Deterministic event-based simulation of quantum phenomena

We propose and analyse simple deterministic algorithms that can be used to construct machines that have primitive learning capabilities. We demonstrate that locally connected networks of these machines can be used to perform blind classification on an event-by-event basis, without storing the information of the individual events. We also demonstrate that properly designed networks of these machines exhibit behavior that is usually only attributed to quantum systems. We present networks that simulate quantum interference on an event-by-event basis. In particular we show that by using simple geometry and the learning capabilities of the machines it is possible to simulate single-photon interference in a Mach-Zehnder interferometer. The interference pattern generated by the network of deterministic learning machines is in perfect agreement with the quantum theoretical result for the single-photon Mach-Zehnder interferometer. To illustrate that networks of these machines are indeed capable of simulating quantum interference we simulate, event-by-event, a setup involving two chained Mach-Zehnder interferometers, and demonstrate that also in this case the simulation results agree with quantum theory.     

Efficient data processing and quantum phenomena: Single-particle systems

We study the relation between the acquisition and analysis of data and quantum theory using a probabilistic and deterministic model for photon polarizers. We introduce criteria for efficient processing of data and then use these criteria to demonstrate that efficient processing of the data contained in single events is equivalent to the observation that Malus' law holds. A strictly deterministic process that also yields Malus' law is analyzed in detail. We present a performance analysis of the probabilistic and deterministic model of the photon polarizer. The latter is an adaptive dynamical system that has primitive learning capabilities. This additional feature has recently been shown to be sufficient to perform event-by-event simulations of interference phenomena, without using concepts of wave mechanics. We illustrate this by presenting results for a system of two chained Mach-Zehnder interferometers, suggesting that systems that perform efficient data processing and have learning capability are able to exhibit behavior that is usually attributed to quantum systems only.     

A computer program for two-particle generalized coefficients of fractional parentage

We present a FORTRAN90 program GCFP for the calculation of the generalized coefficients of fractional parentage (generalized CFPs or GCFP). The approach is based on the observation that the multi-shell CFPs can be expressed in terms of single-shell CFPs, while the latter can be readily calculated employing a simple enumeration scheme of antisymmetric A-particle states and an efficient method of construction of the idempotent matrix eigenvectors. The program provides fast calculation of GCFPs for a given particle number and produces results possessing numerical uncertainties below the desired tolerance. A single j-shell is defined by four quantum numbers, (e,l,j,t).A supplemental C++ program parGCFP allows calculation to be done in batches and/or in parallel.

Program summary

Program title:GCFP, parGCFPCatalogue identifier: AEBI_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEBI_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 17 199No. of bytes in distributed program, including test data, etc.: 88 658Distribution format: tar.gzProgramming language: FORTRAN 77/90 (GCFP), C++ (parGCFP)Computer: Any computer with suitable compilers. The program GCFP requires a FORTRAN 77/90 compiler. The auxiliary program parGCFP requires GNU-C++ compatible compiler, while its parallel version additionally requires MPI-1 standard librariesOperating system: Linux (Ubuntu, Scientific) (all programs), also checked on Windows XP (GCFP, serial version of parGCFP)RAM: The memory demand depends on the computation and output mode. If this mode is not 4, the program GCFP demands the following amounts of memory on a computer with Linux operating system. It requires around 2 MB of RAM for the A=12 system at Ex?2. Computation of the A=50 particle system requires around 60 MB of RAM at Ex=0 and ∼70 MB at Ex=2 (note, however, that the calculation of this system will take a very long time). If the computation and output mode is set to 4, the memory demands by GCFP are significantly larger. Calculation of GCFPs of A=12 system at Ex=1 requires 145 MB. The program parGCFP requires additional 2.5 and 4.5 MB of memory for the serial and parallel version, respectively.Classification: 17.18Nature of problem: The program GCFP generates a list of two-particle coefficients of fractional parentage for several j-shells with isospin.Solution method: The method is based on the observation that multishell coefficients of fractional parentage can be expressed in terms of single-shell CFPs [1]. The latter are calculated using the algorithm [2,3] for a spectral decomposition of an antisymmetrization operator matrix Y. The coefficients of fractional parentage are those eigenvectors of the antisymmetrization operator matrix Y that correspond to unit eigenvalues. A computer code for these coefficients is available [4]. The program GCFP offers computation of two-particle multishell coefficients of fractional parentage. The program parGCFP allows a batch calculation using one input file. Sets of GCFPs are independent and can be calculated in parallel.Restrictions:A<86 when Ex=0 (due to the memory constraints); small numbers of particles allow significantly higher excitations, though the shell with j?11/2 cannot get full (it is the implementation constraint).Unusual features: Using the program GCFP it is possible to determine allowed particle configurations without the GCFP computation. The GCFPs can be calculated either for all particle configurations at once or for a specified particle configuration. The values of GCFPs can be printed out with a complete specification in either one file or with the parent and daughter configurations printed in separate files. The latter output mode requires additional time and RAM memory. It is possible to restrict the (J,T) values of the considered particle configurations. (Here J is the total angular momentum and T is the total isospin of the system.) The program parGCFP produces several result files the number of which equals to the number of particle configurations. To work correctly, the program GCFP needs to be compiled to read parameters from the standard input (the default setting).Running time: It depends on the size of the problem. The minimum time is required, if the computation and output mode (CompMode) is not 4, but the resulting file is larger. A system with A=12 particles at Ex=0 (all 9411 GCFPs) took around 1 sec on a Pentium4 2.8 GHz processor with 1 MB L2 cache. The program required about 14 min to calculate all 1.3×¹⁰⁶ GCFPs of Ex=1. The time for all 5.5×¹⁰⁷ GCFPs of Ex=2 was about 53 hours. For this number of particles, the calculation time of both Ex=0 and Ex=1 with CompMode = 1 and 4 is nearly the same, when no other processes are running. The case of Ex=2 could not be calculated with CompMode = 4, because the RAM memory was insufficient. In general, the latter CompMode requires a longer computation time, although the resulting files are smaller in size. The program parGCFP puts virtually no time overhead. Its parallel version speeds-up the calculation. However, the results need to be collected from several files created for each configuration.References:[1] J. Levinsonas, Works of Lithuanian SSR Academy of Sciences 4 (1957) 17.[2] A. Deveikis, A. Bon?kus, R. Kalinauskas, Lithuanian Phys. J. 41 (2001) 3.[3] A. Deveikis, R.K. Kalinauskas, B.R. Barrett, Ann. Phys. 296 (2002) 287.[4] A. Deveikis, Comput. Phys. Comm. 173 (2005) 186. (CPC Catalogue ID. ADWI_v1_0)     

The analytical Scheme calculator for angular momentum coupling and recoupling coefficients

We describe a Scheme implementation of the interactive environment to calculate analytically the Clebsch-Gordan coefficients, Wigner 6j and 9j symbols, and general recoupling coefficients that are used in the quantum theory of angular momentum. The orthogonality conditions for considered coefficients are implemented. The program provides a fast and exact calculation of the coefficients for large values of quantum angular momenta.

Program summary

Title of program:Scheme2ClebschCatalogue number:ADWCProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADWCProgram obtainable from:CPC Program Library, Queen's University of Belfast, N. IrelandLicensing provisions:noneComputer for which the program is designed:Any Scheme-capable platformOperating systems under which the program has been tested: Windows 2000Programming language used:SchemeMemory required to execute with typical data:50 MB (≈ size of DrScheme, version 204)No. of lines in distributed program, including test data, etc.: 2872No. of bytes in distributed program, including test data, etc.: 109 396Distribution format:tar.gzNature of physical problem:The accurate and fast calculation of the angular momentum coupling and recoupling coefficients is required in various branches of quantum many-particle physics. The presented code provides a fast and exact calculation of the angular momentum coupling and recoupling coefficients for large values of quantum angular momenta and is based on the GNU Library General Public License PLT software http://www.plt-scheme.org/.Method of solution:A direct evaluation of sum formulas. A general angular momentum recoupling coefficient for an arbitrary number of (integer or half-integer) angular momenta is expressed as a sum over products of the Clebsch-Gordan coefficients.Restrictions on the complexity of the problem:Limited only by the DrScheme implementation used to run the program. No limitation inherent in the code.Typical running time:The Clebsch-Gordan coefficients, Wigner 6j and 9j symbols, and general recoupling coefficients with small angular momenta are computed almost instantaneously. The running time for large-scale calculations depends strongly on the number and magnitude of arguments' values (i.e., of the angular momenta).     

A basis-set based Fortran program to solve the Gross-Pitaevskii equation for dilute Bose gases in harmonic and anharmonic traps

Inhomogeneous boson systems, such as the dilute gases of integral spin atoms in low-temperature magnetic traps, are believed to be well described by the Gross-Pitaevskii equation (GPE). GPE is a nonlinear Schrödinger equation which describes the order parameter of such systems at the mean field level. In the present work, we describe a Fortran 90 computer program developed by us, which solves the GPE using a basis set expansion technique. In this technique, the condensate wave function (order parameter) is expanded in terms of the solutions of the simple-harmonic oscillator (SHO) characterizing the atomic trap. Additionally, the same approach is also used to solve the problems in which the trap is weakly anharmonic, and the anharmonic potential can be expressed as a polynomial in the position operators x, y, and z. The resulting eigenvalue problem is solved iteratively using either the self-consistent-field (SCF) approach, or the imaginary time steepest-descent (SD) approach. Iterations can be initiated using either the simple-harmonic-oscillator ground state solution, or the Thomas-Fermi (TF) solution. It is found that for condensates containing up to a few hundred atoms, both approaches lead to rapid convergence. However, in the strong interaction limit of condensates containing thousands of atoms, it is the SD approach coupled with the TF starting orbitals, which leads to quick convergence. Our results for harmonic traps are also compared with those published by other authors using different numerical approaches, and excellent agreement is obtained. GPE is also solved for a few anharmonic potentials, and the influence of anharmonicity on the condensate is discussed. Additionally, the notion of Shannon entropy for the condensate wave function is defined and studied as a function of the number of particles in the trap. It is demonstrated numerically that the entropy increases with the particle number in a monotonic way.

Program summary

Title of program:bose.xCatalogue identifier:ADWZ_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADWZ_v1_0Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format:tar.gzComputers:PC's/Linux, Sun Ultra 10/Solaris, HP Alpha/Tru64, IBM/AIXProgramming language used:mostly Fortran 90Number of bytes in distributed program, including test data, etc.:27 313Number of lines in distributed program, including test data, etc.:28 015Card punching code:ASCIINature of physical problem:It is widely believed that the static properties of dilute Bose condensates, as obtained in atomic traps, can be described to a fairly good accuracy by the time-independent Gross-Pitaevskii equation. This program presents an efficient approach of solving this equation.Method of solution:The solutions of the Gross-Pitaevskii equation corresponding to the condensates in atomic traps are expanded as linear combinations of simple-harmonic oscillator eigenfunctions. Thus, the Gross-Pitaevskii equation which is a second-order nonlinear differential equation, is transformed into a matrix eigenvalue problem. Thereby, its solutions are obtained in a self-consistent manner, using methods of computational linear algebra.Unusual features of the program:None     

Ground state of the time-independent Gross-Pitaevskii equation

We present a suite of programs to determine the ground state of the time-independent Gross-Pitaevskii equation, used in the simulation of Bose-Einstein condensates. The calculation is based on the Optimal Damping Algorithm, ensuring a fast convergence to the true ground state. Versions are given for the one-, two-, and three-dimensional equation, using either a spectral method, well suited for harmonic trapping potentials, or a spatial grid.

Program summary

Program title: GPODACatalogue identifier: ADZN_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADZN_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 5339No. of bytes in distributed program, including test data, etc.: 19 426Distribution format: tar.gzProgramming language: Fortran 90Computer: ANY (Compilers under which the program has been tested: Absoft Pro Fortran, The Portland Group Fortran 90/95 compiler, Intel Fortran Compiler)RAM: From <1 MB in 1D to ∼¹⁰² MB for a large 3D gridClassification: 2.7, 4.9External routines: LAPACK, BLAS, DFFTPACKNature of problem: The order parameter (or wave function) of a Bose-Einstein condensate (BEC) is obtained, in a mean field approximation, by the Gross-Pitaevskii equation (GPE) [F. Dalfovo, S. Giorgini, L.P. Pitaevskii, S. Stringari, Rev. Mod. Phys. 71 (1999) 463]. The GPE is a nonlinear Schrödinger-like equation, including here a confining potential. The stationary state of a BEC is obtained by finding the ground state of the time-independent GPE, i.e., the order parameter that minimizes the energy. In addition to the standard three-dimensional GPE, tight traps can lead to effective two- or even one-dimensional BECs, so the 2D and 1D GPEs are also considered.Solution method: The ground state of the time-independent of the GPE is calculated using the Optimal Damping Algorithm [E. Cancès, C. Le Bris, Int. J. Quantum Chem. 79 (2000) 82]. Two sets of programs are given, using either a spectral representation of the order parameter [C.M. Dion, E. Cancès, Phys. Rev. E 67 (2003) 046706], suitable for a (quasi) harmonic trapping potential, or by discretizing the order parameter on a spatial grid.Running time: From seconds in 1D to a few hours for large 3D grids     

The adjustment-stabilization method for constrained systems

For constrained system which has several independent first integrals, we give a new stabilization method which named adjustment-stabilization method. It can stabilize all known constants of motion for a given dynamical system very well instead of the stabilization and post-stabilization methods which only conserves one of all first integrals. Further more, new method can improve numerical accuracy too. We also point out the post-stabilization is just a simplest case of the new method.     

On stabilization of energy for Hamiltonian systems

We study post-stabilization of numerical integrators for an autonomous Hamiltonian system in the form H=H₁+H₂, where H₁ and H₂ are two constants of motion. It is better to stabilize the two energies H₁ and H₂, respectively. As a particular case, the effectiveness of post-stabilization by the total energy is equivalent to one by the two independent energies when the splitting Hamiltonian becomes isotropic.     

A fast level set framework for large three-dimensional topography simulations

We present fast methods to describe the surface evolution of large three-dimensional structures. Based on the sparse field level set method and the hierarchical run-length encoding level set data structure optimal figures for the computation time and for the memory consumption are achieved. Furthermore, we introduce a new multi-level-set technique, which is able to incorporate multiple material regions, and which can also handle material specific surface speeds accurately. We also describe an optimal algorithm for the visibility check for unidirectional etching. The presented techniques are demonstrated on various examples.     

Massively parallel quantum computer simulator

We describe portable software to simulate universal quantum computers on massive parallel computers. We illustrate the use of the simulation software by running various quantum algorithms on different computer architectures, such as a IBM BlueGene/L, a IBM Regatta p690+, a Hitachi SR11000/J1, a Cray X1E, a SGI Altix 3700 and clusters of PCs running Windows XP. We study the performance of the software by simulating quantum computers containing up to 36 qubits, using up to 4096 processors and up to 1 TB of memory. Our results demonstrate that the simulator exhibits nearly ideal scaling as a function of the number of processors and suggest that the simulation software described in this paper may also serve as benchmark for testing high-end parallel computers.     

Performance of a Lattice Quantum Chromodynamics kernel on the Cell processor

The implementation of a proof-of-concept Lattice Quantum Chromodynamics kernel on the Cell processor is described in detail, illustrating issues encountered in the porting process. The resulting code performs up to 45 GFlop/s per socket (without inter-node parallel communications), indicating that the Cell processor is likely to be a good platform for future Lattice QCD calculations.     

Application of the Feynman-Kac path integral method in finding excited states of quantum systems

Group theory considerations and properties of a continuous path are used to define a failure tree procedure for finding eigenvalues of the Schrödinger equation using stochastic methods. The procedure is used to calculate the lowest excited state eigenvalues of eigenfunctions possessing anti-symmetric nodal regions in configuration space using the Feynman-Kac path integral method. Within this method the solution of the imaginary time Schrödinger equation is approximated by random walk simulations on a discrete grid constrained only by symmetry considerations of the Hamiltonian. The required symmetry constraints on random walk simulations are associated with a given irreducible representation and are found by identifying the eigenvalues for the irreducible representation corresponding to symmetric or antisymmetric eigenfunctions for each group operator. The method provides exact eigenvalues of excited states in the limit of infinitesimal step size and infinite time. The numerical method is applied to compute the eigenvalues of the lowest excited states of the hydrogenic atom that transform as Γ² and Γ⁴ irreducible representations. Numerical results are compared with exact analytical results.     

Theory of dynamic entropy-operator systems and its applications

The phenomenological and mathematical definitions of a class of dynamic systems with an entropy operator are formulated. Dynamic systems with an entropy operator are classified and the main theoretical results pertaining to the properties of entropy operators and these dynamic systems are studied within this classification. By way of examples, restoration of monochromatic images and modeling of labor market are examined.     

Orthogonalising pseudo-potentials in scattering calculations

The Orthogonalising Pseudo-Potential (OPP) introduced by Kukulin and Krasnopolsky is applied to scattering problems. Two simple models admitting analytic solution were used to illustrate the behaviour of the scattering states as a function of the OPP strength parameter. These were a system with a zero-range potential having one bound state, and free particle scattering subject to an orthogonality constraint. Application of the OPP to electron scattering from helium in a modified static exchange model, and Ps scattering from He in an s-wave model, were used to demonstrate that the results derived from the model-potential analysis are also valid in a numerical calculations.     

Highly precise solutions of the one-dimensional Schrödinger equation with an arbitrary potential

This work concerns the obtaining of a highly accurate solution of the one-dimensional Schrödinger equation with a practically arbitrary potential. The approach is based on the power series method and it is implemented in the ultra high precision mode. It is shown that such an approach yields not only highly precise values of the energies but also accurate wave functions.     

On time-step bounds in unitary quantum evolution using the Lanczos method

To solve the non-relativistic time dependent Schrödinger equation using the Lanczos method, Park and Light have provided an approximate expression for the time step for a given accuracy. We provide an exact expression for the time step in terms of the eigenvalues and eigenvectors of the resulting tridiagonal matrix. For two test problems, the values of the time step provided by Park and Light differ significantly from the exact values provided by the present method. We also indicate upper and lower bounds for the time step in terms of the maximum and minimum eigenvalues. These bounds indicate the possibility of using a new time step given by the geometric mean of the eigenvalues of the tridiagonal matrix.     

Linearization methods in classical and quantum mechanics

The applicability and accuracy of linearization methods for initial-value problems in ordinary differential equations are verified on examples that include the nonlinear Duffing equation, the Lane-Emden equation, and scattering length calculations. Linearization methods provide piecewise linear ordinary differential equations which can be easily integrated, and provide accurate answers even for hypersingular potentials, for which perturbation methods diverge. It is shown that the accuracy of linearization methods can be substantially improved by employing variable steps which adjust themselves to the solution.     

The Monte Carlo Event Generator AcerMC version 1.0 with interfaces to PYTHIA 6.2 and HERWIG 6.3

The AcerMC Monte Carlo Event Generator is dedicated for the generation of Standard Model background processes at pp LHC collisions. The program itself provides a library of the massive matrix elements and phase space modules for generation of a set of selected processes: , , , , and complete electroweak process. The hard process event, generated with one of these modules, can be completed by the initial and final state radiation, hadronization and decays, simulated with either PYTHIA or HERWIG Monte Carlo event generator. Interfaces to both of these generators are provided in the distribution version. The matrix element codes have been derived with the help of the MADGRAPH package. The phase-space generation is based on the multi-channel self-optimizing approach as proposed in NEXTCALIBUR event generator. Eventually, additional smoothing of the phase space was obtained by using a modified ac-VEGAS routine in order to improve the generation efficiency.     

CADNA: a library for estimating round-off error propagation

The CADNA library enables one to estimate round-off error propagation using a probabilistic approach. With CADNA the numerical quality of any simulation program can be controlled. Furthermore by detecting all the instabilities which may occur at run time, a numerical debugging of the user code can be performed. CADNA provides new numerical types on which round-off errors can be estimated. Slight modifications are required to control a code with CADNA, mainly changes in variable declarations, input and output. This paper describes the features of the CADNA library and shows how to interpret the information it provides concerning round-off error propagation in a code.

Program summary

Program title:CADNACatalogue identifier:AEAT_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEAT_v1_0.htmlProgram obtainable from:CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions:Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.:53 420No. of bytes in distributed program, including test data, etc.:566 495Distribution format:tar.gzProgramming language:FortranComputer:PC running LINUX with an i686 or an ia64 processor, UNIX workstations including SUN, IBMOperating system:LINUX, UNIXClassification:4.14, 6.5, 20Nature of problem:A simulation program which uses floating-point arithmetic generates round-off errors, due to the rounding performed at each assignment and at each arithmetic operation. Round-off error propagation may invalidate the result of a program. The CADNA library enables one to estimate round-off error propagation in any simulation program and to detect all numerical instabilities that may occur at run time.Solution method:The CADNA library [1] implements Discrete Stochastic Arithmetic [2-4] which is based on a probabilistic model of round-off errors. The program is run several times with a random rounding mode generating different results each time. From this set of results, CADNA estimates the number of exact significant digits in the result that would have been computed with standard floating-point arithmetic.Restrictions:CADNA requires a Fortran 90 (or newer) compiler. In the program to be linked with the CADNA library, round-off errors on complex variables cannot be estimated. Furthermore array functions such as product or sum must not be used. Only the arithmetic operators and the abs, min, max and sqrt functions can be used for arrays.Running time:The version of a code which uses CADNA runs at least three times slower than its floating-point version. This cost depends on the computer architecture and can be higher if the detection of numerical instabilities is enabled. In this case, the cost may be related to the number of instabilities detected.References:

[1]

The CADNA library, URL address: http://www.lip6.fr/cadna.

[2]

J.-M. Chesneaux, L'arithmétique Stochastique et le Logiciel CADNA, Habilitation á diriger des recherches, Université Pierre et Marie Curie, Paris, 1995.

[3]

J. Vignes, A stochastic arithmetic for reliable scientific computation, Math. Comput. Simulation 35 (1993) 233-261.

[4]

J. Vignes, Discrete stochastic arithmetic for validating results of numerical software, Numer. Algorithms 37 (2004) 377-390.

     

The solution of stationary ODE problems in quantum mechanics by Magnus methods with stepsize control

In solid state physics the solution of the Dirac and Schrödinger equation by operator splitting methods leads to differential equations with oscillating solutions for the radial direction. For standard time integrators like Runge-Kutta or multistep methods the stepsize is restricted approximately by the length of the period. In contrast the recently developed Magnus methods allow stepsizes that are substantially larger than one period. They are based on a Lie group approach and incorporate exponential functions and matrix commutators. A stepsize control is implemented and tested. As numerical examples eigenvalue problems for the radial Schrödinger equation and the radial Dirac equation are solved. Further, phase shifts for scattering solutions for hydrogen atoms and copper are computed.