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.
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Yepez, J., Vahala, G. & Vahala, L. Lattice Quantum Algorithm for the Schrödinger Wave Equation in 2+1 Dimensions with a Demonstration by Modeling Soliton Instabilities. Quantum Inf Process 4, 457–469 (2005). https://doi.org/10.1007/s11128-005-0008-8
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DOI: https://doi.org/10.1007/s11128-005-0008-8
Keywords
- Non-linear Schrödinger wave equation
- quantum algorithm
- soliton dynamics
- non-linear quantum mechanical instability
- quantum computing
- computational physics