Some Spectral and Quasi-Spectral Characterizations of Distance-Regular Graphs

In this paper we consider the concept of preintersection numbers of a graph. These numbers are determined by the spectrum of the adjacency matrix of the graph, and generalize the intersection numbers of a distance-regular graph. By using the preintersection numbers we give some new spectral and quasi-spectral characterizations of distance-regularity, in particular for graphs with large girth or large odd-girth

Modeling low order aberrations in laser guide star adaptive optics systems

When using a laser guide star (LGS) adaptive optics (AO) system, quasi-static aberrations are observed between the measured wavefronts from the LGS wavefront sensor (WFS) and the natural guide star (NGS) WFS. These LGS aberrations, which can be as much as 1200 nm RMS on the Keck II LGS AO system, arise due to the finite height and structure of the sodium layer. The LGS aberrations vary significantly between nights due to the difference in sodium structure. In this paper, we successfully model these LGS aberrations for the Keck II LGS AO system. We use this model to characterize the LGS aberrations as a function of pupil angle, elevation, sodium structure, uplink tip/tilt error, detector field of view, the number of detector pixels, and seeing. We also employ the model to estimate the LGS aberrations for the Palomar LGS AO system, the planned Keck I and the Thirty Meter Telescope (TMT) LGS AO systems. The LGS aberrations increase with increasing telescope diameter, but are reduced by central projection of the laser compared to side projection

Hamiltonian Oracles

Hamiltonian oracles are the continuum limit of the standard unitary quantum oracles. In this limit, the problem of finding the optimal query algorithm can be mapped into the problem of finding shortest paths on a manifold. The study of these shortest paths leads to lower bounds of the original unitary oracle problem. A number of example Hamiltonian oracles are studied in this paper, including oracle interrogation and the problem of computing the XOR of the hidden bits. Both of these problems are related to the study of geodesics on spheres with non-round metrics. For the case of two hidden bits a complete description of the geodesics is given. For n hidden bits a simple lower bound is proven that shows the problems require a query time proportional to n, even in the continuum limit. Finally, the problem of continuous Grover search is reexamined leading to a modest improvement to the protocol of Farhi and Gutmann.Comment: 16 pages, REVTeX 4 (minor corrections in v2

Complexes of mercury(II) and zinc(II) with primary aromatic amines

A series of amine complexes has been prepared by reaction of zinc chloride and mercuric chloride with primary aromatic amines. A detailed assignment of the bands in the infra-red spectra of the complexes in the range 4000-625 cm [superscript]-1 is presented. The symmetric and asymmetric N-H stretching frequencies follow the relationship v(sym) = 345.5+ 0.876v(asym). The C-N stretching frequencies exhibit a linear relationship with the Hammett α-functions for the m- and p-substituted amines

Adiabatic Quantum Computation in Open Systems

We analyze the performance of adiabatic quantum computation (AQC) under the effect of decoherence. To this end, we introduce an inherently open-systems approach, based on a recent generalization of the adiabatic approximation. In contrast to closed systems, we show that a system may initially be in an adiabatic regime, but then undergo a transition to a regime where adiabaticity breaks down. As a consequence, the success of AQC depends sensitively on the competition between various pertinent rates, giving rise to optimality criteria.Comment: v2: 4 pages, 1 figure. Published versio

Faster Base64 Encoding and Decoding Using AVX2 Instructions

Web developers use base64 formats to include images, fonts, sounds and other resources directly inside HTML, JavaScript, JSON and XML files. We estimate that billions of base64 messages are decoded every day. We are motivated to improve the efficiency of base64 encoding and decoding. Compared to state-of-the-art implementations, we multiply the speeds of both the encoding (~10x) and the decoding (~7x). We achieve these good results by using the single-instruction-multiple-data (SIMD) instructions available on recent Intel processors (AVX2). Our accelerated software abides by the specification and reports errors when encountering characters outside of the base64 set. It is available online as free software under a liberal license.Comment: software at https://github.com/lemire/fastbase6

On SIC-POVMs in Prime Dimensions

The generalized Pauli group and its normalizer, the Clifford group, have a rich mathematical structure which is relevant to the problem of constructing symmetric informationally complete POVMs (SIC-POVMs). To date, almost every known SIC-POVM fiducial vector is an eigenstate of a "canonical" unitary in the Clifford group. I show that every canonical unitary in prime dimensions p > 3 lies in the same conjugacy class of the Clifford group and give a class representative for all such dimensions. It follows that if even one such SIC-POVM fiducial vector is an eigenvector of such a unitary, then all of them are (for a given such dimension). I also conjecture that in all dimensions d, the number of conjugacy classes is bounded above by 3 and depends only on d mod 9, and I support this claim with computer computations in all dimensions < 48.Comment: 6 pages, no figures. v3 Refs added, improved discussion of previous work. Ref to a proof of the main conjecture also adde

Eigenlevel statistics of the quantum adiabatic algorithm

We study the eigenlevel spectrum of quantum adiabatic algorithm for 3-satisfiability problem, focusing on single-solution instances. The properties of the ground state and the associated gap, crucial for determining the running time of the algorithm, are found to be far from the predictions of random matrix theory. The distribution of gaps between the ground and the first excited state shows an abundance of small gaps. Eigenstates from the central part of the spectrum are, on the other hand, well described by random matrix theory.Comment: 8 pages, 10 ps figure

Bipartite entangled stabilizer mutually unbiased bases as maximum cliques of Cayley graphs

We examine the existence and structure of particular sets of mutually unbiased bases (MUBs) in bipartite qudit systems. In contrast to well-known power-of-prime MUB constructions, we restrict ourselves to using maximally entangled stabilizer states as MUB vectors. Consequently, these bipartite entangled stabilizer MUBs (BES MUBs) provide no local information, but are sufficient and minimal for decomposing a wide variety of interesting operators including (mixtures of) Jamiolkowski states, entanglement witnesses and more. The problem of finding such BES MUBs can be mapped, in a natural way, to that of finding maximum cliques in a family of Cayley graphs. Some relationships with known power-of-prime MUB constructions are discussed, and observables for BES MUBs are given explicitly in terms of Pauli operators.Comment: 8 pages, 1 figur