## Lattice quantum algorithm for the Schrödinger wave equation in $$2+1$$ dimensions with a demonstration by modeling soliton instabilities.(English)Zbl 1130.81027

Summary: A lattice-based quantum algorithm is presented to model the nonlinear 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 nonlinear 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.

### MSC:

 81P68 Quantum computation 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 37K60 Lattice dynamics; integrable lattice equations 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 35Q55 NLS equations (nonlinear Schrödinger equations) 35Q51 Soliton equations
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