Two-dimensional quantum lattice gas algorithm for anisotropic Burger-like equations

Abstract

Building on hybrid quantum lattice gas algorithm, we revisit the possibilities of this quantum lattice model. By deriving a correction to the predicted viscosity, we provide analytical and numerical results that refine original formulation. We introduce a minimal 2D generalization of the algorithm, which allows to simulate anisotropic Burgerlike equations while retaining only two lattice velocities. This approach opens a promising route toward embedding momentum conservation and advancing toward NavierStokes dynamics in 2D, going beyond Frisch, Hasslacher and Pomeau (FHP) with a quantum native model.

Figures (9)

• Figure 1: Evolution of a one-dimensional lattice gas cellular automaton with three discrete velocities (D1Q3). Each cell is represented by three bits $[b_2 b_1 b_0]$, indicating the presence of particles moving with velocities $[-1, 0, +1]$. The collision rule that conserves both mass and momentum (assuming the rest particle has mass 2) is $[101] \leftrightarrow [010]$, which occurs at positions $x = 0, 1, 3,$ and $5$. All other sites remain unaffected by the collision. During the streaming step, particles propagate according to their respective velocities under periodic boundary conditions.
• Figure 2: Algorithmic scheme for Q-D1Q2 model
• Figure 3: Different possible velocity sets. Continuous (dashed) arrows correspond to $f_1$ ($f_0$) streaming direction. On the left we have $c_{0,i}=-c_{1,i}=c_i$. At the center we have $c_{0,x}=1$,$c_{0,y}=0$,$c_{1,x}=0$,$c_{1,y}=-1$. On the right we have a triangular lattice and $\vec{c}_0=(-1/2,\sqrt{3}/2)$ and $\vec{c}_4=(1/2,\sqrt{3}/2)$
• Figure 4: Simulation viscosities calculated according to Eq. \ref{['eq:experimental_nu']}, initializing the system with $N_x=64$ and $\rho_a=0.005$, doing the average for $T=200$ timesteps for the plot above, and $T=2000$ for the plot below. The blue line is the viscosities in Eq. \ref{['eq:nu_1d']}, while the red line is the viscosity of yepez2006open$\cot^2(\theta)/2$
• Figure 5: Maximal spatial gradient of $w(x,t)/\alpha$ for different parameters.