Geometric theory of nonlocal two-qubit operations

By Jun Zhang, J. Vala, S. Sastry, K. Birgitta Whaley. 2002

386 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/26641f1e4ca1af4d3c1d28a6119e0ea77dc2df74

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In Atomic and Molecular Physics, and Optics

Citations in the corpus

Efficient symmetry-preserving state preparation circuits for the variational quantum eigensolver algorithm (2019, background): “up to an irrelevant global phase.(30) Building upon the example of Fig.”
Geometry of Quantum States: Index (2007, neutral): “30) – see [277, 394] – and other techniques [684, 693, 1000].”
Quantum approximate optimization of non-planar graph problems on a planar superconducting processor (2020, neutral): “Any two qubit gate is locally equivalent to standard KAK form [40].”
Realization of a programmable two-qubit quantum processor (2009, methodology): “Since realising gates acting on two or more qubits tends to be more experimentally challenging[8], much attention has been focused on using them optimally in the creation of entanglement[11] and on finding decompositions minimizing the number of times they are applied[7, 12, 13, 14, 15].”
Geometric (2019, background): “Furthermore, the expansion of the neural manifold’s embedding dimensions required neurons exhibiting nonlinear mixed selectivity in networks with heterogeneous connectivity patterns, thereby echoing the longstanding dictum that structure determines function.”
Synthesis and optimization of reversible circuits—a survey (2011, methodology): “A physical implementation of an elementary operation depends on the Hamiltonian10 of a given quantum system [Zhang et al. 2003].”