Modeling The Gross-Pitaevskii Equation using The Quantum Lattice Gas Method

We present an improved Quantum Lattice Gas (QLG) algorithm as a mesoscopic unitary perturbative representation of the mean field Gross Pitaevskii (GP) equation for Bose–Einstein Condensates (BECs). The method employs an interleaved sequence of unitary collide and stream operators. QLG is applicable to many different scalar potentials in the weak interaction regime and has been used to model the Korteweg–de Vries (KdV), Burgers and GP equations. It can be implemented on both quantum and classical computers and is extremely scalable. We present results for 1D soliton solutions with positive and negative internal interactions, as well as vector solitons with inelastic scattering. In higher dimensions we look at the behavior of vortex ring reconnection. A further improvement is considered with a proper operator splitting technique via a Fourier transformation. This is great for quantum computers since the quantum FFT is exponentially faster than its classical counterpart which involves non-local data on the entire lattice (Quantum FFT is the backbone of the Shor algorithm for quantum factorization). We also present an imaginary time method in which we transform the Schrödinger equation into a diffusion equation for recovering ground state initial conditions of a quantum system suitable for the QLG algorithm

Benchmarking the Dirac-generated unitary lattice qubit collision-stream algorithm for 1D vector Manakov soliton collisions

The unitary quantum lattice gas (QLG) algorithm is a mesoscopic unitary perturbative representation that can model the mean field Gross Pitaevskii equation for the evolution of the ground state wave function of Bose Einstein Condensates (BECs). The QLG considered here consists of an interleaved sequence of unitary collide-stream operators, with the collision operator being deduced from that for the Dirac equation, with the nonlinear potentials of the BECs being the mass term in the Dirac equation. Since the unitary collision operator is more accurate one obtains a more accurate representation of the nonlinear terms. Further benchmark QLG simulations are reported here: that for the exactly soluble 1D vector Manakov soliton collisions. It is found that this Dirac-based unitary algorithm permits simulations with vector soliton parameters (soliton amplitudes and speeds) that are considerably greater than those achieved under our previous root swap QLG algorithm. (C) 2015 Elsevier Ltd. All rights reserved