Quantum Lattice Gas Research Articles

Quantum computing classical physics

In the past decade, quantum algorithms have been found which outperform the best classical solutions known for certain classical problems as well as the best classical methods known for simulation of certain quantum systems. This suggests that they may also speed up the simulation of some classical systems. I describe one class of discrete quantum algorithms which do so--quantum lattice-gas automata--and show how to implement them efficiently on standard quantum computers.

PARRONDO GAMES AS LATTICE GAS AUTOMATA

Parrondo games are coin flipping games with the surprising property that alternating plays of two losing games can produce a winning game. We show that this phenomenon can be modelled by probabilistic lattice gas automata. Furthermore, motivated by the recent introduction of quantum coin flipping games, we show that quantum lattice gas automata provide an interesting definition for quantum Parrondo games.

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.

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.

From gauge transformations to topology computation in quantum lattice gas automata

The evolution of a quantum lattice gas automaton (LGA) for a single charged particle is invariant under multiplication of the wave function by a global phase. Requiring invariance under the corresponding local gauge transformations determines the rule for minimal coupling to an arbitrary external electromagnetic field. We develop the Aharonov-Bohm effect in the resulting model into a constant time algorithm to distinguish a one dimensional periodic lattice from one with boundaries; any classical deterministic LGA algorithm distinguishing these two spatial topologies would have expected running time on the order of the cardinality of the lattice.

Quantum lattice-gas model for computational fluid dynamics.

Quantum-computing ideas are applied to the practical and ubiquitous problem of fluid dynamics simulation. Hence, this paper addresses two separate areas of physics: quantum mechanics and fluid dynamics (or specifically, the computational simulation of fluid dynamics). The quantum algorithm is called a quantum lattice gas. An analytical treatment of the microscopic quantum lattice-gas system is carried out to predict its behavior at the mesoscopic scale. At the mesoscopic scale, a lattice Boltzmann equation with a nonlocal collision term that depends on the entire system wave function, governs the dynamical system. Numerical results obtained from an exact simulation of a one-dimensional quantum lattice model are included to illustrate the formalism. A symbolic mathematical method is used to implement the quantum mechanical model on a conventional workstation. The numerical simulation indicates that classical viscous damping is not present in the one-dimensional quantum lattice-gas system.

Cnoidal Waves and Solitons in the Quantum Lattice Gas Model in a Ring and Effects of the Aharonov-Bohm Flux on Them

Cnoidal wave solutions in the quantum lattice gas model in the presence of the Aharonov-Bohm flux in a ring are found. The energy gap, the effective mass and the binding energy of cnoidal waves are given. It is pointed out that the spin-wave-like excitation and the soliton are special cases of the cnoidal wave. Effects of the Aharonov-Bohm flux on the soliton are considered. It is shown that the width, the peak, the binding energy and the effective mass of the soliton and the energy gap are affected and a Josephson-like current is induced by the Aharonov-Bohm flux.

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 counterpart's.

Exact Solution for the Ground State of a One-Dimensional Quantum Lattice Gas with Coulomb-Like Interaction

We present exact wave functions for the ground state of a lattice gas of spinless one-dimensional fermions, at filling factors of $1/M$. The one-body hopping operator decreases exponentially. The two-body interaction is Coulomb-like: A neutralizing background is required for the system to be extensive. There is a parameter in the problem, which can be taken to be the strength of the charge. All correlation functions are evaluated, and the system is shown to have long-ranged crystalline order. For filling factor $1/2$, the momentum distribution and pair correlation function are explicitly evaluated. There is no Fermi surface. The crystalline order parameter is evaluated.

Quantum mechanics of lattice gas automata: boundary conditions and other inhomogeneities

We continue our analysis of the physics of quantum lattice gas automata (QLGA). Previous work has been restricted to periodic or infinite lattices; simulation of more realistic physical situations requires finite sizes and nonperiodic boundary conditions. Furthermore, envisioning a QLGA as a nanoscale computer architecture motivates consideration of inhomogeneities in the `substrate'; this translates into inhomogeneities in the local evolution rules. Concentrating on the one-particle sector of the model, we determine the various boundary conditions and rule inhomogeneities which are consistent with unitary global evolution. We analyse the reflection of plane waves from boundaries, simulate wavepacket refraction across inhomogeneities, and conclude by discussing the extension of these results to multiple particles.

Quantum lattice-gas model for the many-particle Schrödinger equation inddimensions

We consider a general class of discrete unitary dynamical models on the lattice. We show that generically such models give rise to a wave function satisfying a Schr\odinger equation in the continuum limit, in any number of dimensions. There is a simple mathematical relationship between the mass of the Schr\odinger particle and the eigenvalues of a unitary matrix describing the local evolution of the model. Second quantized versions of these unitary models can be defined, describing in the continuum limit the evolution of a nonrelativistic quantum many-body theory. An arbitrary potential is easily incorporated into these systems. The models we describe fall in the class of quantum lattice-gas automata and can be implemented on a quantum computer with a speedup exponential in the number of particles in the system. This gives an efficient algorithm for simulating general nonrelativistic interacting quantum many-body systems on a quantum computer.

Field-Induced Phase Transition in S=1 XXZ Chain with Biquadratic Exchange Interaction

We study the magnetization process of the S=1 XXZ chain with biquadratic exchange interaction at zero temperature. This system can be regarded as a quantum lattice gas composed of "particles" with Sz=0, ±1. We introduce an approximate nested Bethe ansatz (ANBA) to examine the possibility of phase transition induced by the magnetic field. In the high-magnetization region σ≥σ0 for certain σ0, the ANBA gives the exact ground state energies. For σ<σ0 the ANBA gives approximate solutions. The analysis shows that a phase transition occurs at σ0 across which the magnetic susceptibility changes discontinuously. Comparing with the finite-size diagonalization, we argue that σ0 is the exact critical magnetization.

Quantum Lattice Gases and Their Invariants

The one-particle sector of the simplest one-dimensional quantum lattice-gas automaton has been observed to simulate both the (relativistic) Dirac and (nonrelativistic) Schrödinger equations, in different continuum limits. By analyzing the discrete analogues of plane waves in this sector we find conserved quantities corresponding to energy and momentum. We show that the Klein paradox obtains so that in some regimes the model must be considered to be relativistic and the negative energy modes interpreted as positive energy modes of antiparticles. With a formally similar approach — the Bethe ansatz — we find the evolution eigenfunctions in the two-particle sector of the quantum lattice-gas automaton and conclude by discussing consequences of these calculations and their extension to more particles, additional velocities, and higher dimensions.

Quantum mechanics of lattice gas automata: One-particle plane waves and potentials

Classical lattice gas automata effectively simulate physical processes such as diffusion and fluid flow (in certain parameter regimes) despite their simplicity at the microscale. Motivated by current interest in quantum computation we recently defined quantum lattice gas automata; in this paper we initiate a project to analyze which physical processes these models can effectively simulate. Studying the single particle sector of a one dimensional quantum lattice gas we find discrete analogues of plane waves and wave packets, and then investigate their behaviour in the presence of inhomogeneous potentials.

From quantum cellular automata to quantum lattice gases

A natural architecture for nanoscale quantum computation is that of a quantum cellular automaton. Motivated by this observation, in this paper we begin an investigation of exactly unitary cellular automata. After proving that there can be no nontrivial, homogeneous, local, unitary, scalar cellular automaton in one dimension, we weaken the homogeneity condition and show that there are nontrivial, exactly unitary, partitioning cellular automata. We find a one parameter family of evolution rules which are best interpreted as those for a one particle quantum automaton. This model is naturally reformulated as a two component cellular automaton which we demonstrate to limit to the Dirac equation. We describe two generalizations of this automaton, the second of which, to multiple interacting particles, is the correct definition of a quantum lattice gas.

Bethe Ansatz Calculations for Quantum Spin Chains with Partial Integrability

We study the S matrix of a spin- s quantum spin chain which can be interpreted as a multicomponent quantum lattice gas. We show that this system contains integrable subsystems where the Bethe ansatz method is applicable. We demonstrate that for some ranges of parameters of a spin-1 model, the ground states and low-lying excited states are exactly given by those of integrable subsystems; exact calculation is possible even if the system is not strictly integrable as a whole. We also analyze the magnetization curves for the spin chain, with exact saturation fields.

Bethe Ansatz Calculations for Quantum Spin Chains with Partial Integrability

We study the S matrix of a spin- s quantum spin chain which can be interpreted as a multicomponent quantum lattice gas. We show that this system contains integrable subsystems where the Bethe ansatz method is applicable. We demonstrate that for some ranges of parameters of a spin-1 model, the ground states and low-lying excited states are exactly given by those of integrable subsystems; exact calculation is possible even if the system is not strictly integrable as a whole. We also analyze the magnetization curves for the spin chain, with exact saturation fields.

An Approximation Scheme of the Cluster Variation Method for Quantum Lattice Gases

A new scheme of approximations is proposed in the cluster variation method, by which we can discuss the properties of quantum lattice gases which have interactions, near zero temperature as well as at high temperatures, including the transition temperature. The site and the pair approximation in this scheme are discussed for uniform systems mainly for a boson system, and some discussion is given also of the pair approximation applied to a boson lattice gas which is equivalent to the Heisenberg ferromagnet. Possibility of superftuidity or superconductivity is included only in the general formulation.

The long-range ordering in a quantum lattice gas

The mean-field-like theory of the equilibrium state of an ordered quantum lattice gas is developed within the framework of the density matrix formalism. This theory is the generalization of the Gorsky-Bragg-Williams theory of atomic ordering in binary alloys. The density matrix is determined to describe the ordered state of a system of identical particles performing random walks on the sites of a cubic lattice. The temperature dependence of the long-range order parameter and other characteristics of the phase transition are established.

Berry's phase in a one-dimensional quantum many-body system.

We study an interacting one-dimensional quantum lattice gas of massive fermions on a ring with L lattice sites. The ring is threaded by a magnetic flux corresponding to a twist in boundary conditions. We compute the periodicity of the ground state under an adiabatically increasing flux and the associated Berry's phase occurring in this process. The model has a second-order phase transition line which coincides with a line where the Berry phase changes nonanalytically.