Measurement-based quantum lattice gas model of fluid dynamics in 2+1 dimensions

Presented are quantum simulation results using a measurement-based quantum lattice gas algorithm for Navier-Stokes fluid dynamics in 2+1 dimensions. Numerical prediction of the kinematic viscosity was measured by the decay rate of an initial sinusoidal flow profile. Due to local quantum entanglement in the quantum lattice gas, the minimum kinematic viscosity in the measurement-based quantum lattice gas is lower than achievable in a classical lattice gas. The numerically predicted viscosities precisely match the theoretical predictions obtained with a mean field approximation. Uniform flow profile with double shear layers, on a $16K\times 8K$ lattice, leads to the Kelvin-Helmholtz instability, breaking up the shear layer into pairs of counter-rotating vortices that eventually merge via vortex fusion and dissipate because of the nonzero shear viscosity.

Figure 1

Examples of particle-particle collisions in a classical lattice gas confined to a two-dimensional hexagonal lattice. The collisions show particle momenta with incoming states (solid) transitioning to outgoing states (dotted). All single particle momentum vectors are unit magnitude and are along the hexagonal lattice directions. The value within the ket is the number of the shown state out of the possible $2^6=64$ states in the Hilbert space as given by (5). The other particle-particle collisions are obtained by cyclic permutations of these.

Figure 2

Kinematic viscosity as a function of reduced density $d$. Dashed lines show the mean field approximation results. Blue (dark) represents two- and three-body collisions, red (middle) represents two-, three-, and four-body collisions, and orange (light) represents two-, three-, four-, and two-body with a spectator particle collisions.

Figure 3

Examples of particle-particle collisions in a quantum gas confined to a two-dimensional hexagonal lattice. The collisions show particle momenta with incoming states (solid lines) transitioning to outgoing states (dotted lines). All single particle momentum vectors are unit magnitude and are along the hexagonal lattice directions.

Figure 4

Kinematic viscosity as a function of reduced density $d$. Lines show the mean field approximation results, dashed for classical lattice gas and solid for quantum lattice gas. Blue (top pair) represents two- and three-body collisions, red (middle pair) represents two-, three-, and four-body collisions, and orange (bottom pair) represents two-, three-, four-, and two-body with a spectator particle collisions.

Figure 5

Contour plot of vorticity at initialization. Center flow is upward at a velocity of 0.28, and outer flow is downward at a velocity of 0.04 with a background density of $d=0.17$. Start of the KH instability occurs around $100000$ time steps. By $150000$ time steps, the KH instability causes the breakup of the shear layer into separate vortex pairs. By $400000$ time steps, dissipation begins to dominate the flow. By $700000$ time steps, merging of the vortices has started.