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.
Quantum information theory is undergoing rapid development and recently there has been much progress in
mapping out its relationship to low dimensional gravity, primarily through Chern-Simons topological quantum
field theory and conformal field theory, with the prime application being topological quantum computation.
Less attention has been paid to the relationship of quantum information theory to the long established and well
tested theory of gravitational dynamics of 3+1 dimensional spacetime. Here we discuss this question in the
weak field approximation of the 4-space metric tensor. The proposed approach considers a quantum algorithmic
scheme suitable for simulating physical curved space dynamics that is traditionally described by the well known
Einstein-Hilbert action. The quantum algorithmic approach builds upon Einstein's veirbein representation of
gravity, which Einstein originally developed back in 1928 in his search for a unified field theory and, moreover,
which is presently widely accepted as the preferred theoretical approach for representing dynamical relativistic
Dirac fields in curved space. Although the proposed quantum algorithmic scheme is regular-lattice based it
nevertheless recovers both the Einstein equation of motion as an effective field theory and invariance of the
gravitational gauge field (i.e., the spin connection) with respect to Lorentz transformations as the local symmetry
group in the low energy limit.
Spinor Bose Einstein Condensates are intriguing because of their vast range of different topological vortices. These
states occur when a BEC gas is trapped in an optical lattice rather than in a magnetic well (which would result in
scalar BEC vortices). A spinor BEC states also occur in a quantum gas when several hyperfine states of the atom
co-exist in the same trap. A unitary quantum lattice algorithm that is ideally parallelized to all available processors is
used to solve the evolution of non-eigenstate Skyrmions in a coupled BEC system. The incompressible kinetic energy
spectrum of the inner quantum vortex ring core rapidly deviates from the k^{-3} spectrum found in the evolution of
scalar BECs.
A quantum lattice gas algorithm, based on interleaved unitary collide-stream operators, is
used to study quantum turbulence of the ground state wave function of a Bose-Einstein
condensate (BEC). The Gross-Pitaevskii equation is a Hamiltonian system for a
compressible, inviscid quantum fluid. From simulations on a 5760^{3} grid it was observed
that a multi-cascade existed for the incompressible kinetic energy spectrum with universal
features: the large spatial scales exhibit a classical Kolmogorov k ^{-5/3} spectrum while the
very small scales exhibit a quantum Kelvin wave cascade k^{-3} spectrum. Under certain
conditions one can explicitly determine the Poincare recurrence of initial conditions as
well as the intermittent destruction of the Kelvin wave cascade.
Presented is a second quantized technology for representing fermionic and bosonic entanglement in terms of
generalized joint ladder operators, joint number operators, interchangers, and pairwise entanglement operators.
The joint number operators generate conservative quantum logic gates that are used for pairwise entanglement
in quantum dynamical systems. These are most useful for quantum computational physics. The generalized joint
operator approach provides a pathway to represent the Temperley-Lieb algebra and to represent braid group
operators for either fermionic or bosonic many-body quantum systems. Moreover, the entanglement operators
allow for a representation of quantum measurement, quantum maps (associated with quantum Boltzmann equation
dynamics), and for a way to completely and efficiently extract all accessible bits of joint information from
entangled quantum systems in terms of quantum propositions.
The dynamics of vortex solitons is studied in a BEC superfluid. A quantum lattice-gas algorithm (measurementbased
quantum computation) is employed to examine the dynamical behavior vortex soliton solutions of the
Gross-Pitaevskii equation (φ^{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 new k^{-5.9} power law emerges, due to vortex filamentary reconnections
associated with Kelvin wave instabilities (vortex twisting) coupling to sound modes and the exchange of
intermediate vortex rings. 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. Poincaré recurrence is studied: in the
free non-interacting system, a fast Poincaré recurrence occurs for regular arrays of line vortices. The recurrence
period is used to demarcate dynamics driving a 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 and, without modification, should allow for higher resolution
simulations (with many components) on future quantum computers.
The ground state wave function for a Bose Einstein condensate is well described by the
Gross-Pitaevskii equation. A Type-II quantum algorithm is devised that is ideally
parallelized even on a classical computer. Only 2 qubits are required per spatial node.
With unitary local collisions, streaming of entangled states and a spatially
inhomogeneous unitary gauge rotation one recovers the Gross-Pitaevskii equation.
Quantum vortex reconnection is simulated - even without any viscosity or resistivity
(which are needed in classical vortex reconnection).
A quantum lattice algorithm is developed for the solution of the coupled Gross-Pitaevskii equations, which are just
the nonlinear Schrodinger equations with an external potential. In the language of solitons, a sufficiently strong
external potential can localize the solitons and even the radiation in the case of Manikov solitons. For non-integrable
potential cases the radiation is no longer localized and one can also lose the soliton structures themselves.
One also finds locked mode structures if the systems is close to being integrable.
The quantum Boltzmann equation method is demonstrated by numerically predicting the time-dependent solutions of the velocity and magnetic fields governed by nonintegrable magnetohydrodynamic equations in one spatial dimension. The method allows arbitrary tuning of the value of the viscosity and resistivity transport coefficients without compromising numerical integrity even near the zero dissipation and turbulent regime where shock front discontinuities emerge.
The nonlinear Schrodinger (NLS) equation in a self-defocusing Kerr medium supports dark solitons. Moreover the mean field description of a dilute Bose-Einstein condensate (BEC) is described by the Gross-Pitaevskii equation, which for a highly anisotropic (cigar-shaped) magnetic trap reduces to a one-dimensional (1D) cubic NLS in an external potential. A quantum lattice algorithm is developed for the dark solitons. Simulations are presented for both black (stationary) solitons as well as (moving) dark solitons. Collisions of dark solitons are compared with the exact analytic solutions and coupled dark-bright vector solitons are examined. The quantum algorithm requires 2 qubits per scalar field at each spatial node. The unitary collision operator quantum mechanically entangles the on-site qubits, and this transitory entanglement is spread throughout the lattice by the streaming operators. These algorithms are suitable for a Type-II quantum computers, with wave function collapse induced by quantum measurements required to determine the coupling potentials.
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(ε^{2}).
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