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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., Comment: 24 figures

Presented is a quantum computing model of a quantum field theory for a system of fermions interacting via a massive gauge field. The model describes a relativistic superconducting fluid and uses a metric tensor field to both encode the fermion's intrinsic spin in the torsion of curved space and encode the coupling of fermions via a massive 4-potential field. The quantum computing model is a lattice model whose cell size is a deformation parameter: the equivalent lattice and curved-space gauge field theory models both reduce to quantum field theory in flat Minkowski space at zero cell size. The low-energy expansions of the lattice model and Euler-Lagrange equations of the curved-space gauge field theory are the same equations of motion. The fermion and gauge fields obey the Dirac and Proca equations, and the gauge field strength is determined by the fermion field., Comment: 4 pages, 2 figures, Supplemental Material. arXiv admin note: text overlap with arXiv:1609.02225. The new version includes the analytical device of using a curved-space metric tensor to model the nonlinear interactions in the superconducting fluid and derives the equivalent lattice model

Presented is a quantum lattice gas algorithm to efficiently model a system of Dirac particles interacting through an intermediary gauge field. The algorithm uses a fixed qubit array to represent both the spacetime and the particles contained in the spacetime. Despite being a lattice based algorithm, Lorentz invariance is preserved down to the grid scale, with the continuum Dirac Hamiltonian generating the local unitary evolution even at that scale: there is nonlinear scaling between the smallest observable time and that time measured in the quantum field theory limit, a kind of time dilation effect that emerges on small scales but has no effect on large scales. The quantum lattice gas algorithm correctly accounts for the anticommutative braiding of indistinguishable fermions---it does not suffer the Fermi-sign problem. It provides a highly convergent numerical simulation for strongly-correlated fermions equal to a covariant path integral, presented here for the case when a Dirac particle's Compton wavelength is large compared to the grid scale of the qubit array., Comment: SPIE Quantum Information Science and Technology, Paper 9996-22 (2016)

Dirac particle dynamics is encoded as a unitary path summation rule and implemented on a qubit array, where the qubit array represents both spacetime and the fermions contained therein. The unitary path summation rule gives a quantum algorithm to model a many-body system of Dirac particles in a gauge field with Lorentz invariance down to the grid scale (Planck scale)--the lattice-based model neither suffers the Fermi-sign problem nor breaks Lorentz invariance. Yet, for the Dirac Hamiltonian to generate the unitary evolution of the 4-spinor field at the Planck scale, there is time dilation between the shortest observable time near a single space point and that time measured at long-wavelength scales. We find gravitational time dilation where the model space around each point (with an even number of qubits) is curved like the space around a Schwarzschild black hole., Comment: Version of the path summation rule with 4-vector spin variable notation better suited to relativistic quantum mechanics and with an updated title

Presented are several example quantum computing representations of quantum systems with a relativistic energy relation. Basic unitary representations of free Dirac particles and BCS superconductivity are given. Then, these are combined into a novel unitary representation of a Fermi condensate superfluid. The modeling approach employs an operator splitting method that is an analytically closed-form product decomposition of the unitary evolution operator, applied in the high-energy limit. This allows the relativistic wave equations to be cast as unitary finite-difference equations. The split evolution operators (comprising separate kinetic and interaction energy evolution terms) serve as quantum lattice gas models useful for efficient quantum simulation.

Presented is a quantum computing representation of Dirac particle dynamics. The approach employs an operator splitting method that is an analytically closed-form product decomposition of the unitary evolution operator. This allows the Dirac equation to be cast as a unitary finite-difference equation in a high-energy limit. The split evolution operator (with separate kinetic and interaction terms) is useful for efficient quantum simulation. For pedagogical purposes, here we restrict the treatment to Dirac particle dynamics in 1+1 spacetime dimensions. Independent derivations of the quantum algorithm are presented and the model's validity is tested in several quantum simulations by comparing the numerical results against analytical predictions. Using the relativistic quantum algorithm in the case when mc^2 >> pc, quantum simulations of a nonrelativistic particle in an external scalar square well and parabolic potential is presented., Comment: 33 figures

We revisit the quantum lattice gas model of a spinor quantum field theory-the smallest scale particle dynamics is partitioned into unitary collide and stream operations. The construction is covariant (on all scales down to a small length {\ell} and small time {\tau} = c {\ell}) with respect to Lorentz transformations. The mass m and momentum p of the modeled Dirac particle depend on {\ell} according to newfound relations m = mo cos (2{\pi}{\ell}/{\lambda}) and p = (h/2{\pi}{\ell}) sin(2{\pi}{\ell}/{\lambda}), respectively, where {\lambda} is the Compton wavelength of the modeled particle. These relations represent departures from a relativistically invariant mass and the de Broglie relation-when taken as quantifying numerical errors the model is physically accurate when {\ell} {\ll} {\lambda}. Calculating the vacuum energy in the special case of a massless spinor field, we find that it vanishes (or can have a small positive value) for a sufficiently large wave number cutoff. This is a marked departure from the usual behavior of such a massless field., Comment: 5 pages, 2 figures; 66 ABW review on 01 Dec 2010: Ref. No. RVB10.015-198, Case No. 66ABW-2010-1430

The evolution of the ground state wave function of a zero-temperature Bose-Einstein condensate (BEC) is well described by the Hamiltonian Gross-Pitaevskii (GP) equation. Using a set of appropriately interleaved unitary collision-streaming operators, a quantum lattice gas algorithm is devised which on taking moments recovers the Gross-Pitaevskii (GP) equation in diffusion ordering (time scales as square of length). Unexpectedly, there is a class of initial conditions in which their Poincare recurrence is extremely short. Further it is shown that the Poincare recurrence time scales with diffusion ordering as the the grid is increased. The spectral results of Yepez et.al. [1] for quantum turbulence are corrected and it is found that it is the compressible kinetic energy spectrum that exhibits the 3 cascade regions: a small k classical Kolmogorov k^(-5/3) spectrum, a steep semi-classical cascade region, and a large k quantum Kelvin wave cascade k^(-3) spectrum. The incompressible kinetic energy spectrum exhibits basically a single cascade power law of k^(-3). For winding number 1 linear vortices it is also shown that there is an intermittent loss of Kelvin wave cascade with its signature seen in the time evolution of the kinetic energy, the loss of the k^(-3) spectrum in the incompressible kinetic energy spectrum as well as the minimization of the vortex core isosurfaces that inhibits the Kelvin wave cascade.

Presented is a prediction, based on the Frenet-Serret differential geometry of space curves, that the wave number dependence of the average kinetic energy per unit length of two mutually interacting highly curved quantum vortex scales as k^-3. The interacting quantum vortices are helical in shape, supporting circularly polarized counter-propagating waves, with arbitrary curvature and torsion. This power-law spectrum agrees with the high-k spectrum found in precise quantum simulations of turbulent superfluidity with tangle of highly curved and excited quantum vortices.

Presented is a topological representation of quantum logic that views entangled qubit spacetime histories (or qubit world lines) as a generalized braid, referred to as a superbraid. The crossing of world lines is purely quantum in nature, most conveniently expressed analytically with ladder-operator-based quantum gates. At a crossing, independent world lines can become entangled. Complicated superbraids are systematically reduced by recursively applying novel quantum skein relations. If the superbraid is closed (e.g. representing quantum circuits with closed-loop feedback, quantum lattice gas algorithms, loop or vacuum diagrams in quantum field theory), then one can decompose the resulting superlink into an entangled superposition of classical links. In turn, for each member link, one can compute a link invariant, e.g. the Jones polynomial. Thus, a superlink possesses a unique link invariant expressed as an entangled superposition of classical link invariants., Comment: 4 pages

The dynamics of vortex solitons in a BEC superfluid is studied. A quantum lattice-gas algorithm (localization-based quantum computation) is employed to examine the dynamical behavior of vortex soliton solutions of the Gross-Pitaevskii equation (phi^4 interaction nonlinear Schroedinger equation). Quantum turbulence is studied in large grid numerical simulations: Kolmogorov spectrum associated with a Richardson energy cascade occurs on large flow scales. At intermediate scales a k^{-6} power law emerges, in a classical-quantum transition from vortex filament reconnections to Kelvin wave-acoustic wave coupling. The spontaneous exchange of intermediate vortex rings is observed. Finally, at very small spatial scales a k^{-3} power law emerges, characterizing fluid dynamics occurring within the scale size of the vortex cores themselves, a characteristic Kelvin wave cascade region. Poincare recurrence is studied: in the free non-interacting system, a fast Poincare recurrence occurs for regular arrays of line vortices. The recurrence period is used to demarcate dynamics driving the nonlinear quantum fluid towards turbulence, since fast recurrence is an approximate symmetry of the nonlinear quantum fluid at early times. This class of quantum algorithms is useful for studying BEC superfluid dynamics over a broad range of wave numbers, from quantum flow to a pseudo-classical inviscid flow regime to a Kolmogorov inertial subrange., Comment: 10 pages, 6 figures

A novel unitary quantum lattice gas algorithm is used to simulate quantum turbulence of a BEC 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 much greater than the individual quantum vortex cores, and a quantum Kelvin wave cascade spectrum k^{-3} on scales of order the vortex cores. In the semiclassical regime between these two spectra there is a pronounced steeper spectral decay, with non-universal exponent. The Kelvin k^{-3} spectrum is very robust, even on small grids, while the Kolmogorov k^{-5/3} spectrum becomes more and more apparent as the grids increase from 2048^3 grids to 5760^3., Comment: 4 pages, 2 figures

We report on the implementation of Burgers equation as a type-II quantum computation on an NMR quantum information processor. Since the flow field evolving under the Burgers equation develops sharp features over time, this is a better test of liquid state NMR implementations of type-II quantum computers than the previous examples using the diffusion equation. In particular, we show that Fourier approximations used in the encoding step are not the dominant error. Small systematic errors in the collision operator accumulate and swamp all other errors. We propose, and demonstrate, that the accumulation of this error can be avoided to a large extent by replacing the single collision operator with a set of operators with random errors and similar fidelities. Experiments have been implemented on 16 two-qubit sites for eight successive time steps for the Burgers equation., Comment: 5 pages, 3 figures

We report an ensemble nuclear magnetic resonance (NMR) implementation of a quantum lattice gas algorithm for the diffusion equation. The algorithm employs an array of quantum information processors sharing classical information, a novel architecture referred to as a type-II quantum computer. This concrete implementation provides a test example from which to probe the strengths and limitations of this new computation paradigm. The NMR experiment consists of encoding a mass density onto an array of 16 two-qubit quantum information processors and then following the computation through 7 time steps of the algorithm. The results show good agreement with the analytic solution for diffusive dynamics. We also describe numerical simulations of the NMR implementation. The simulations aid in determining sources of experimental errors, and they help define the limits of the implementation., Comment: 20 pages, 3 figures

An efficient quantum algorithm for the many-body three-dimensional Dirac equation is presented. Its computational complexity is dominantly linear in the number of qubits used to spatially resolve the 4-spinor wave function.

We analyze one-dimensional classical and quantum microscopic lattice-gas models governed by a lattice Boltzmann equation at the mesoscopic scale, achieved by ensemble averaging over microscopic realizations. The models are governed by the Burgers equation at the macroscopic scale, achieved by taking the limit where the grid size and time step both approach zero and by performing a perturbative Chapman-Enskog expansion. The quantum algorithm exploiting superposition and entanglement is more efficient than the classical one because the quantum algorithm requires less memory. Furthermore, its viscosity can be made arbitrarily small., Comment: 4 pages, 2 figures