Quadratic Dynamical Decoupling with Non-Uniform Error Suppression

We analyze numerically the performance of the near-optimal quadratic dynamical decoupling (QDD) single-qubit decoherence errors suppression method [J. West et al., Phys. Rev. Lett. 104, 130501 (2010)]. The QDD sequence is formed by nesting two optimal Uhrig dynamical decoupling sequences for two orthogonal axes, comprising N1 and N2 pulses, respectively. Varying these numbers, we study the decoherence suppression properties of QDD directly by isolating the errors associated with each system basis operator present in the system-bath interaction Hamiltonian. Each individual error scales with the lowest order of the Dyson series, therefore immediately yielding the order of decoherence suppression. We show that the error suppression properties of QDD are dependent upon the parities of N1 and N2, and near-optimal performance is achieved for general single-qubit interactions when N1=N2.Comment: 17 pages, 22 figure

Optimal Dynamical Decoherence Control of a Qubit

A theory of dynamical control by modulation for optimal decoherence reduction is developed. It is based on the non-Markovian Euler-Lagrange equation for the energy-constrained field that minimizes the average dephasing rate of a qubit for any given dephasing spectrum.Comment: 6 pages, including 2 figures and an appendi

Observability inequalities for transport equations through Carleman estimates

We consider the transport equation \ppp_t u(x,t) + H(t)\cdot \nabla u(x,t) = 0 in \OOO\times(0,T), where $T>0$ and \OOO\subset \R^d is a bounded domain with smooth boundary \ppp\OOO. First, we prove a Carleman estimate for solutions of finite energy with piecewise continuous weight functions. Then, under a further condition which guarantees that the orbits of $H$ intersect \ppp\OOO, we prove an energy estimate which in turn yields an observability inequality. Our results are motivated by applications to inverse problems.Comment: 18 pages, 3 figure

Rediscovering Manning’S Equation Using Genetic Programming

Open-channel hydraulics’ (OCH) research traditionally links empirical formulas to observational data. One of the most common equations in OCH is Manning’s formula for open channel flow (Q) driven by gravity (also known as the Gauckler-Manning-Strickler formula). The formula relates the cross-sectional average velocity (V=Q/A), the hydraulic radius (R), and the slope of the water surface (S) with a friction coefficient n, characteristic of the channel’s surface. Here we show a practical example where Genetic Programming (GP), a technique derived from Bioinformatics, can be used to derive an empirical relationship based on different synthetic datasets of the aforementioned parameters. Specifically, we evaluated if Manning’s formula could be retrieved from datasets with 300 pentads of A, n, R, S, and Q (from Manning’s equation) using GP. The cross-validated results show success retrieving the functional form from the synthetic data and encourage the application of GP on problems where traditional empirical relationships show high biases, like sediment transport. The results also show alternative flow equations that can be used in the absence of one of the predictors and approximate Manning’s equation

Minimal and Robust Composite Two-Qubit Gates with Ising-Type Interaction

We construct a minimal robust controlled-NOT gate with an Ising-type interaction by which elementary two-qubit gates are implemented. It is robust against inaccuracy of the coupling strength and the obtained quantum circuits are constructed with the minimal number (N=3) of elementary two-qubit gates and several one-qubit gates. It is noteworthy that all the robust circuits can be mapped to one-qubit circuits robust against a pulse length error. We also prove that a minimal robust SWAP gate cannot be constructed with N=3, but requires N=6 elementary two-qubit gates.Comment: 7 pages, 2 figure

Microscopic Analysis of the Non-Dissipative Force on a Line Vortex in a Superconductor: Berry's Phase, Momentum Flows and the Magnus Force

A microscopic analysis of the non-dissipative force ${\bf F}_{nd}$ acting on a line vortex in a type-II superconductor at $T=0$ is given. We first examine the Berry phase induced in the true superconducting ground state by movement of the vortex and show how this induces a Wess-Zumino term in the hydrodynamic action $S_{hyd}$ of the superconducting condensate. Appropriate variation of $S_{hyd}$ gives ${\bf F}_{nd}$ and variation of the Wess-Zumino term is seen to contribute the Magnus (lift) force of classical hydrodynamics to ${\bf F}_ {nd}$. This first calculation confirms and strengthens earlier work by Ao and Thouless which was based on an ansatz for the many-body ground state. We also determine ${\bf F}_{nd}$ through a microscopic derivation of the continuity equation for the condensate linear momentum. This equation yields the acceleration equation for the superflow and shows that the vortex acts as a sink for the condensate linear momentum. The rate at which momentum is lost to the vortex determines ${\bf F}_{nd}$ and the result obtained agrees with the Berry phase calculation. The Magnus force contribution to ${\bf F}_{nd}$ is seen to be a consequence of the vortex topology. Preliminary remarks are made regarding finite temperature extensions, with emphasis on its relevance to the sign anomaly occurring in Hall effect experiments done in the flux flow regime.Comment: 40 pages, RevTex, UBCTP-94-00