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In a recent paper we have suggested that the finite temperature density matrix can be computed efficiently by a combination of polynomial expansion and iterative inversion techniques. We present here significant improvements over this scheme. The original complex-valued formalism is turned into a purely real one. In addition, we use Chebyshev polynomials expansion and fast summation techniques. This drastically reduces the scaling of the algorithm with the width of the Hamiltonian spectrum, which is now of the order of the cubic root of such parameter. This makes our method very competitive for applications to ab-initio simulations, when high energy resolution is required. [ABSTRACT FROM AUTHOR]

Beginning with its introduction by W. Heisenberg, quantum mechanics was often seen as an overly abstract theory, mathematically and physically, vis-à-vis classical physics or relativity. This perception was amplified by the fact that, while the quantum-mechanical formalism provided effective predictive algorithms for the probabilistic predictions concerning quantum experiments, it appeared unable to describe, even by way idealization, quantum processes themselves in space and time, in the way classical mechanics or relativity did. The aim of the present paper is to reconsider the nature of mathematical and physical abstraction in modern physics by offering an analysis of the concept of 'physical fact' and of the concept of 'physical concept,' in part by following G. W. F. Hegel's and G. Deleuze's arguments concerning the nature of conceptual thinking. In classical physics, relativity, and quantum physics alike, I argue, physical concepts are defined by the following main features-1) their multi-component multiplicity; 2) their essential relations to problems; 3) and the interactions between physical, mathematical, and philosophical components within each concept. It is the particular character of these interactions in quantum mechanics, as defined by its essentially predictive (rather than descriptive) nature, that distinguishes it from classical physics and relativity. [ABSTRACT FROM AUTHOR]

We have studied a quantum algorithm for several years, and developed some applications for difficult problems, NPC problems and NP intermidiate problems. In order to discuss the computational complexity of quantum algorithm, we defined a generalized quantum Turing machine using a density operator on a Hilbert space and quantum channels on it. Since the properity of quantum channel, this mathematical model can describe not only unitary process of quantum algorithm but also measurement process and non-linear dynamical process. Then we can calculate the computational complexity of quantum algorithm more regorously. In this paper, we review our results, and discuss why quantum algorithms are more effective than classical ones. Moreover, we propose a quantum algorotihm for EXPTIME problem. [ABSTRACT FROM AUTHOR]

The DMRG (Density Matrix Renormalizarion Group) is a powerful method to study the low energy properties of many body systems which are difficult to treat by other methods due to the large dimension of their Hilbert space and their strongly correlated nature. Usually these systems are not analitically solvable and numerical approaches become indispensable to shed some light upon their physical properties. In this paper we will review the staggered Heisenberg antiferromagnetic chain with S = 1 and S = 1/2 and will use the DMRG algorithm to obtain some properties relative to their critical behavior, in particular concerning the critical points appearing in the phase diagram of these systems. © 2007 American Institute of Physics [ABSTRACT FROM AUTHOR]

This paper is a presentation of how to implement quantum algorithms (namely, Shor’s algorithm ) on a classical computer by using the well-known Mathematica package. It will give us a lucid connection between mathematical formulation of quantum mechanics and computational methods. © 2007 American Institute of Physics [ABSTRACT FROM AUTHOR]

One of the most basic tasks in quantum information processing, communication and security (QIPCC) research, theoretically deep and practically important, is to find bounds on how really important are inherently quantum resources for speeding up computations. This area of research is bringing a variety of results that imply, often in a very unexpected and counter-intuitive way, that: (a) surprisingly large classes of quantum circuits and algorithms can be efficiently simulated on classical computers; (b) the border line between quantum processes that can and cannot be efficiently simulated on classical computers is often surprisingly thin; (c) the addition of a seemingly very simple resource or a tool often enormously increases the power of available quantum tools. These discoveries have put also a new light on our understanding of quantum phenomena and quantum physics and on the potential of its inherently quantum and often mysteriously looking phenomena. The paper motivates and surveys research and its outcomes in the area of de-quantisation, especially presents various approaches and their outcomes concerning efficient classical simulations of various families of quantum circuits and algorithms. To motivate this area of research some outcomes in the area of de-randomization of classical randomized computations. [ABSTRACT FROM AUTHOR]

The 1-way quantum finite automata and 2-way quantum finite automata are the first ever developed quantum automatons. But they suffer from many disadvantages which make them less powerful and impracticable. To remove these drawbacks there are many ways to modify and aggrandize the quantum finite automatons. Including 1-counter with 1-way QFA significantly improves its power. Likely quantum finite automata with quantum and classical states (QCFA) and 2-way quantum finite 1-counter automata (2-QF1-CA) are the quintessence of these modifications. In our paper we first discuss about 1-QFA and 2-QFA and their drawbacks. In the section IV we introduce an algorithm by implementing 1-way quantum finite k-counter automata (1-QFk-CA). Then we discuss another algorithm in section V to recognize another language. These algorithms prove that 1-QFk-CA significantly improves the power of QFA. [ABSTRACT FROM AUTHOR]

Quantum information processing and communication (QIPC) theory has developed recently very fast and brought a variety of interesting and important results of the great value also for the whole area of quantum physics. One can also say that the field of QIPC has been so far mainly theory driven and the experiments have been mostly done to show that it is indeed possible, and how difficult is to make it, what theory shows as possible. One of the main reasons for such a fast and successful development of the QIPC science is the fact that paradigms, models, concepts, value system, as well as methods and results of the (theoretical) informatics have been intensively used. The goal of this paper is to go behind this successful crusade and applications of the informatics for QIPC and to present, analyse and illustrate the main ideas, concepts, methods and tools that have been involved. All that should help more physics-oriented researchers in QIPC to understand that in order to explore the quantum world, new paradigms, concepts, models and so on are now available, and they could and should be used, due to the progress in (theoretical) informatics. Our concentration will be not only on what has been achieved, but even more on the main new challenges. In doing that we will concentrate more on the backgrounds, motivations, goals and implications than on the very technical results. © 2006 American Institute of Physics [ABSTRACT FROM AUTHOR]

In this paper, we will define a quantum operator that performs the inversion about the mean only on a sub space of the system (partial diffusion operator). This operator is used in a quantum search algorithm that runs in O(&radicN/M;) for searching an unstructured list of size N with M matches such that, 1 ≤5 M ≤ N. We will show that the performance of the algorithm is more reliable than known fixed operator quantum search algorithms especially for multiple matches where we can get a solution after a single iteration with probability over 90% if the number of matches is approximately more than one-third of the search space. A performance comparison with Grover's algorithm will be provided. [ABSTRACT FROM AUTHOR]

Since its first description fifty years ago, the Metropolis Monte Carlo method has been used in a variety of different ways for the simulation of continuum quantum many-body systems. This paper will consider some of the generalizations of the Metropolis algorithm employed in quantum Monte Carlo: Variational Monte Carlo, dynamical methods for projector monte carlo (i.e. diffusion Monte Carlo with rejection), multilevel sampling in path integral Monte Carlo, the sampling of permutations, cluster methods for lattice models, the penalty method for coupled electron-ionic systems and the Bayesian analysis of imaginary time correlation functions. © 2003 American Institute of Physics [ABSTRACT FROM AUTHOR]

This paper employs statistical and quantum mechanics to develop a model for the mechanism underlying the Seebeck effect. The conventional view of the equilibrium criterion for valence electrons in a material is that the Fermi Energy should be constant throughout the system. However, this criterion is an approximation and it is shown to be inadequate for thermocouple systems. An improved equilibrium criterion is developed by applying statistical and quantum mechanics to determine the total flow of electrons across an arbitrary boundary within a system. Dynamic equilibrium is then considered to be the situation where the Fermi Energy either side of the boundary is such that the flow of electrons in each direction is the same. This equilibrium criterion is then applied to the conditions along the thermocouple wires and at the junctions in order to generate a model for the Seebeck effect. The equations involved for calculating the electronic structure of a material cannot be solved analytically, so a solution is achieved using numeric models employing CASTEP code running on a Sun Beowulf cluster and iterative algorithms written in the Excel™ VBA language on a PC. The model is used to calculate the EMF versus temperature function for the gold versus platinum thermocouple, which is then compared with established experimental data. © 2003 American Institute of Physics [ABSTRACT FROM AUTHOR]

Most of physical experiments are usually described as repeated measurements of some random variables. Experimental data registered by on-line computers form time series of outcomes. The frequencies of different outcomes are compared with the probabilities provided by the algorithms of quantum theory (QT). In spite of statistical predictions of QT a claim was made that it provided the most complete description of the data and of the underlying physical phenomena. This claim could be easily rejected if some fine structures, averaged out in the standard descriptive statistical analysis, were found in time series of experimental data. To search for these structures one has to use more subtle statistical tools which were developed to study time series produced by various stochastic processes. In this talk we review some of these tools. As an example we show how the standard descriptive statistical analysis of the data is unable to reveal a fine structure in a simulated sample of AR (2) stochastic process. We emphasize once again that the violation of Bell inequalities gives no information on the completeness or the non locality of QT. The appropriate way to test the completeness of quantum theory is to search for fine structures in time series of the experimental data by means of the purity tests or by studying the autocorrelation and partial autocorrelation functions. [ABSTRACT FROM AUTHOR]

Matrix product states (MPS) are considered to be the most efficient parametrization of quantum states that emerge as ground states or low-lying excitations of short-ranged Hamiltonians in one spatial dimension. The most powerful simulation algorithms for strongly correlated quantum systems in one dimension, the family of density-matrix renormalization-group(DMRG) algorithms, can be understood as acting in this very particular state class and find a particularly transparent formulation if expressed in terms of matrix product states. In this set of lectures, I will introduce matrix product states, discuss their properties, the typical quantum mechanical operations expressed in their language, matrix product operators, and present both static and dynamic simulation algorithms. I will also make a connection to elements of entanglement theory. [ABSTRACT FROM AUTHOR]

We calculate time complexity and gate complexity of two Quantum Fourier Transforms (QFT) circuit designs. Then we compare gate (time) complexity of the two designs of the QFT. [ABSTRACT FROM AUTHOR]

Bob chooses a function and gives to Alice the black box that computes it. Alice, without knowing Bob's choice, should find a character of the function (e. g. its period) by computing its value for different arguments. There is naturally correlation between Bob's choice and the solution found by Alice. We show that, in quantum algorithms, this correlation becomes quantum. This highlights an overlooked measurement problem: sharing between two completely or partly redundant measurements the determination of completely or partly correlated measurement outcomes. All is like Alice, by reading the solution at the end of the algorithm, contributed to the initial choice of Bob, with half of it in quantum superposition for all the possible ways of taking this half. This contribution, back evolved to before running the algorithm, where Bob's choice is located, becomes Alice knowing in advance half of the choice. The quantum algorithm is the quantum superposition of all the possible ways of taking half of Bob's choice and, given the advanced knowledge of it, classically computing the missing half. The quantum speed-up comes from comparing two classical algorithms, with and without advanced knowledge of half of Bob's choice. [ABSTRACT FROM AUTHOR]

These lecture notes introduce quantum spin systems and several computational methods for studying their ground-state and finite-temperature properties. Symmetry-breaking and critical phenomena are first discussed in the simpler setting of Monte Carlo studies of classical spin systems, to illustrate finite-size scaling at continuous and first-order phase transitions. Exact diagonalization and quantum Monte Carlo (stochastic series expansion) algorithms and their computer implementations are then discussed in detail. Applications of the methods are illustrated by results for some of the most essential models in quantum magnetism, such as the S = 1/2 Heisenberg antiferromagnet in one and two dimensions, as well as extended models useful for studying quantum phase transitions between antiferromagnetic and magnetically disordered states. [ABSTRACT FROM AUTHOR]

The resonant control techniques are susceptible of assisting error accumulation in quantum control protocols consisting of large sequences of external fields. It is shown that for some algorithms the information about the final state of the system may be partially or completely lost. Non trivial operations in the non resonant regime are also discussed. [ABSTRACT FROM AUTHOR]

Ray tracing methods are commonly used to estimate the propagation of RF fields in plasmas. By going to higher order in the approximations used to derive the ray equations, it is possible to also reconstruct an approximate field from a family of rays. However, this field reconstruction is often neglected, primarily because of the difficulty of automatically dealing with focusing and reflections (caustics) in computational algorithms. In this work, we have been examining the feasibility of using tools developed for semi-classical quantum approximations to do RF field reconstructions. Specifically, it is possible to obtain a set of ODE’s which approximately describe the dynamics of a wave packet centered on a ray[1]. From this information, an approximate field can be reconstructed. We here report preliminary results for the reconstruction of lower hybrid fields using this technique. [ABSTRACT FROM AUTHOR]

We present examples of macroscopic systems entailing a quantum mechanical structure. One of our examples has a structure which is isomorphic to the spin structure for a spin 1/2 and another system entails a structure isomorphic to the structure of two spin 1/2 in the entangled singlet state. We elaborate this system by showing that an arbitrary tensor product state representing two entangled qubits can be described in a complete way by a specific internal constraint between the ray or density states of the two qubits, which describes the behavior of the state of one of the spins if measurements are executed on the other spin. Since any n-qubit unitary operation can be decomposed into 2-qubit gates and unary operations, we argue that our representation of 2-qubit entanglement contributes to a better understanding of the role of n-qubit entanglement in quantum computation. We illustrate our approach on two 2-qubit algorithms proposed by Deutsch, respectively Arvind et al. [ABSTRACT FROM AUTHOR]

We present a trivial probabilistic illustration for representation of quantum mechanics as an algorithm for approximative calculation of averages. © 2007 American Institute of Physics [ABSTRACT FROM AUTHOR]

Origin of the speed-up in quantum algorithms is not yet fully understood. But there are indications that entanglement does play an important role in quantum computation. Algorithms that do not involve entanglement can be efficiently simulated on a classical computer. Here I make a simple observation. I show that infinitesimal change in multi-particle pure product state always gives rise to an entangled state. During quantum computation even though at each time instant the state is not entangled, it does pass through entangled states at infinitesimal time steps. This applies to multi-particle pure, pseudo-pure and general mixed states as well. This suggests that even though unitary operators do not produce any entanglement, quantum entanglement guides the process of quantum computation. © 2006 American Institute of Physics [ABSTRACT FROM AUTHOR]

A short introduction to Quantum Computing, regarded as a paradigm of “parallel computing based on exterior algebras”, is presented from the point of view of its main algorithms including the primitive functions, the compositional schemes and the input and output data coding. In this study the Exterior Algebra notation is used in lieu of the more conventional Dirac’s ket notation. The basic notions of Exterior Algebra and Hilbert Spaces are sketched in order to show the qubits as points in the unit circle in the two-dimensional complex Hilbert space H1, and any word consisting of qubits as a point in the unit sphere of a external product of H1. The computing procedure is illustrated through the classical Deutsch-Josza’s algorithm, followed by the quantum algorithm to compute the Discrete Fourier Transform in linear time and the famous polynomial-time Shor’s Algorithm for integer factorization. Finally, basic notions in Quantum Cryptography and Quantum Communication Complexity are discussed. © 2006 American Institute of Physics [ABSTRACT FROM AUTHOR]

We review basic elements of the single-reference coupled-cluster theory and discuss large scale ab initio calculations of ground and excited states of 15O, 16O, and 17O using coupled-cluster methods and algorithms developed in quantum chemistry. By using realistic two-body interactions and the renormalized form of the Hamiltonian obtained with a no-core G-matrix approach, we obtain the converged results for 16O and promising preliminary results for 15O and 17O at the level of two-body interactions. The calculated properties other than energies include matter density, charge radius, and charge form factor. The relatively low costs of coupled-cluster calculations, which are characterized by the low-order polynomial scaling with the system size, enable us to probe large model spaces with up to 7 or 8 major oscillator shells, for which non-truncated shell-model calculations for nuclei with A = 15 17 active particles are presently not possible. We argue that the use of coupled-cluster methods and computer algorithms developed by quantum chemists to calculate properties of nuclei is an important step toward the development of accurate and affordable many-body theories that cross the boundaries of various physical sciences. © 2005 American Institute of Physics [ABSTRACT FROM AUTHOR]