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Radiation effects and defects in solids

Imaginary time integration method using a quantum lattice gas approach

Armen Oganesov, Christopher Flint, George Vahala, Linda L. Vahala, Jeffrey Yepez, Min Soe

Published Feb 1, 2016• 7 citations
Publisher

Abstract

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.

DOI
10.1080/10420150.2015.1137916

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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.
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.
A new unitary quantum lattice gas (QLG) algorithm is proposed as a mesoscopic unitary perturbative representation of the mean field Gross Pitaevskii equation for Bose–Einstein Condensates (BECs). This consists of an interleaved sequence of unitary collide-stream operators and is tested on the 1D nonlinear Schrödinger (NLS) equation since exact soliton solutions are well known. An earlier QLG algorithm, based on the collision operator has been found to have limited application to spinor-BECs. Here, a new unitary collision operator, based on the recent QLG of Yepez for the Dirac particle, is used to model the 1D NLS soliton–soliton problem. It is found that this new unitary algorithm can handle parameters (soliton amplitudes and speeds) a factor of over 20 greater than those under the previous algorithm.[Re: (v.07) To be submitted to special issue ‘Plasma and Fluid Dynamics: Computational, Experimental and Theoretical Advancement’]
The multidimensional formulation of the quantum lattice Boltzmann (qLB) scheme is extended to the case of nonlinear quantum wave equations. More specifically, imaginary-time formulations of the qLB scheme are developed and applied to the numerical computation of the ground state of the Gross-Pitaevskii equation in one and two spatial dimensions. The calculation is validated through detailed comparison with other numerical methods, as well as with analytical results based on the Thomas-Fermi approximation. The linear scaling of the time-step size with the spatial mesh spacing, a distinctive feature of the present quantum kinetic approach, is also numerically demonstrated.
The nonlinear Schrodinger (NLS) equation in a self-defocusing Kerr medium supports dark solitons. Moreover the mean field description of a dilute Bose-Einstein condensate (BEC) is described by the Gross-Pitaevskii equation, which for a highly anisotropic (cigar-shaped) magnetic trap reduces to a one-dimensional (1D) cubic NLS in an external potential. A quantum lattice algorithm is developed for the dark solitons. Simulations are presented for both black (stationary) solitons as well as (moving) dark solitons. Collisions of dark solitons are compared with the exact analytic solutions and coupled dark-bright vector solitons are examined. The quantum algorithm requires 2 qubits per scalar field at each spatial node. The unitary collision operator quantum mechanically entangles the on-site qubits, and this transitory entanglement is spread throughout the lattice by the streaming operators. These algorithms are suitable for a Type-II quantum computers, with wave function collapse induced by quantum measurements required to determine the coupling potentials.
Lattice Boltzmann (LB) methods have proven to be powerful tools for simulating classical physical systems such as waves, diffusion and fluid dynamics. For quantum mechanics in one dimension there are two main schemes: one proposed by Meyer, where the Schrödinger equation is interpreted as a diffusion equation with imaginary diffusion constant; the other, proposed by Succi, is based on the Dirac equation, which in the mesoscopic limit reproduces the non-relativistic quantum behaviour in a similar way to lattice Boltzmann models for fluids. In the present work we compare these models by testing the quantum time evolution of a free Gaussian wavepacket. It is found to be in close correspondence with the results of the two models. Time oscillations are found both for the mean and variance of the wavepacket. Also we found their dependences from factors not present in the predicted values. These elements could provide clues for developing a good adaptation of lattice Boltzmann method for the quantum world.