Dynamically corrected gates from geometric space curves

Quantum information technologies demand highly accurate control over quantum systems. Achieving this requires control techniques that perform well despite the presence of decohering noise and other adverse effects. Here, we review a general technique for designing control fields that dynamically correct errors while performing operations using a close relationship between quantum evolution and geometric space curves. This approach provides access to the global solution space of control fields that accomplish a given task, facilitating the design of experimentally feasible gate operations for a wide variety of applications.

A Different View on Dynamics of Space Curves Geometry

In this study, we define the X-torque curves, $$X-$$ X - equilibrium curves, X-moment conservative curves, $$X-$$ X - gyroscopic curves as new curves derived from a regular space curve by using the Frenet vectors of a space curve and its position vector, where $$X\in \left\{ T\left( s\right) , N\left( s\right) , B\left( s\right) \right\} $$ X ∈ T s , N s , B s and we examine these curves and we give their properties.

Geometric Flows of Curves, Two-Component Camassa-Holm Equation and Generalized Heisenberg Ferromagnet Equation

In this paper, we study the generalized Heisenberg ferromagnet equation, namely, the M-CVI equation. This equation is integrable. The integrable motion of the space curves induced by the M-CVI equation is presented. Using this result, the Lakshmanan (geometrical) equivalence between the M-CVI equation and the two-component Camassa-Holm equation is established.

Visual space curves before eye movements

Despite the incessant retinal disruptions that necessarily accompany eye movements, our percept of the visual world remains continuous and stable—a phenomenon referred to as spatial constancy. How the visual system achieves spatial constancy remains unclear despite almost four centuries worth of experimentation. Here I measured visual sensitivity at geometrically symmetric locations, observing transient sensitivity differences between them where none should be observed if cells that support spatial constancy indeed faithfully translate or converge. These differences, recapitulated by a novel neurobiological mechanical model, reflect an overriding influence of putative visually transient error signals that curve visual space. Intermediate eccentric locations likely to contain retinal disruptions are uniquely affected by curved visual space, suggesting that visual processing at these locations is transiently turned off before an eye movement, and with the gating off of these error signals, turned back on after an eye-movement— a possible mechanism underlying spatial constancy.

Surfaces and Curves Induced by Nonlinear Schrödinger-Type Equations and Their Spin Systems

In recent years, symmetry in abstract partial differential equations has found wide application in the field of nonlinear integrable equations. The symmetries of the corresponding transformation groups for such equations make it possible to significantly simplify the procedure for establishing equivalence between nonlinear integrable equations from different areas of physics, which in turn open up opportunities to easily find their solutions. In this paper, we study the symmetry between differential geometry of surfaces/curves and some integrable generalized spin systems. In particular, we investigate the gauge and geometrical equivalence between the local/nonlocal nonlinear Schrödinger type equations (NLSE) and the extended continuous Heisenberg ferromagnet equation (HFE) to investigate how nonlocality properties of one system are inherited by the other. First, we consider the space curves induced by the nonlinear Schrödinger-type equations and its equivalent spin systems. Such space curves are governed by the Serret–Frenet equation (SFE) for three basis vectors. We also show that the equation for the third of the basis vectors coincides with the well-known integrable HFE and its generalization. Two other equations for the remaining two vectors give new integrable spin systems. Finally, we investigated the relation between the differential geometry of surfaces and integrable spin systems for the three basis vectors.

Some New Characterizations of The Harmonic and Harmonic 1-Type Curves in Euclidean 3-Space

A Laplace operator and harmonic curve have very important uses in various engineering science such as quantum mechanics, wave propagation, diffusion equation for heat, and fluid flow. Additionally, the differential equation characterizations of the harmonic curves play an important role in estimating the geometric properties of these curves. Hence, this paper proposes to compute some new differential equation characterizations of the harmonic curves in Euclidean 3-space by using an alternative frame named the N-Bishop frame. Firstly, we investigated some new differential equation characterizations of the space curves due to the N-Bishop frame. Secondly, we firstly introduced some new space curves which have the harmonic and harmonic 1-type vectors due to alternative frame N-Bishop frame. Finally, we compute new differential equation characterizations using the N-Bishop Darboux and normal Darboux vectors. Thus, using these differential equation characterizations we have proved in which conditions the curve indicates a helix.

Space curves, X-ranks and cuspidal projections

Let $$X\subset \mathbb {P}^3$$ X ⊂ P 3 be an integral and non-degenerate curve. We say that $$q\in \mathbb {P}^3\setminus X$$ q ∈ P 3 \ X has X-rank 3 if there is no line $$L\subset \mathbb {P}^3$$ L ⊂ P 3 such that $$q\in L$$ q ∈ L and $$\#(L\cap X)\ge 2$$ # ( L ∩ X ) ≥ 2 . We prove that for all hyperelliptic curves of genus $$g\ge 5$$ g ≥ 5 there is a degree $$g+3$$ g + 3 embedding $$X\subset \mathbb {P}^3$$ X ⊂ P 3 with exactly $$2g+2$$ 2 g + 2 points with X-rank 3 and another embedding without points with X-rank 3 but with exactly $$2g+2$$ 2 g + 2 points $$q\in \mathbb {P}^3$$ q ∈ P 3 such that there is a unique pair of points of X spanning a line containing q. We also prove the non-existence of points of X-rank 3 for general curves of bidegree (a, b) in a smooth quadric (except in known exceptional cases) and we give lower bounds for the number of pairs of points of X spanning a line containing a fixed $$q\in \mathbb {P}^3\setminus X$$ q ∈ P 3 \ X . For all integers $$g\ge 5$$ g ≥ 5 , $$x\ge 0$$ x ≥ 0 we prove the existence of a nodal hyperelliptic curve X with geometric genus g, exactly x nodes, $$\deg (X) = x+g+3$$ deg ( X ) = x + g + 3 and having at least $$x+2g+2$$ x + 2 g + 2 points of X-rank 3.