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.

When visualized as an operation on the Bloch sphere, the qubit π/8 gate corresponds to 1/8 of a complete rotation about the vertical axis. This simple gate often plays an important role in quantum information theory, typically in situations for which Pauli and Clifford gates are insufficient. Most notably, if it supplements the set of Clifford gates, then universal quantum computation can be achieved. The π/8 gate is the simplest example of an operation from the third level of the Clifford hierarchy (i.e., it maps Pauli operations to Clifford operations under conjugation). Here we derive explicit expressions for all qudit (d-level, where d is prime) versions of this gate and analyze the resulting group structure that is generated by these diagonal gates. This group structure differs depending on whether the dimensionality of the qudit is two, three, or greater than three. We then discuss the geometrical relationship of these gates (and associated states) with respect to Clifford gates and stabilizer states. We present evidence that these gates are maximally robust to depolarizing and phase-damping noise, in complete analogy with the qubit case. Motivated by this and other similarities, we conjecture that these gates could be useful for the task of qudit magic-state distillation and, by extension, fault-tolerant quantum computing. Very recently, independent work by Campbell et al. confirmed the correctness of this intuition, and we build upon their work to characterize noise regimes for which noisy implementations of these gates can (or provably cannot) supplement Clifford gates to enable universal quantum computation.

Enhancement of the production of cold molecules via photoassociation is considered for the Cs 2 system. The employment of chirped picosecond pulses is proposed and studied theoretically. The analysis is based on the ability to achieve impulsive excitation which is given by the ultracold initial conditions where the nuclei are effectively stationary during the interaction with a field. The appropriate theoretical framework is the coordinate-dependent two-level system. Matching the pulse parameters to the potentials and initial conditions results in full Rabi cycling between the electronic potentials. By chirping the laser pulse, adiabatic transfer leading to the population inversion from the ground to the excited state is possible in a broad and tunable range of internuclear distance. Numerical simulations based on solving the time-dependent Schrödinger equation ͑TDSE͒ were performed. The simulation of the photoassociation of Cs 2 from the ground 3 ⌺ u ϩ to the excited 0 g Ϫ state under ultracold conditions verifies the qualitative picture. The ability to control the population transfer is employed to optimize molecular formation. Transfer of population to the excited 0 g Ϫ surface leaves a void in the nuclear density of the ground 3 ⌺ u ϩ surface. This void is either filled by thermal motion or by quantum ''pressure'' and it is the rate-determining step in the photoassociation. The spontaneous-emission process leading to cold-molecules is simulated by including an optical potential in the TDSE. Consequently, the rate of cold molecule formation in a pulsed mode is found to be larger than that obtained in a continuous-wave mode.