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By modifying the collision operator in the quantum lattice gas (QLG) algorithm one can develop an imaginary time (IT) integration to determine the ground state solutions of the Schrödinger equation and its variants. These solutions are compared to those found by other methods (in particular the backward-Euler finite-difference scheme and the quantum lattice Boltzmann). In particular, the ground state of the quantum harmonic oscillator is considered as well as bright solitons in the one-dimensional (1D) non-linear Schrödinger equation. The dark solitons in an external potential are then determined. An advantage of the QLG IT algorithm is the avoidance of any real/complex matrix inversion and that its extension to arbitrary dimensions is straightforward. [ABSTRACT FROM PUBLISHER]

A lattice-based quantum algorithm is presented to model the non-linear Schrödinger-like equations in 2 + 1 dimensions. In this lattice-based model, using only 2 qubits per node, a sequence of unitary collide (qubit–qubit interaction) and stream (qubit translation) operators locally evolve a discrete field of probability amplitudes that in the long-wavelength limit accurately approximates a non-relativistic scalar wave function. The collision operator locally entangles pairs of qubits followed by a streaming operator that spreads the entanglement throughout the two dimensional lattice. The quantum algorithmic scheme employs a non-linear potential that is proportional to the moduli square of the wave function. The model is tested on the transverse modulation instability of a one dimensional soliton wave train, both in its linear and non-linear stages. In the integrable cases where analytical solutions are available, the numerical predictions are in excellent agreement with the theory. [ABSTRACT FROM AUTHOR]

A quantum lattice gas representation is determined for both the non-linear Schro¨dinger (NLS) and Korteweg–de Vries (KdV) equations. There is excellent agreement with the solutions from these representations to the exact soliton–soliton collisions of the integrable NLS and KdV equations. These algorithms could, in principle, be simulated on a hybrid quantum-classical computer. [Copyright &y& Elsevier]

Abstract: A quantum lattice algorithm is developed to examine the effect of an external potential well on exactly integrable vector Manakov solitons. It is found that the exact solutions to the coupled nonlinear Schrodinger equations act like quasi-solitons in weak potentials, leading to mode-locking, trapping and untrapping. Stronger potential wells will lead to the emission of radiation modes from the quasi-soliton initial conditions. If the external potential is applied to that particular mode polarization, then the radiation will be trapped within the potential well. The algorithm developed leads to a finite difference scheme that is unconditionally stable. The Manakov system in an external potential is very closely related to the Gross–Pitaevskii equation for the ground state wave functions of a coupled BEC state at . [Copyright &y& Elsevier]