Fidelity of optimally controlled quantum gates with randomly coupled multiparticle environments

This work studies the feasibility of optimal control of high-fidelity quantum gates in a model of interacting two-level particles. One particle (the qubit) serves as the quantum information processor, whose evolution is controlled by a time-dependent external field. The other particles are not directly controlled and serve as an effective environment, coupling to which is the source of decoherence. The control objective is to generate target one-qubit gates in the presence of strong environmentally-induced decoherence and under physically motivated restrictions on the control field. It is found that interactions among the environmental particles have a negligible effect on the gate fidelity and require no additional adjustment of the control field. Another interesting result is that optimally controlled quantum gates are remarkably robust to random variations in qubit-environment and inter-environment coupling strengths. These findings demonstrate the utility of optimal control for management of quantum-information systems in a very precise and specific manner, especially when the dynamics complexity is exacerbated by inherently uncertain environmental coupling.Comment: tMOP LaTeX, 9 pages, 3 figures; Special issue of the Journal of Modern Optics: 37th Winter Colloquium on the Physics of Quantum Electronics, 2-6 January 200

Encoding a qubit into multilevel subspaces

We present a formalism for encoding the logical basis of a qubit into subspaces of multiple physical levels. The need for this multilevel encoding arises naturally in situations where the speed of quantum operations exceeds the limits imposed by the addressability of individual energy levels of the qubit physical system. A basic feature of the multilevel encoding formalism is the logical equivalence of different physical states and correspondingly, of different physical transformations. This logical equivalence is a source of a significant flexibility in designing logical operations, while the multilevel structure inherently accommodates fast and intense broadband controls thereby facilitating faster quantum operations. Another important practical advantage of multilevel encoding is the ability to maintain full quantum-computational fidelity in the presence of mixing and decoherence within encoding subspaces. The formalism is developed in detail for single-qubit operations and generalized for multiple qubits. As an illustrative example, we perform a simulation of closed-loop optimal control of single-qubit operations for a model multilevel system, and subsequently apply these operations at finite temperatures to investigate the effect of decoherence on operational fidelity.Comment: IOPart LaTeX, 2 figures, 31 pages; addition of a numerical simulatio

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

House Prices and Economic Growth

Economic growth, House prices, Wealth effect, Collateral effect, Common correlated effects estimators, Long horizon predictability, E23, E24, R11,

Numerical Implementation and Test of the Modified Variational Multiconfigurational Gaussian Method for High-Dimensional Quantum Dynamics

In this paper, a new numerical implementation and a test of the modified variational multiconfigurational Gaussian (vMCG) equations are presented. In vMCG, the wave function is represented as a superposition of trajectory guided Gaussian coherent states, and the time derivatives of the wave function parameters are found from a system of linear equations, which in turn follows from the variational principle applied simultaneously to all wave function parameters. In the original formulation of vMCG, the corresponding matrix was not well-behaved and needed regularization, which required matrix inversion. The new implementation of the modified vMCG equations seems to have improved the method, which now enables straightforward solution of the linear system without matrix inversion, thus achieving greater efficiency, stability and robustness. Here the new version of the vMCG approach is tested against a number of benchmarks, which previously have been studied by split-operator, multiconfigurational time-dependent Hartree (MCTDH) and multilayer MCTDH (ML-MCTDH) techniques. The accuracy and efficiency of the new implementation of vMCG is directly compared with the method of coupled coherent states (CCS), another technique that uses trajectory guided grids. More generally we demonstrate that trajectory guided Gaussian based methods are capable of simulating quantum systems with tens or even hundreds of degrees of freedom previously accessible only for MCTDH and ML-MCTDH