Jeffrey Yepez

Entropic lattice Boltzmann methods

We present a general methodology for constructing lattice Boltzmann models of hydrodynamics with certain desired features of statistical physics and kinetic theory. We show how a methodology of linear programming theory, known as Fourier‐Motzkin elimination, provides an important tool for visualizing the state‐space of lattice Boltzmann algorithms that conserve a given set of moments of the distribution function. We show how such models can be endowed with a Lyapunov functional, analogous to Boltzmanns H, resulting in unconditional numerical stability. Using the Chapman‐Enskog analysis and numerical simulation, we demonstrate that such entropically stabilized lattice Boltzmann algorithms, while fully explicit and perfectly conservative, may achieve remarkably low values for transport coefficients, such as viscosity. Indeed, the lowest such attainable values are limited only by considerations of accuracy, rather than stability. The method thus holds promise for high‐Reynolds‐number simulations of the Navier‐Stokes equations.

Galilean-Invariant Lattice-Boltzmann Models with H-Theorem

We demonstrate that the requirement of Galilean invariance determines the choice of H function for a wide class of entropic lattice-Boltzmann models for the incompressible Navier-Stokes equations. The required H function has the form of the Burg entropy for D=2, and of a Tsallis entropy with q=1-(2/D) for D>2, where D is the number of spatial dimensions. We use this observation to construct a fully explicit, unconditionally stable, Galilean-invariant, lattice-Boltzmann model for the incompressible Navier-Stokes equations, for which attainable Reynolds number is limited only by grid resolution.

An efficient and accurate quantum lattice-gas model for the many-body Schrödinger wave equation

Presented is quantum lattice-gas model for simulating the time-dependent evolution of a many-body quantum mechanical system of particles governed by the non-relativistic Schrodinger wave equation with an external scalar potential. A variety of computational demonstrations are given where the numerical predictions are compared with exact analytical solutions. In all cases, the model results accurately agree with the analytical predictions and we show that the models error is second order in the temporal discretization and fourth order in the spatial discretization. The difficult problem of simulating a system of fermionic particles is also treated and a general computational formulation of this problem is given. For pedagogical purposes, the two-particle case is presented and the numerical dispersion of the simulated wave packets is compared with the analytical solutions.

Quantum Lattice-Gas Model for the Burgers Equation

A quantum algorithm is presented for modeling the time evolution of a continuous field governed by the nonlinear Burgers equation in one spatial dimension. It is a microscopic-scale algorithm for a type-II quantum computer, a large lattice of small quantum computers interconnected in nearest neighbor fashion by classical communication channels. A formula for quantum state preparation is presented. The unitary evolution is governed by a conservative quantum gate applied to each node of the lattice independently. Following each quantum gate operation, ensemble measurements over independent microscopic realizations are made resulting in a finite-difference Boltzmann equation at the mesoscopic scale. The measured values are then used to re-prepare the quantum state and one time step is completed. The procedure of state preparation, quantum gate application, and ensemble measurement is continued ad infinitum. The Burgers equation is derived as an effective field theory governing the behavior of the quantum computer at its macroscopic scale where both the lattice cell size and the time step interval become infinitesimal. A numerical simulation of shock formation is carried out and agrees with the exact analytical solution.

TYPE-II QUANTUM COMPUTERS

This paper discusses a computing architecture that uses both classical parallelism and quantum parallelism. We consider a large parallel array of small quantum computers, connected together by classical communication channels. This kind of computer is called a type-II quantum computer, to differentiate it from a globally phase-coherent quantum computer, which is the first type of quantum computer that has received nearly exclusive attention in the literature. Although a hybrid, a type-II quantum computer retains the crucial advantage allowed by quantum mechanical superposition that its computational power grows exponentially in the number of phase-coherent qubits per node, only short-range and short time phase-coherence is needed, which significantly reduces the level of engineering facility required to achieve its construction. Therefore, the primary factor limiting its computational power is an economic one and not a technological one, since the volume of its computational medium can in principle scale indefinitely.

Superfluid Turbulence from Quantum Kelvin Wave to Classical Kolmogorov Cascades

The main topological feature of a superfluid is a quantum vortex with an identifiable inner and outer radius. A novel unitary quantum lattice gas algorithm is used to simulate quantum turbulence of a Bose-Einstein condensate superfluid described by the Gross-Pitaevskii equation on grids up to 5760(3). For the first time, an accurate power-law scaling for the quantum Kelvin wave cascade is determined: k(-3). The incompressible kinetic energy spectrum exhibits very distinct power-law spectra in 3 ranges of k space: a classical Kolmogorov k(-(5/3)) spectrum at scales greater than the outer radius of individual quantum vortex cores and a quantum Kelvin wave cascade spectrum k(-3) on scales smaller than the inner radius of the quantum vortex core. The k(-3) quantum Kelvin wave spectrum due to phonon radiation is robust, while the k(-(5/3)) classical Kolmogorov spectrum becomes robust on large grids.

Relativistic Path Integral as a Lattice-based Quantum Algorithm

We demonstrate the equivalence of two representations of many-body relativistic quantum mechanics: the quantum lattice-gas method and the path integral method. The former serves as an efficient lattice-based quantum algorithm to simulate the space-time dynamics of a system of Dirac particles.

Lattice-Gas Quantum Computation

We present a quantum lattice-gas model for a quantum computer operating with continual wavefunction collapse; entanglement of the wavefunction occurs locally over small spatial regions between nearby qubits for only a short time period. The quantum lattice-gas is a noiseless method that directly models the lattice-gas particle dynamics at the mesoscopic scale. The system behaves like a viscous Navier–Stokes fluid. Numerical simulations indicate the viscosity of the quantum lattice-gas fluid is lower than its classical lattice-gas counterparts.

QUANTUM LATTICE-GAS MODEL FOR THE DIFFUSION EQUATION

Presented is a factorized quantum lattice-gas algorithm to model the diffusion equation. It is a minimal model with two qubits per node of a one-dimensional lattice and it is suitable for implementation on a large array of small quantum computers interconnected by nearest-neighbor classical communication channels. The quantum lattice-gas system is described at the mesoscopic scale by a lattice-Boltzmann equation whose collision term is unconditionally stable and obeys the principle of detailed balance. An analytical treatment of the model is given to predict a macroscopic effective field theory. The numerical simulations are in excellent agreement with the analytical results. In particular, numerical simulations confirm the value of the analytically calculated diffusion constant. The algorithm is time-explicit with numerical convergence that is first-order accurate in time and second-order accurate in space.

Quantum lattice gas representation of some classical solitons

A quantum lattice gas representation is determined for both the non-linear Schrodinger (NLS) and Korteweg–de Vries (KdV) equations. There is excellent agreement with the solutions from these representations to the exact soliton–soliton collisions of the integrable NLS and KdV equations. These algorithms could, in principle, be simulated on a hybrid quantum-classical computer.