On the efficiency of Hamiltonian-based quantum computation for low-rank matrices

We present an extension of Adiabatic Quantum Computing (AQC) algorithm for the unstructured search to the case when the number of marked items is unknown. The algorithm maintains the optimal Grover speedup and includes a small counting subroutine. Our other results include a lower bound on the amount of time needed to perform a general Hamiltonian-based quantum search, a lower bound on the evolution time needed to perform a search that is valid in the presence of control error and a generic upper bound on the minimum eigenvalue gap for evolutions. In particular, we demonstrate that quantum speedup for the unstructured search using AQC type algorithms may only be achieved under very rigid control precision requirements.Comment: 17 pages, no figures, to appear in JM

Local and global statistical distances are equivalent on pure states

The statistical distance between pure quantum states is obtained by finding a measurement that is optimal in a sense defined by Wootters. As such, one may expect that the statistical distance will turn out to be different if the set of possible measurements is restricted in some way. It nonetheless turns out that if the restriction is to local operations and classical communication (LOCC) on any multipartite system, then the statistical distance is the same as it is without restriction, being equal to the angle between the states in Hilbert space.Comment: 5 pages, comments welcom

Improved Lower Bounds for Locally Decodable Codes and Private Information Retrieval

We prove new lower bounds for locally decodable codes and private information retrieval. We show that a 2-query LDC encoding n-bit strings over an l-bit alphabet, where the decoder only uses b bits of each queried position of the codeword, needs code length m = exp(Omega(n/(2^b Sum_{i=0}^b {l choose i}))) Similarly, a 2-server PIR scheme with an n-bit database and t-bit queries, where the user only needs b bits from each of the two l-bit answers, unknown to the servers, satisfies t = Omega(n/(2^b Sum_{i=0}^b {l choose i})). This implies that several known PIR schemes are close to optimal. Our results generalize those of Goldreich et al. who proved roughly the same bounds for linear LDCs and PIRs. Like earlier work by Kerenidis and de Wolf, our classical lower bounds are proved using quantum computational techniques. In particular, we give a tight analysis of how well a 2-input function can be computed from a quantum superposition of both inputs.Comment: 12 pages LaTeX, To appear in ICALP '0

Neurotransmitter profile of saccadic omnipause neurons in nucleus raphe interpositus

Saccadic omnipause neurons (OPNs) are essential for the generation of saccadic eye movements. In primates OPNs are located near the midline within the nucleus raphe interpositus (rip). In the present study we used several different neuroanatomical methods to investigate the transmitters associated with OPNs in the monkey. Immunolabeling for the calcium-binding protein parvalbumin was employed to mark OPNs in the monkey and define the homologous cell group in cat and human. The use of antibodies against GABA, glycine (GLY), glutamate (GLU), serotonin (5-HT), and tyrosine hydroxylase revealed that the somata of OPNs are GLY immunoreactive, but they are devoid of GABA and 5-HT immunostaining. In situ hybridization with the GAD67 mRNA probe confirmed the negative GABA immunostaining of OPNs. 3H-GLY was injected into a projection field of OPNs, the rostral interstitial nucleus of the medial longitudinal fascicle (riMLF)--the vertical saccadic burst neuron area. This resulted in selective retrograde labeling of the OPNs in rip, while no labeling was found in the superior colliculus, which sends an excitatory projection to the riMLF. The somata and dendrites of putative burst neurons in the riMLF were contacted by numerous GLY- immunoreactive terminals. The quantitative analysis of immunoreactive terminal-like structures contacting OPNs revealed a strong input from GLY- and GABA-positive terminals on somata and dendrites, whereas GLU- positive puncta were mainly confined to the dendrites. Very few 5-HT and catecholaminergic terminals contacted OPN somata. Our findings suggest that OPNs use GLY as a neurotransmitter, and they receive numerous contacts from GABAergic, glycinergic, and glutaminergic afferents, and significantly fewer from monoaminergic inputs.</jats:p

Solution to the Equations of the Moment Expansions

We develop a formula for matching a Taylor series about the origin and an asymptotic exponential expansion for large values of the coordinate. We test it on the expansion of the generating functions for the moments and connected moments of the Hamiltonian operator. In the former case the formula produces the energies and overlaps for the Rayleigh-Ritz method in the Krylov space. We choose the harmonic oscillator and a strongly anharmonic oscillator as illustrative examples for numerical test. Our results reveal some features of the connected-moments expansion that were overlooked in earlier studies and applications of the approach

Character of Locally Inequivalent Classes of States and Entropy of Entanglement

In this letter we have established the physical character of pure bipartite states with the same amount of entanglement in the same Schmidt rank that either they are local unitarily connected or they are incomparable. There exist infinite number of deterministically locally inequivalent classes of pure bipartite states in the same Schmidt rank (starting from three) having same amount of entanglement. Further, if there exists incomparable states with same entanglement in higher Schmidt ranks (greater than three), then they should differ in at least three Schmidt coefficients.Comment: 4 pages, revtex4, no figure, accepted in Physical Review A (rapid communications

Single-qubit optical quantum fingerprinting

We analyze and demonstrate the feasibility and superiority of linear optical single-qubit fingerprinting over its classical counterpart. For one-qubit fingerprinting of two-bit messages, we prepare `tetrahedral' qubit states experimentally and show that they meet the requirements for quantum fingerprinting to exceed the classical capability. We prove that shared entanglement permits 100% reliable quantum fingerprinting, which will outperform classical fingerprinting even with arbitrary amounts of shared randomness.Comment: 4 pages, one figur

Classification of the Entangled states L\times N\times N

We presented a general classification scheme for the tripartite $L\times N\times N$ entangled system under stochastic local operation and classical communication. The whole classification procedure consists of two correlated parts: the simultaneous similarity transformation of a commuting matrix pair into a canonical form and the study of internal symmetry of parameters in the canonical form. Based on this scheme, a concrete example of entanglement classification for a $3\times N\times N$ system is given.Comment: 21 pages; published in Phys. Rev.