Exactness of the Original Grover Search Algorithm

It is well-known that when searching one out of four, the original Grover's search algorithm is exact; that is, it succeeds with certainty. It is natural to ask the inverse question: If we are not searching one out of four, is Grover's algorithm definitely not exact? In this article we give a complete answer to this question through some rationality results of trigonometric functions.Comment: 8 pages, 2 figure

Lie, Cheat, and Steal: How Harmful Brands Motivate Consumers to Act Unethically

While brand punishment—through either individual or collective action—has received ample attention by consumer psychologists, absent from this literature is that such punishment can take the form of unethical actions that can occur even when the consumer is not personally harmed. Across three studies, we examine consumers’ propensity to act unethically towards a brand that they perceive to be harmful. We document that when consumers come to see brands as harmful—even in the absence of a direct, personal transgression—they can be motivated to seek retribution in the form of unethical intentions and behaviors. That is, consumers are more likely to lie, cheat, or steal to punish a harmful brand. Drawing on these findings, we advance implications for consumer psychologists and marketing practitioners and provide avenues for future research in the area

Bound on the multiplicity of almost complete intersections

Let $R$ be a polynomial ring over a field of characteristic zero and let $I \subset R$ be a graded ideal of height $N$ which is minimally generated by $N+1$ homogeneous polynomials. If $I=(f_1,...,f_{N+1})$ where $f_i$ has degree $d_i$ and $(f_1,...,f_N)$ has height $N$, then the multiplicity of $R/I$ is bounded above by $\prod_{i=1}^N d_i - \max\{1, \sum_{i=1}^N (d_i-1) - (d_{N+1}-1) \}$.Comment: 7 pages; to appear in Communications in Algebr

On Dijkgraaf-Witten Type Invariants

We explicitly construct a series of lattice models based upon the gauge group $Z_{p}$ which have the property of subdivision invariance, when the coupling parameter is quantized and the field configurations are restricted to satisfy a type of mod-$p$ flatness condition. The simplest model of this type yields the Dijkgraaf-Witten invariant of a $3$-manifold and is based upon a single link, or $1$-simplex, field. Depending upon the manifold's dimension, other models may have more than one species of field variable, and these may be based on higher dimensional simplices.Comment: 18 page

Perceiving the agency of harmful agents: A test of dehumanization versus moral typecasting accounts

It is clear that harmful agents are targets of severe condemnation, but it is much less clear how perceivers conceptualize the agency of harmful agents. The current studies tested two competing predictions made by moral typecasting theory and the dehumanization literature. Across six studies, harmful agents were perceived to possess less agency than neutral (non-offending) and benevolent agents, consistent with a dehumanization perspective but inconsistent with the assumptions of moral typecasting theory. This was observed for human targets (Studies 1–2b and 4–5) and corporations (Study 3), and across various gradations of harmfulness (Studies 3 and 4). Importantly, denial of agency to harmful agents occurred even when controlling for perceptions of the agent’s likeability (Studies 2a and 2b) and while using two different operationalizations of agency (Study 2a). Study 5 showed that harmful agents are denied agency primarily through an inferential process, and less through motivations to see the agent punished. Across all six studies, harmful agents were deemed less worthy of moral standing as a consequence of their harmful conduct and this reduction in moral standing was mediated through reductions in agency. Our findings clarify a current tension in the moral cognition literature, which have direct implications for the moral typecasting framework

Displacement energy of unit disk cotangent bundles

We give an upper bound of a Hamiltonian displacement energy of a unit disk cotangent bundle $D^*M$ in a cotangent bundle $T^*M$, when the base manifold $M$ is an open Riemannian manifold. Our main result is that the displacement energy is not greater than $C r(M)$, where $r(M)$ is the inner radius of $M$, and $C$ is a dimensional constant. As an immediate application, we study symplectic embedding problems of unit disk cotangent bundles. Moreover, combined with results in symplectic geometry, our main result shows the existence of short periodic billiard trajectories and short geodesic loops.Comment: Title slightly changed. Close to the version published online in Math Zei

Quantum walks on quotient graphs

A discrete-time quantum walk on a graph is the repeated application of a unitary evolution operator to a Hilbert space corresponding to the graph. If this unitary evolution operator has an associated group of symmetries, then for certain initial states the walk will be confined to a subspace of the original Hilbert space. Symmetries of the original graph, given by its automorphism group, can be inherited by the evolution operator. We show that a quantum walk confined to the subspace corresponding to this symmetry group can be seen as a different quantum walk on a smaller quotient graph. We give an explicit construction of the quotient graph for any subgroup of the automorphism group and illustrate it with examples. The automorphisms of the quotient graph which are inherited from the original graph are the original automorphism group modulo the subgroup used to construct it. We then analyze the behavior of hitting times on quotient graphs. Hitting time is the average time it takes a walk to reach a given final vertex from a given initial vertex. It has been shown in earlier work [Phys. Rev. A {\bf 74}, 042334 (2006)] that the hitting time can be infinite. We give a condition which determines whether the quotient graph has infinite hitting times given that they exist in the original graph. We apply this condition for the examples discussed and determine which quotient graphs have infinite hitting times. All known examples of quantum walks with fast hitting times correspond to systems with quotient graphs much smaller than the original graph; we conjecture that the existence of a small quotient graph with finite hitting times is necessary for a walk to exhibit a quantum speed-up.Comment: 18 pages, 7 figures in EPS forma

State Sum Models and Simplicial Cohomology

We study a class of subdivision invariant lattice models based on the gauge group $Z_{p}$, with particular emphasis on the four dimensional example. This model is based upon the assignment of field variables to both the $1$- and $2$-dimensional simplices of the simplicial complex. The property of subdivision invariance is achieved when the coupling parameter is quantized and the field configurations are restricted to satisfy a type of mod-$p$ flatness condition. By explicit computation of the partition function for the manifold $RP^{3} \times S^{1}$, we establish that the theory has a quantum Hilbert space which differs from the classical one.Comment: 28 pages, Latex, ITFA-94-13, (Expanded version with two new sections

Quantum Fourier transform, Heisenberg groups and quasiprobability distributions

This paper aims to explore the inherent connection among Heisenberg groups, quantum Fourier transform and (quasiprobability) distribution functions. Distribution functions for continuous and finite quantum systems are examined first as a semiclassical approach to quantum probability distribution. This leads to studying certain functionals of a pair of "conjugate" observables, connected via the quantum Fourier transform. The Heisenberg groups emerge naturally from this study and we take a rapid look at their representations. The quantum Fourier transform appears as the intertwining operator of two equivalent representation arising out of an automorphism of the group. Distribution functions correspond to certain distinguished sets in the group algebra. The marginal properties of a particular class of distribution functions (Wigner distributions) arise from a class of automorphisms of the group algebra of the Heisenberg group. We then study the reconstruction of Wigner function from the marginal distributions via inverse Radon transform giving explicit formulas. We consider applications of our approach to quantum information processing and quantum process tomography.Comment: 39 page