dorsal/arxiv
View SchemaA geometric theory of non-local two-qubit operations
| Authors | Jun Zhang, Jiri Vala, K. Birgitta Whaley, Shankar Sastry |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0209120 |
| URL | https://arxiv.org/abs/quant-ph/0209120 |
| DOI | 10.1103/PhysRevA.67.042313 |
| Journal | Phys.Rev.A67:042313,2003 |
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 non-local 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.
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"abstract": "We study non-local two-qubit operations from a geometric perspective. By\napplying a Cartan decomposition to su(4), we find that the geometric structure\nof non-local gates is a 3-Torus. We derive the invariants for local\ntransformations, and connect these local invariants to the coordinates of the\n3-Torus. Since different points on the 3-Torus may correspond to the same local\nequivalence class, we use the Weyl group theory to reduce the symmetry. We show\nthat the local equivalence classes of two-qubit gates are in one-to-one\ncorrespondence with the points in a tetrahedron except on the base. We then\nstudy the properties of perfect entanglers, that is, the two-qubit operations\nthat can generate maximally entangled states from some initially separable\nstates. We provide criteria to determine whether a given two-qubit gate is a\nperfect entangler and establish a geometric description of perfect entanglers\nby making use of the tetrahedral representation of non-local gates. We find\nthat exactly half the non-local gates are perfect entanglers. We also\ninvestigate the non-local operations generated by a given Hamiltonian. We first\nstudy the gates that can be directly generated by a Hamiltonian. Then we\nexplicitly construct a quantum circuit that contains at most three non-local\ngates generated by a two-body interaction Hamiltonian, together with at most\nfour local gates generated by single qubit terms. We prove that such a quantum\ncircuit can simulate any arbitrary two-qubit gate exactly, and hence it\nprovides an efficient implementation of universal quantum computation and\nsimulation.",
"arxiv_id": "quant-ph/0209120",
"authors": [
"Jun Zhang",
"Jiri Vala",
"K. Birgitta Whaley",
"Shankar Sastry"
],
"categories": [
"quant-ph"
],
"doi": "10.1103/PhysRevA.67.042313",
"journal_ref": "Phys.Rev.A67:042313,2003",
"title": "A geometric theory of non-local two-qubit operations",
"url": "https://arxiv.org/abs/quant-ph/0209120"
},
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