Molecular Quantum Computing by an Optimal Control Algorithm for Unitary Transformations

Quantum computation is based on implementing selected unitary transformations which represent algorithms. A generalized optimal control theory is used to find the driving field that generates a prespecified unitary transformation. The approach is illustrated in the implementation of one and two qubits gates in model molecular systems.Comment: 10 pages, 2 figure

Negativity as a distance from a separable state

The computable measure of the mixed-state entanglement, the negativity, is shown to admit a clear geometrical interpretation, when applied to Schmidt-correlated (SC) states: the negativity of a SC state equals a distance of the state from a pertinent separable state. As a consequence, a SC state is separable if and only if its negativity vanishes. Another remarkable consequence is that the negativity of a SC can be estimated "at a glance" on the density matrix. These results are generalized to mixtures of SC states, which emerge in certain quantum-dynamical settings.Comment: 9 pages, 1 figur

Dynamic Matter-Wave Pulse Shaping

In this paper we discuss possibilities to manipulate a matter-wave with time-dependent potentials. Assuming a specific setup on an atom chip, we explore how one can focus, accelerate, reflect, and stop an atomic wave packet, with, for example, electric fields from an array of electrodes. We also utilize this method to initiate coherent splitting. Special emphasis is put on the robustness of the control schemes. We begin with the wave packet of a single atom, and extend this to a BEC, in the Gross-Pitaevskii picture. In analogy to laser pulse shaping with its wide variety of applications, we expect this work to form the base for additional time-dependent potentials eventually leading to matter-wave pulse shaping with numerous application

New, Highly Accurate Propagator for the Linear and Nonlinear Schr\"odinger Equation

A propagation method for the time dependent Schr\"odinger equation was studied leading to a general scheme of solving ode type equations. Standard space discretization of time-dependent pde's usually results in system of ode's of the form u_t -Gu = s where G is a operator (matrix) and u is a time-dependent solution vector. Highly accurate methods, based on polynomial approximation of a modified exponential evolution operator, had been developed already for this type of problems where G is a linear, time independent matrix and s is a constant vector. In this paper we will describe a new algorithm for the more general case where s is a time-dependent r.h.s vector. An iterative version of the new algorithm can be applied to the general case where G depends on t or u. Numerical results for Schr\"odinger equation with time-dependent potential and to non-linear Schr\"odinger equation will be presented.Comment: 14 page

Optimal control theory for unitary transformations

The dynamics of a quantum system driven by an external field is well described by a unitary transformation generated by a time dependent Hamiltonian. The inverse problem of finding the field that generates a specific unitary transformation is the subject of study. The unitary transformation which can represent an algorithm in a quantum computation is imposed on a subset of quantum states embedded in a larger Hilbert space. Optimal control theory (OCT) is used to solve the inversion problem irrespective of the initial input state. A unified formalism, based on the Krotov method is developed leading to a new scheme. The schemes are compared for the inversion of a two-qubit Fourier transform using as registers the vibrational levels of the $X^1\Sigma^+_g$ electronic state of Na$_2$. Raman-like transitions through the $A^1\Sigma^+_u$ electronic state induce the transitions. Light fields are found that are able to implement the Fourier transform within a picosecond time scale. Such fields can be obtained by pulse-shaping techniques of a femtosecond pulse. Out of the schemes studied the square modulus scheme converges fastest. A study of the implementation of the $Q$ qubit Fourier transform in the Na$_2$ molecule was carried out for up to 5 qubits. The classical computation effort required to obtain the algorithm with a given fidelity is estimated to scale exponentially with the number of levels. The observed moderate scaling of the pulse intensity with the number of qubits in the transformation is rationalized.Comment: 32 pages, 6 figure

Noise and Controllability: suppression of controllability in large quantum systems

A closed quantum system is defined as completely controllable if an arbitrary unitary transformation can be executed using the available controls. In practice, control fields are a source of unavoidable noise. Can one design control fields such that the effect of noise is negligible on the time-scale of the transformation? Complete controllability in practice requires that the effect of noise can be suppressed for an arbitrary transformation. The present study considers a paradigm of control, where the Lie-algebraic structure of the control Hamiltonian is fixed, while the size of the system increases, determined by the dimension of the Hilbert space representation of the algebra. We show that for large quantum systems, generic noise in the controls dominates for a typical class of target transformations i.e., complete controllability is destroyed by the noise.Comment: 4 pages, no figure

A numerical boundary integral equation method for elastodynamics. I

The boundary initial value problems of elastodynamics are formulated as boundary integral equations. It is shown that these integral equations may be solved by time-stepping numerical methods for the unknown boundary values. A specific numerical scheme is presented for antiplane strain problems and a numerical example is given