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884
Elementary Gates for Quantum Computation
, 1995
"... We show that a set of gates that consists of all onebit quantum gates (U(2)) and the twobit exclusiveor gate (that maps Boolean values (x,y) to (x,x⊕y)) is universal in the sense that all unitary operations on arbitrarily many bits n (U(2 n)) can be expressed as compositions of these gates. We in ..."
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Cited by 215 (11 self)
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We show that a set of gates that consists of all onebit quantum gates (U(2)) and the twobit exclusiveor gate (that maps Boolean values (x,y) to (x,x⊕y)) is universal in the sense that all unitary operations on arbitrarily many bits n (U(2 n)) can be expressed as compositions of these gates. We investigate the number of the above gates required to implement other gates, such as generalized DeutschToffoli gates, that apply a specific U(2) transformation to one input bit if and only if the logical AND of all remaining input bits is satisfied. These gates play a central role in many proposed constructions of quantum computational networks. We derive upper and lower bounds on the exact number of elementary gates required to build up a variety of two and threebit quantum gates, the asymptotic number required for nbit DeutschToffoli gates, and make some observations about the number required for arbitrary nbit unitary operations.
A kernel between sets of vectors
 In International Conference on Machine Learning
, 2003
"... In various application domains, including image recognition, it is natural to represent each example as a set of vectors. With a base kernel we can implicitly map these vectors to a Hilbert space and fit a Gaussian distribution to the whole set using Kernel PCA. We define our kernel between examples ..."
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Cited by 102 (8 self)
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In various application domains, including image recognition, it is natural to represent each example as a set of vectors. With a base kernel we can implicitly map these vectors to a Hilbert space and fit a Gaussian distribution to the whole set using Kernel PCA. We define our kernel between examples as Bhattacharyya’s measure of affinity between such Gaussians. The resulting kernel is computable in closed form and enjoys many favorable properties, including graceful behavior under transformations, potentially justifying the vector set representation even in cases when more conventional representations also exist. 1.
Kontsevich’s universal formula for deformation quantization
 and the CBH formula, I, math.QA/9811174
"... Abstract. We relate a universal formula for the deformation quantization of Poisson structures (⋆products) on R d proposed by Maxim Kontsevich to the CampbellBakerHausdorff formula. Our basic thesis is that exponentiating a suitable deformation of the Poisson structure provides a prototype for su ..."
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Cited by 70 (0 self)
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Abstract. We relate a universal formula for the deformation quantization of Poisson structures (⋆products) on R d proposed by Maxim Kontsevich to the CampbellBakerHausdorff formula. Our basic thesis is that exponentiating a suitable deformation of the Poisson structure provides a prototype for such formulae. For the dual of a Lie algebra, the ⋆product given by the universal enveloping algebra via symmetrization is shown to be of this type. In fact this ⋆product is essentially given by the CampbellBakerHausdorff (CBH) formula. We call this the CBHquantization. Next we limn Kontsevich’s construction using a graphical representation for differential calculus. We outline a structure theory for the weighted graphs which encode bidifferential operators in his formula and compute certain weights. We then establish that the Kontsevich and CBH quantizations are identical for the duals of nilpotent Lie algebras. Consequently part of Kontsevich’s ⋆product is determined by the CBH formula. Working the other way, we have a graphical encoding for the
Decoherence, einselection, and the quantum origins of the classical
 REVIEWS OF MODERN PHYSICS 75, 715. AVAILABLE ONLINE AT HTTP://ARXIV.ORG/ABS/QUANTPH/0105127
, 2003
"... The manner in which states of some quantum systems become effectively classical is of great significance for the foundations of quantum physics, as well as for problems of practical interest such as quantum engineering. In the past two decades it has become increasingly clear that many (perhaps all) ..."
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Cited by 54 (1 self)
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The manner in which states of some quantum systems become effectively classical is of great significance for the foundations of quantum physics, as well as for problems of practical interest such as quantum engineering. In the past two decades it has become increasingly clear that many (perhaps all) of the symptoms of classicality can be induced in quantum systems by their environments. Thus decoherence is caused by the interaction in which the environment in effect monitors certain observables of the system, destroying coherence between the pointer states corresponding to their eigenvalues. This leads to environmentinduced superselection or einselection, a quantum process associated with selective loss of information. Einselected pointer states are stable. They can retain correlations with the rest of the universe in spite of the environment. Einselection enforces classicality by imposing an effective ban on the vast majority of the Hilbert space, eliminating especially the flagrantly nonlocal "Schrödingercat states." The classical structure of phase space emerges from the quantum Hilbert space in the appropriate macroscopic limit. Combination of einselection with dynamics leads to the idealizations of a point and of a classical trajectory. In measurements, einselection replaces quantum entanglement between the apparatus and the measured system with the classical correlation. Only the preferred pointer observable of the apparatus can store information
A Universal Reduction Procedure for Hamiltonian Group Actions
 in The Geometry of Hamiltonian systems, T. Ratiu, ed., MSRI Series
, 1991
"... We give a universal method of inducing a Poisson structure on a singular reduced space from the Poisson structure on the orbit space for the group action. For proper actions we show that this reduced Poisson structure is nondegenerate. Furthermore, in cases where the MarsdenWeinstein reduction ..."
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Cited by 50 (3 self)
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We give a universal method of inducing a Poisson structure on a singular reduced space from the Poisson structure on the orbit space for the group action. For proper actions we show that this reduced Poisson structure is nondegenerate. Furthermore, in cases where the MarsdenWeinstein reduction is welldefined, the action is proper, and the preimage of a coadjoint orbit under the momentum mapping is closed, we show that universal reduction and MarsdenWeinstein reduction coincide. As an example, we explicitly construct the reduced spaces and their Poisson algebras for the spherical pendulum.
Rieffel induction as generalized quantum MarsdenWeinstein reduction
 J. Geom. Phys
, 1995
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Universe of Fluctuations
 Int.J. of Mod.Phys.A
, 1998
"... Dedicated to the memory of my parents We discuss a recent model of a Quantum Mechanical Black Hole (QMBH) which describes the most fundamental known particles the leptons and approximately the quarks in terms of the KerrNewman Black Hole with a naked singularity shielded by Zitterbewegung effects. ..."
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Cited by 43 (40 self)
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Dedicated to the memory of my parents We discuss a recent model of a Quantum Mechanical Black Hole (QMBH) which describes the most fundamental known particles the leptons and approximately the quarks in terms of the KerrNewman Black Hole with a naked singularity shielded by Zitterbewegung effects. This goes beyond the Zitterbewegung and self interaction models of Barut and Bracken, Hestenes, Chacko and others and provides a unified picture which amongst other things gives a rationale for and an insight into: 1. The apparently inexplicable reason why complex space time transformations lead to the KerrNewman metric in General Relativity. 2. The value of the fine structure constant. 3. The ratio between electromagnetic and gravitational interaction strengths. 4. The anomalous gyromagnetic ratio for the electron. 5. Why the neutrino is left handed. 6. Why the charge is discrete. In the spirit of Effective Field Theories, this model provides an alternative formalism for Quantum Theory and also for its combination with General Relativity. Finally a mechanism for the formation of these QMBH or particles is explored within the framework of Stochastic Electrodynamics, QED and Quantum Statistical Mechanics. The cosmological implications 0 International Journal of Modern Physics ’A’ 1 are then examined. It turns out that a surprisingly large number of facts, including some which were hitherto inexplicable, follow as a consequence of the model. These include a theoretical deduction of the Mass, Radius and Age of the Universe, as also the values of Hubble’s constant and the Cosmological constant. 1
Quantum chromodynamics and other field theories on the light cone
, 1997
"... In recent years lightcone quantization of quantum field theory has emerged as a promising method for solving problems in the strong coupling regime. The approach has a number of unique features that make it particularly appealing, most notably, the ground state of the free theory is also a ground s ..."
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Cited by 37 (10 self)
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In recent years lightcone quantization of quantum field theory has emerged as a promising method for solving problems in the strong coupling regime. The approach has a number of unique features that make it particularly appealing, most notably, the ground state of the free theory is also a ground state of the full theory. We discuss the lightcone quantization of gauge theories from two perspectives: as a calculational tool for representing hadrons as QCD boundstates of relativistic quarks and gluons, and also as a novel method for simulating quantum field theory on a computer. The lightcone Fock state expansion of wavefunctions provides a precise definition of the parton model and a general calculus for hadronic matrix elements. We present several new applications of lightcone Fock methods, including calculations of exclusive weak decays of heavy hadrons, and intrinsic heavyquark contributions to structure functions. A general nonperturbative method for numerically solving quantum field theories, “discretized lightcone quantization”, is outlined and applied to several gauge theories. This method is invariant under the
Time dependent resonance theory
, 1995
"... An important class of resonance problems involves the study of perturbations of systems having embedded eigenvalues in their continuous spectrum. Problems with this mathematical structure arise in the study of many physical systems, e.g. the coupling of an atom or molecule to a photonradiation fiel ..."
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Cited by 33 (10 self)
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An important class of resonance problems involves the study of perturbations of systems having embedded eigenvalues in their continuous spectrum. Problems with this mathematical structure arise in the study of many physical systems, e.g. the coupling of an atom or molecule to a photonradiation field, and Auger states of the helium atom, as well as in spectral geometry and number theory. We present a dynamic (timedependent) theory of such quantum resonances. The key hypotheses are (i) a resonance condition which holds generically (nonvanishing of the Fermi golden rule) and (ii) local decay estimates for the unperturbed dynamics with initial data consisting of continuum modes associated with an interval containing the embedded eigenvalue of the unperturbed Hamiltonian. No assumption of dilation analyticity of the potential is made. Our method explicitly demonstrates the flow of energy from the resonant discrete mode to continuum modes due to their coupling. The approach is also applicable to nonautonomous linear problems and to nonlinear problems. We derive the time behavior of the resonant states for intermediate and long times. Examples and applications are presented. Among them is a proof of the instability of an embedded eigenvalue at a threshold energy under suitable hypotheses.
Quantum Equilibrium and the Role of Operators as Observables in Quantum Theory
, 2003
"... Bohmian mechanics is the most naively obvious embedding imaginable of Schrödinger’s equation into a completely coherent physical theory. It describes a world in which particles move in a highly nonNewtonian sort of way, one which may at first appear to have little to do with the spectrum of predict ..."
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Cited by 31 (14 self)
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Bohmian mechanics is the most naively obvious embedding imaginable of Schrödinger’s equation into a completely coherent physical theory. It describes a world in which particles move in a highly nonNewtonian sort of way, one which may at first appear to have little to do with the spectrum of predictions of quantum mechanics. It turns out, however, that as a consequence of the defining dynamical equations of Bohmian mechanics, when a system has wave function ψ its configuration is typically random, with probability density ρ given by ψ², the quantum equilibrium distribution. It also turns out that the entire quantum formalism, operators as observables and all the rest, naturally emerges in Bohmian mechanics from the analysis of “measurements. ” This analysis reveals the status of operators as observables in the description of quantum phenomena, and facilitates a clear view of the range of applicability of the usual quantum mechanical formulas.