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

On Local Equivalence, Surface Code States and Matroids

Recently, Ji et al disproved the LU-LC conjecture and showed that the local unitary and local Clifford equivalence classes of the stabilizer states are not always the same. Despite the fact this settles the LU-LC conjecture, a sufficient condition for stabilizer states that violate the LU-LC conjecture is missing. In this paper, we investigate further the properties of stabilizer states with respect to local equivalence. Our first result shows that there exist infinitely many stabilizer states which violate the LU-LC conjecture. In particular, we show that for all numbers of qubits $n\geq 28$, there exist distance two stabilizer states which are counterexamples to the LU-LC conjecture. We prove that for all odd $n\geq 195$, there exist stabilizer states with distance greater than two which are LU equivalent but not LC equivalent. Two important classes of stabilizer states that are of great interest in quantum computation are the cluster states and stabilizer states of the surface codes. To date, the status of these states with respect to the LU-LC conjecture was not studied. We show that, under some minimal restrictions, both these classes of states preclude any counterexamples. In this context, we also show that the associated surface codes do not have any encoded non-Clifford transversal gates. We characterize the CSS surface code states in terms of a class of minor closed binary matroids. In addition to making connection with an important open problem in binary matroid theory, this characterization does in some cases provide an efficient test for CSS states that are not counterexamples.Comment: LaTeX, 13 pages; Revised introduction, minor changes and corrections mainly in section V

Respiratory symptoms and cross-shift lung function in relation to cotton dust and endotoxin exposure in textile workers in Nepal: a cross-sectional study

Objectives: Inhalation of a cotton-based particulates has previously been associated with respiratory symptoms and impaired lung function. This study investigates the respiratory health of Nepalese textile workers in relation to dust and endotoxin exposure. Methods: A total of 938 individuals from four sectors (garment, carpet, weaving and recycling) of the textile industry in Kathmandu, Nepal completed a health questionnaire and performed spirometry. A subset (n=384) performed cross-shift spirometry. Personal exposure to inhalable dust and airborne endotoxin was measured during a full shift for 114 workers. Results: The overall prevalence of persistent cough, persistent phlegm, wheeze ever, breathlessness ever and chest tightness ever was 8.5%, 12.5%, 3.2%, 6.5% and 12.3%, respectively. Symptoms were most common among recyclers and least common among garment workers. Exposure to inhalable dust significantly predicted persistent cough and chest tightness. Exposure to endotoxin did not have any independent predictive effect. Significant cross-shift reduction in forced expiratory volume in 1 s (FEV1) and forced vital capacity (FVC) were found (p<0.001 for both) being largest for FEV1 in the recyclers (−143 mL), and least in the garment workers (−38 mL; p=0.012). Exposure to inhalable dust predicted a cross-shift reduction in FEV1. Conclusions: This study is the first to investigate the respiratory health of Nepalese cotton workers. The measured association between inhalable dust exposure and reporting of respiratory symptoms and across-shift decrement in FEV1 and FVC indicates that improved dust control measures should be instituted, particularly in the recycling and carpet sectors. The possible role of other biologically active agents of cotton dust beyond endotoxin should be further explored

Quantum bound states for a derivative nonlinear Schrodinger model and number theory

A derivative nonlinear Schrodinger model is shown to support localized N-body bound states for several ranges (called bands) of the coupling constant eta. The ranges of eta within each band can be completely determined using number theoretic concepts such as Farey sequences and continued fractions. For N > 2, the N-body bound states can have both positive and negative momentum. For eta > 0, bound states with positive momentum have positive binding energy, while states with negative momentum have negative binding energy.Comment: Revtex, 7 pages including 2 figures, to appear in Mod. Phys. Lett.

Grover's Quantum Search Algorithm and Diophantine Approximation

In a fundamental paper [Phys. Rev. Lett. 78, 325 (1997)] Grover showed how a quantum computer can find a single marked object in a database of size N by using only O(N^{1/2}) queries of the oracle that identifies the object. His result was generalized to the case of finding one object in a subset of marked elements. We consider the following computational problem: A subset of marked elements is given whose number of elements is either M or K, M<K, our task is to determine which is the case. We show how to solve this problem with a high probability of success using only iterations of Grover's basic step (and no other algorithm). Let m be the required number of iterations; we prove that under certain restrictions on the sizes of M and K the estimation m < (2N^{1/2})/(K^{1/2}-M^{1/2}) obtains. This bound sharpens previous results and is known to be optimal up to a constant factor. Our method involves simultaneous Diophantine approximations, so that Grover's algorithm is conceptualized as an orbit of an ergodic automorphism of the torus. We comment on situations where the algorithm may be slow, and note the similarity between these cases and the problem of small divisors in classical mechanics.Comment: 8 pages, revtex, Title change

Maximum-Entropy Weighting of Multi-Component Earth Climate Models

A maximum entropy-based framework is presented for the synthesis of projections from multiple Earth climate models. This identifies the most representative (most probable) model from a set of climate models -- as defined by specified constraints -- eliminating the need to calculate the entire set. Two approaches are developed, based on individual climate models or ensembles of models, subject to a single cost (energy) constraint or competing cost-benefit constraints. A finite-time limit on the minimum cost of modifying a model synthesis framework, at finite rates of change, is also reported.Comment: Inspired by discussions at the Mathematical and Statistical Approaches to Climate Modelling and Prediction workshop, Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, 11 Aug. to 22 Dec. 2010. Accepted for publication in Climate Dynamics, 8 August 201

Electron-electron interaction corrections to the thermal conductivity in disordered conductors

We evaluate the electron-electron interaction corrections to the electronic thermal conductivity in a disordered conductor in the diffusive regime. We use a diagrammatic many-body method analogous to that of Altshuler and Aronov for the electrical conductivity. We derive results in one, two and three dimensions for both the singlet and triplet channels, and in all cases find that the Wiedemann-Franz law is violated.Comment: 8 pages, 2 figures Typos corrected in formulas (15) and (A.4) and Table 1; discussion of previous work in introduction extended; reference clarifying different definitions of parameter F adde

The f-vector of the descent polytope

For a positive integer n and a subset S of [n-1], the descent polytope DP_S is the set of points x_1, ..., x_n in the n-dimensional unit cube [0,1]^n such that x_i >= x_{i+1} for i in S and x_i <= x_{i+1} otherwise. First, we express the f-vector of DP_S as a sum over all subsets of [n-1]. Second, we use certain factorizations of the associated word over a two-letter alphabet to describe the f-vector. We show that the f-vector is maximized when the set S is the alternating set {1,3,5, ...}. We derive a generating function for the f-polynomial F_S(t) of DP_S, written as a formal power series in two non-commuting variables with coefficients in Z[t]. We also obtain the generating function for the Ehrhart polynomials of the descent polytopes.Comment: 14 pages; to appear in Discrete & Computational Geometr

Quenching across quantum critical points in periodic systems: dependence of scaling laws on periodicity

We study the quenching dynamics of a many-body system in one dimension described by a Hamiltonian that has spatial periodicity. Specifically, we consider a spin-1/2 chain with equal xx and yy couplings and subject to a periodically varying magnetic field in the z direction or, equivalently, a tight-binding model of spinless fermions with a periodic local chemical potential, having period 2q, where q is a natural number. For a linear quench of the magnetic field strength (or potential strength) at rate 1/\tau across a quantum critical point, we find that the density of defects thereby produced scales as 1/\tau^{q/(q+1)}, deviating from the 1/\sqrt{\tau} scaling that is ubiquitous to a range of systems. We analyze this behavior by mapping the low-energy physics of the system to a set of fermionic two-level systems labeled by the lattice momentum k undergoing a non-linear quench as well as by performing numerical simulations. We also find that if the magnetic field is a superposition of different periods, the power law depends only on the smallest period for very large values of \tau although it may exhibit a cross-over at intermediate values of \tau. Finally, for the case where a zz coupling is also present in the spin chain, or equivalently, where interactions are present in the fermionic system, we argue that the power associated with the scaling law depends on a combination of q and interaction strength.Comment: 13 pages including 11 figure