of Chemistry and Pitzer Center for Theoretical Chemistry, University of California, Berkeley, California 94720 (United States)

We study nonlocal two-qubit operations from a geometric perspective. By applying a Cartan decomposition to su(4), we find that the geometric structure of nonlocal 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 nonlocal gates. We find that exactly half the nonlocal 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 nonlocal 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.

• OSTI ID: 20633958
• Journal Information: Physical Review. A, Journal Name: Physical Review. A Journal Issue: 4 Vol. 67; ISSN 1050-2947; ISSN PLRAAN
• Country of Publication: United States
• Language: English

Related Subjects

74 ATOMIC AND MOLECULAR PHYSICS
BASIC INTERACTIONS
COMPUTERIZED SIMULATION
EIGENFUNCTIONS
EIGENVALUES
GROUP THEORY
HAMILTONIANS
INFORMATION THEORY
QUANTUM MECHANICS
SYMMETRY
TRANSFORMATIONS
TWO-BODY PROBLEM