Related papers: On Hamiltonian operators in differential algebras
We present a new approach to simulating Hamiltonian dynamics based on implementing linear combinations of unitary operations rather than products of unitary operations. The resulting algorithm has superior performance to existing simulation…
Following the definitions of the algebras of differential operators, $\beta$-differential operators, and the quantum differential operators on a noncommutative (graded) algebra given in \cite{LR}, we describe these operators on the free…
Time-driven quantum systems are important in many different fields of physics like cold atoms, solid state, optics, etc. Many of their properties are encoded in the time evolution operator which is calculated by using a time-ordered product…
Two models of candidates for hereditary symmetry operators are proposed and thus many nonlinear systems of evolution equations possessing infinitely many commutative symmetries may be generated. Some concrete structures of hereditary…
We investigate the properties of some recently developed variable-order differential operators involving order transition functions of exponential type. Since the characterisation of such operators is performed in the Laplace domain it is…
In this paper we study the algebraic properties of a new integrable differential-difference equation. This equation can be seen as a deformation of the modified Narita-Itoh-Bogoyavlensky equation and has the Kaup-Kupershmidt equation in its…
This paper introduces a notion of gradient and an infimal-convolution operator that extend properties of solutions of Hamilton Jacobi equations to more general spaces, in particular to graphs. As a main application, the hypercontractivity…
Theory of differential operators on associative algebras is not extended to the non-associative ones in a straightforward way. We consider differential operators on Lie algebras. A key point is that multiplication in a Lie algebra is its…
We construct all (2+1)-dimensional PDEs depending only on 2nd-order derivatives of unknown which have the Euler-Lagrange form and determine the corresponding Lagrangians. We convert these equations and their Lagrangians to two-component…
We describe a quantum algorithm for preparing states that encode solutions of non-homogeneous linear partial differential equations. The algorithm is a continuous-variable version of matrix inversion: it efficiently inverts differential…
We propose a distinct approach to solving linear and nonlinear differential equations (DEs) on quantum computers by encoding the problem into ground states of effective Hamiltonian operators. Our algorithm relies on constructing such…
We study some classes of symmetric operators for the discrete series representations of the quantum algebra U_q(su_{1,1}), which may serve as Hamiltonians of various physical systems. The problem of diagonalization of these operators…
In this paper we work with the approximation of unitary groups of operators of the form $e^{-itH}$ where $H\in\mathscr{L}(\mathcal{H})$ is the Hamiltonian of a given quantum dynamical system modeled in the discretizable Hilbert space…
For a general self-adjoint Hamiltonian operator $H_0$ on the Hilbert space $L^2(\RE^d)$, we determine the set of all self-adjoint Hamiltonians $H$ on $L^2(\RE^d)$ that dynamically confine the system to an open set $\Omega \subset \RE^d$…
An overview of Hamiltonian systems with noncanonical Poisson structures is given. Examples of bi-Hamiltonian ode's, pde's and lattice equations are presented. Numerical integrators using generating functions, Hamiltonian splitting,…
The exchange operator formalism in polar coordinates, previously considered for the Calogero-Marchioro-Wolfes problem, is generalized to a recently introduced, infinite family of exactly solvable and integrable Hamiltonians $H_k$, $k=1$, 2,…
It is well-known that if a symplectic integrator is applied to a Hamiltonian system, then the modified equation, whose solutions interpolate the numerical solutions, is again Hamiltonian. We investigate this property from the variational…
A mechanism deriving new well-posed evolutionary equations from given ones is inspected. It turns out that there is one particular spatial operator from which many of the standard evolutionary problems of mathematical physics can be…
We present the basic formulation of Hamilton dynamics in complex phase space. We extend the Hamilton's function by including the imaginary part and find out the corresponding Hamilton's canonical equation of motion. Example of simple…
Recent advances in the field of adiabatic quantum computing and the closely related field of quantum annealers has centered around using more advanced and novel Hamiltonian representations to solve optimization problems. One of these…