Application of unitary transformations to fourth order perturbation theory
Unitary transformations applied to fourth order perturbation theory for energy correction to nondegenerate stationary system
QUANTUM MECHANICS↗
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A note on parallel and pipeline computation of fast unitary transforms
The parallel and pipeline organization of fast unitary transform algorithms such as the Fast Fourier Transform are discussed. The efficiency is pointed out of a combined parallel-pipeline processor of a transform such as the Haar transform in which 2 to the n minus 1 power hardware butterflies generate a transform of order 2 to the n power every computation cycle.
Fino, B. J.↗
Parallel and pipeline computation of fast unitary transforms
The letter discusses the parallel and pipeline organization of fast-unitary-transform algorithms such as the fast Fourier transform, and points out the efficiency of a combined parallel-pipeline processor of a transform such as the Haar transform, in which (2 to the n-th power) -1 hardware 'butterflies' generate a transform of order 2 to the n-th power every computation cycle.
Fino, B. J.↗
Computation of transform domain covariance matrices
It is often of interest in applications to compute the covariance matrix of a random process transformed by a fast unitary transform. Here, the recursive definition of fast unitary transforms is used to derive recursive relations for the covariance matrices of the transformed process. These relations lead to fast methods of computation of covariance matrices and to substantial reductions of the number of arithmetic operations required.
Fino, B. J.↗
Source encoding of images for efficient transmission
A set of recursive rules which generate unitary transforms with a fast algorithm are presented. For each rule, simple relations give the number of elementary operations required by the fast algorithm. The common Fourier, Walsh-Hadamard, Slant, and Haar transforms are expressed with these rules. The framework developed allows the introduction of generalized transforms which include all common transforms in a large class of identical computation transforms. A systematic and unified view is provided for unitary transforms which have appeared in the literature and for a number of new transforms of potential interest. Generalization to complex and multidimensional unitary transforms is considered.
Algazi, V. R.↗
Non-gauge phase transformations in quantum transition amplitudes
The prescription for introducing a gauge transformation into a quantum transition amplitude, nominally well known, contains an ambiguous feature. It is presumed by some authors that an appropriate transformation of the phase of a wave function will generate the associated gauge transformation. It is shown that this is a necessary but not sufficient step. Examples from the literature are cited to show the consequences of the failure of this procedure. One must distinguish between true gauge transformations and unitary transformations within a fixed gauge.
Reiss, H. R.↗
Simultaneous Stoquasticity
Stoquastic Hamiltonians play a role in the computational complexity of the local Hamiltonian problem as well as the study of classical simulability. In particular, stoquastic Hamiltonians can be straightforwardly simulated using Monte Carlo techniques. We address the question of whether two or more Hamiltonians may be made simultaneously stoquastic via a unitary transformation. This question has important implications for the complexity of simulating quantum annealing where quantum advantage is related to the stoquasticity of the Hamiltonians involved in the anneal. We find that for almost all problems no such unitary exists and show that the problem of determining the existence of such a unitary is equivalent to identifying if there is a solution to a system of polynomial (in)equalities in the matrix elements of the initial and transformed Hamiltonians. Solving such a system of equations is NP-hard. We highlight a geometric understanding of this problem in terms of a collection of generalized Bloch vectors.
Jacob Bringewatt↗
Simultaneous Stoquasticity
Stoquastic Hamiltonians play a role in the computational complexity of the local Hamiltonian problem as well as the study of classical simulability. In particular, stoquastic Hamiltonians can be straightforwardly simulated using Monte Carlo techniques. We address the question of whether two or more Hamiltonians may be made simultaneously stoquastic via a unitary transformation. This question has important implications for the complexity of simulating quantum annealing where quantum advantage is related to the stoquasticity of the Hamiltonians involved in the anneal. We find that for almost all problems no such unitary exists and show that the problem of determining the existence of such a unitary is equivalent to identifying if there is a solution to a system of polynomial (in)equalities in the matrix elements of the initial and transformed Hamiltonians. Solving such a system of equations is NP-hard. We highlight a geometric understanding of this problem in terms of a collection of generalized Bloch vectors.
Monte Carlo↗
General relation between electron trapping and the normal modes of a lattice.
Adiabatic harmonic unitary transformations and relation between electron trapping energy and normal lattice modes associated with F center ground state
ELECTRON-PHONON INTERACTION↗
On the use of corresponding orbitals in the calculation of nonorthogonal transition moments
Full valence and first-order CI wave functions are invariant with respect to unitary transformations among the valence orbitals. We exploit this degree of freedom and show that by transforming the valence orbitals into a corresponding orbital basis, nonorthogonal transition moment calculations become an easily managed task. Sample full valence calculations on several states of O2(+) and OF are also presented.
Lengsfield, B. H., III↗
Kinematics of the CS method for the treatment of molecular collisions
The problem of the quantum treatment of nonreactive collisions of two simple molecules or of an atom and a molecule is considered mathematically. Kinematical structure theorems analogous to the Wigner-Eckart theorem for scalar operators are derived for the transition operator (T-operator) of a two-molecule system, where, in addition to the usual conservation laws, certain additional conservation laws are presumed satisfied in a collision. The additional conservation laws were examined in order to develop a testable approximation scheme (CS method) for the specification of the kinematics of a system of two rigid diatomic molecules. The rand p-helicity basis states for a two-rigid rotor system are defined and unitary transformations established between these and conventional bases. It is found that only questionable criteria exist for testing the on-the-energy-shell matrix of a T(energy) operator for the presence of properties of two of the additional conservation laws.
Hahne, G. E.↗
Parallel solution of closely coupled systems
The odd-even permutation and associated unitary transformations for reordering the matrix coefficient A are employed as means of breaking the strong seriality which is characteristic of closely coupled systems. The nested dissection technique is also reviewed, and the equivalence between reordering A and dissecting its network is established. The effect of transforming A with odd-even permutation on its topology and the topology of its Cholesky factors is discussed. This leads to the construction of directed graphs showing the computational steps required for factoring A, their precedence relationships and their sequential and concurrent assignment to the available processors. Expressions for the speed-up and efficiency of using N processors in parallel relative to the sequential use of a single processor are derived from the directed graph. Similar expressions are also derived when the number of available processors is fewer than required.
Utku, S.↗
The wave function and minimum uncertainty function of the bound quadratic Hamiltonian system
The bound quadratic Hamiltonian system is analyzed explicitly on the basis of quantum mechanics. We have derived the invariant quantity with an auxiliary equation as the classical equation of motion. With the use of this invariant it can be determined whether or not the system is bound. In bound system we have evaluated the exact eigenfunction and minimum uncertainty function through unitary transformation.
Yeon, Kyu Hwang↗
Using deflation in the pole assignment problem with output feedback
A direct algorithm is suggested for the computation of a linear output feedback for a multi input, multi output system such that the resultant closed-loop matrix has eigenvalues that include a specified set of eigenvalues. The algorithm uses deflation based on unitary similarity transformations. Thus researchers hope the algorithm is numerically stable; however, this has not been proven as yet.
Miminis, George↗
A restricted signature normal form for Hermitian matrices, quasi-spectral decompositions, and applications
In recent years, a number of results on the relationships between the inertias of Hermitian matrices and the inertias of their principal submatrices appeared in the literature. We study restricted congruence transformation of Hermitian matrices M which, at the same time, induce a congruence transformation of a given principal submatrix A of M. Such transformations lead to concept of the restricted signature normal form of M. In particular, by means of this normal form, we obtain short proofs of most of the known inertia theorems and also derive some new results of this type. For some applications, a special class of almost unitary restricted congruence transformations turns out to be useful. We show that, with such transformations, M can be reduced to a quasi-diagonal form which, in particular, displays the eigenvalues of A. Finally, applications of this quasi-spectral decomposition to generalize inverses and Hermitian matrix pencils are discussed.
Freund, Roland W.↗
Automated Rapid Prototyping of 3D Ceramic Parts
An automated system of manufacturing equipment produces three-dimensional (3D) ceramic parts specified by computational models of the parts. The system implements an advanced, automated version of a generic rapid-prototyping process in which the fabrication of an object having a possibly complex 3D shape includes stacking of thin sheets, the outlines of which closely approximate the horizontal cross sections of the object at their respective heights. In this process, the thin sheets are made of a ceramic precursor material, and the stack is subsequently heated to transform it into a unitary ceramic object. In addition to the computer used to generate the computational model of the part to be fabricated, the equipment used in this process includes: 1) A commercially available laminated-object-manufacturing machine that was originally designed for building woodlike 3D objects from paper and was modified to accept sheets of ceramic precursor material, and 2) A machine designed specifically to feed single sheets of ceramic precursor material to the laminated-object-manufacturing machine. Like other rapid-prototyping processes that utilize stacking of thin sheets, this process begins with generation of the computational model of the part to be fabricated, followed by computational sectioning of the part into layers of predetermined thickness that collectively define the shape of the part. Information about each layer is transmitted to rapid-prototyping equipment, where the part is built layer by layer. What distinguishes this process from other rapid-prototyping processes that utilize stacking of thin sheets are the details of the machines and the actions that they perform. In this process, flexible sheets of ceramic precursor material (called "green" ceramic sheets) suitable for lamination are produced by tape casting. The binder used in the tape casting is specially formulated to enable lamination of layers with little or no applied heat or pressure. The tape is cut into individual sheets, which are stacked in the sheet-feeding machine until used. The sheet-feeding machine can hold enough sheets for about 8 hours of continuous operation.
McMillin, Scott G.↗
Non-unitary probabilistic quantum computing circuit and method
A quantum circuit performing quantum computation in a quantum computer. A chosen transformation of an initial n-qubit state is probabilistically obtained. The circuit comprises a unitary quantum operator obtained from a non-unitary quantum operator, operating on an n-qubit state and an ancilla state. When operation on the ancilla state provides a success condition, computation is stopped. When operation on the ancilla state provides a failure condition, computation is performed again on the ancilla state and the n-qubit state obtained in the previous computation, until a success condition is obtained.
Williams, Colin P.↗
Covariant harmonic oscillators: 1973 revisited
Using the relativistic harmonic oscillator, a physical basis is given to the phenomenological wave function of Yukawa which is covariant and normalizable. It is shown that this wave function can be interpreted in terms of the unitary irreducible representations of the Poincare group. The transformation properties of these covariant wave functions are also demonstrated.
Noz, M. E.↗