The Symmetric Group Defies Strong Fourier Sampling

The dramatic exponential speedups of quantum algorithms over their best existing classical counterparts were ushered in by the technique of Fourier sampling, introduced by Bernstein and Vazirani and developed by Simon and Shor into an approach to the hidden subgroup problem. This approach has proved successful for abelian groups, leading to efficient algorithms for factoring, extracting discrete logarithms, and other number-theoretic problems. We show, however, that this method cannot resolve the hidden subgroup problem in the symmetric groups, even in the weakest, information-theoretic sense. In particular, we show that the Graph Isomorphism problem cannot be solved by this approach. Our work implies that any quantum approach based upon the measurement of coset states must depart from the original framework by using entangled measurements on multiple coset states

The Power of Strong Fourier Sampling: Quantum Algorithms for Affine Groups and Hidden Shifts

Many quantum algorithms, including Shor's celebrated factoring and discrete log algorithms, proceed by reduction to a hidden subgroup problem, in which an unknown subgroup $H$ of a group $G$ must be determined from a quantum state $\psi$ over $G$ that is uniformly supported on a left coset of $H$. These hidden subgroup problems are typically solved by Fourier sampling: the quantum Fourier transform of $\psi$ is computed and measured. When the underlying group is nonabelian, two important variants of the Fourier sampling paradigm have been identified: the weak standard method, where only representation names are measured, and the strong standard method, where full measurement (i.e., the row and column of the representation, in a suitably chosen basis, as well as its name) occurs. It has remained open whether the strong standard method is indeed stronger, that is, whether there are hidden subgroups that can be reconstructed via the strong method but not by the weak, or any other known, method. In this article, we settle this question in the affirmative. We show that hidden subgroups $H$ of the $q$-hedral groups, i.e., semidirect products ${\mathbb Z}_q \ltimes {\mathbb Z}_p$, where $q \mid (p-1)$, and in particular the affine groups $A_p$, can be information-theoretically reconstructed using the strong standard method. Moreover, if $|H| = p/ {\rm polylog}(p)$, these subgroups can be fully reconstructed with a polynomial amount of quantum and classical computation. We compare our algorithms to two weaker methods that have been discussed in the literature—the “forgetful” abelian method, and measurement in a random basis—and show that both of these are weaker than the strong standard method. Thus, at least for some families of groups, it is crucial to use the full power of representation theory and nonabelian Fourier analysis, namely, to measure the high-dimensional representations in an adapted basis that respects the group's subgroup structure. We apply our algorithm for the hidden subgroup problem to new families of cryptographically motivated hidden shift problems, generalizing the work of van Dam, Hallgren, and Ip on shifts of multiplicative characters. Finally, we close by proving a simple closure property for the class of groups over which the hidden subgroup problem can be solved efficiently

Predicting Intake from Indigestible Fibre

Dry matter intake (DMI) of forages is often estimated as a reciprocal function of fibre concentration: DMI = fibre intake capacity / dietary fibre concentration (Mertens, 1987). This theoretical relationship is based on the concept that consumption of forage diets is limited by fill and that fibre represents the bulk of forage diets. This model, however, does not account for differences in DMI which should occur among forages with similar fibre concentrations but differing fibre digestibility. To account for these differences, we proposed an intake model where DMI is a reciprocal function of indigestible fibre concentration: DMI = c / CI, where c = intake capacity for indigestible fibre and CI is the concentration of indigestible fibre. This model assumes that livestock consuming forage diets of similar physical form but differing digestibility will consume a constant level of indigestible fibre. It applies only when DMI intake is regulated by fill and is not affected by digestible protein or energy concentrations. Since true DM digestibility is the inverse of indigestible fibre concentration, true digestible DM intake can be estimated as DDMI = c(1 - CI)/CI

Boundary-detection algorithm for locating edges in digital imagery

The author has identified the following significant results. Initial development of a computer program which implements a boundary detection algorithm to detect edges in digital images is described. An evaluation of the boundary detection algorithm was conducted to locate boundaries of lakes from LANDSAT-1 imagery. The accuracy of the boundary detection algorithm was determined by comparing the area within boundaries of lakes located using digitized LANDSAT imagery with the area of the same lakes planimetered from imagery collected from an aircraft platform

Soil moisture and evapotranspiration predictions using Skylab data

The author has identified the following significant results. Multispectral reflectance and emittance data from the Skylab workshop were evaluated for prediction of evapotranspiration and soil moisture for an irrigated region of southern Texas. Wavelengths greater than 2.1 microns were required to spectrally distinguish between wet and dry fallow surfaces. Thermal data provided a better estimate of soil moisture than did data from the reflective bands. Thermal data were dependent on soil moisture but not on the type of agricultural land use. The emittance map, when used in conjunction with existing models, did provide an estimate of evapotranspiration rates. Surveys of areas of high soil moisture can be accomplished with space altitude thermal data. Thermal data will provide a reliable input into irrigation scheduling