Radiation effects and defects in solids

Unitary quantum lattice gas algorithm generated from the Dirac collision operator for 1D soliton–soliton collisions

Armen Oganesov, George Vahala, Linda L. Vahala, Jeffrey Yepez, Min Soe, Bo Zhang

Published Jan 2, 2015• 6 citations

Abstract

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’]

DOI: 10.1080/10420150.2014.988625

Related papers

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

Armen Oganesov

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.

Jan 1, 20171 citation

Unitary Quantum Lattice Algorithms for Turbulence

George Vahala, Linda L. Vahala

Abstract : A qubit unitary lattice algorithm is developed for the mean field evolution of the ground state of a Bose-Einstein condensate whether in a magnetic or optical trap. The algorithm consists of interleaved collide-stream operators which in diffusionordering recovers the spinor Gross Pitaevskii equations and is ideally parallelized. The algorithm is benchmarked againstexact one dimensional vector inelastic soliton collision solutions. Three dimensional quantum turbulence is examined forscalar Bose Einstein condensates and the compressible kinetic energy spectrum exhibits three different cascades

May 23, 20160 citations

Quantum lattice gas representation for vector solitons

George Vahala, Linda L. Vahala, Jeffrey Yepez

Quantum lattice gas algorithms are developed for the coupled-nonlinear Schrodinger (coupled-NLS) equations, equations that describe the propagation of pulses in birefringent fibers. When the cross-phase modulation factor is unity, the coupled-NLS reduce to the Manakov equations. The quantum lattice gas algorithm yields vector solitons for the fully integrable Manakov system that are in excellent agreement with exact results. Simulations are also presented for the interaction between a turbulent 2-soliton mode and a simple NLS 2-soliton mode. The quantum algorithm requires 4 qubits for each spatial node, with quantum entanglement required only between pairs of qubits through a unitary collision operator. The coupling between the qubits is achieved through a local phase change in the absolute value of the paired qubit wave functions. On symmetrizing the unitary streaming operators, the resulting quantum algorithm, which is unconditionally stable, is accurate to O(ε2).

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIEAug 5, 20033 citations

Spin-2 BEC spinor superfluid soliton-soliton scattering in one and two space dimensions

Jasper Smead Taylor, Smith, Steven, Jeffrey Yepez

Presented is a study of a spin-2 Bose-Einstein condensate (BEC) by unitary quantum simulations of time-dependent soliton-soliton scattering. The quantum simulation method is based on a quantum lattice algorithm which is designed for implementation on a future digital quantum computer but is tested today using a parallel computing architecture based on graphical processing units (GPUs). We analytically solve the spin-2 BEC equations of motion, a nonlinear system of 5 coupled Gross-Pitiaevskii (GP) equations, in one- and two-spatial dimensions. In 1D there are 16 bright soliton and 16 dark soliton solutions. In 2D there are 3 dark soliton solutions Pade approximation solutions, for m_f=+/-2, m_f=+/-1 and m_f=0, corresponding to quantum vortices. We report on the implementation the unitary quantum lattice gas algorithm for spinor superfluid and establish its efficacy by validating the stability of the 1D and 2D energy eigenstate solutions of the spin-2 BEC Hamiltonian. Using the calibrated quantum lattice gas algorithm, the highly nonlinear physics in the nonintegrable regime of the spin-2 BEC is studied by performing soliton-soliton scattering experiments. The scattering of topological solitons produces breathers and complex quantum vortices characterized by local entanglement across multiple m_f-hyperfine states of the Zeeman manifold.

arXiv:1907.12834Jul 30, 20191 citation

Effect of Fourier transform on the streaming in quantum lattice gas algorithms

Armen Oganesov, George Vahala, Linda L. Vahala +1 more

All our previous quantum lattice gas algorithms for nonlinear physics have approximated the kinetic energy operator by streaming sequences to neighboring lattice sites. Here, the kinetic energy can be treated to all orders by Fourier transforming the kinetic energy operator with interlaced Dirac-based unitary collision operators. Benchmarking against exact solutions for the 1D nonlinear Schrodinger equation shows an extended range of parameters (soliton speeds and amplitudes) over the Dirac-based near-lattice-site streaming quantum algorithm.

Radiation effects and defects in solidsApr 3, 201812 citations

Qubit unitary lattice algorithm for spin-2 Bose–Einstein Condensates. I – Theory and Pade initial conditions

George Vahala, Linda L. Vahala, Min Soe

A qubit unitary lattice algorithm is developed for the evolution of five coupled Gross–Pitaesvkii equations that govern the dynamics of a spin-2 Bose–Einstein Condensate. On introducing 2 qubits/scalar component of the order parameter, 10 qubits/lattice node are required. The interleaved sequence of unitary collide-stream operators relies on their non-commutivity in order that their be non-trivial evolution equations. This collisional entanglement followed by streaming of this entanglement throughout the lattice is a hallmark of the algorithm. The qubit algorithm for spin-2 is summarized and new Pade initial conditions are determined. Being unitary, the algorithm can be directly encoded onto a quantum-gate computer.

Radiation effects and defects in solidsJan 2, 20208 citations