Quantum computers can search rapidly by using almost any transformation

A quantum computer has a clear advantage over a classical computer for exhaustive search. The quantum mechanical algorithm for exhaustive search was originally derived by using subtle properties of a particular quantum mechanical operation called the Walsh-Hadamard (W-H) transform. This paper shows that this algorithm can be implemented by replacing the W-H transform by almost any quantum mechanical operation. This leads to several new applications where it improves the number of steps by a square-root. It also broadens the scope for implementation since it demonstrates quantum mechanical algorithms that can readily adapt to available technology.Comment: This paper is an adapted version of quant-ph/9711043. It has been modified to make it more readable for physicists. 9 pages, postscrip

A case for post-purchase support programs as part of Minnesota's emerging markets homeownership initiative

The State of Minnesota’s Emerging Markets Homeownership Initiative (EMHI) seeks to boost homeownership rates among Minnesota’s “emerging markets,” defined as households of color, non-English speaking households, and households in which English is a second language. Many of the implementation strategies in the EMHI Business Plan address general barriers to homeownership and should increase the number of emerging market households that become first-time homeowners. EMHI doesn’t stop there, however. It also recognizes the need to sustain homeownership after initial purchase, in keeping with growing evidence that the cliché “once an owner, always an owner” is far from true, especially for minority and low-income households. In particular, the EMHI Business Plan includes a strategy for developing and implementing a post-purchase services network that will enhance their prospects for successful, sustainable homeownership. As a foundation for the implementation effort, this report explains why Minnesota is in a good position to use post-purchase support programs to pursue EMHI’s goals.

Quantum search algorithms on a regular lattice

Quantum algorithms for searching one or more marked items on a d-dimensional lattice provide an extension of Grover's search algorithm including a spatial component. We demonstrate that these lattice search algorithms can be viewed in terms of the level dynamics near an avoided crossing of a one-parameter family of quantum random walks. We give approximations for both the level-splitting at the avoided crossing and the effectively two-dimensional subspace of the full Hilbert space spanning the level crossing. This makes it possible to give the leading order behaviour for the search time and the localisation probability in the limit of large lattice size including the leading order coefficients. For d=2 and d=3, these coefficients are calculated explicitly. Closed form expressions are given for higher dimensions

Grover's quantum searching algorithm is optimal

I improve the tight bound on quantum searching by Boyer et al. (quant-ph/9605034) to a matching bound, thus showing that for any probability of success Grovers quantum searching algorithm is optimal. E.g. for near certain success we have to query the oracle pi/4 sqrt{N} times, where N is the size of the search space. I also show that unfortunately quantum searching cannot be parallelized better than by assigning different parts of the search space to independent quantum computers. Earlier results left open the possibility of a more efficient parallelization.Comment: 13 pages, LaTeX, essentially published versio

Realization of generalized quantum searching using nuclear magnetic resonance

According to the theoretical results, the quantum searching algorithm can be generalized by replacing the Walsh-Hadamard(W-H) transform by almost any quantum mechanical operation. We have implemented the generalized algorithm using nuclear magnetic resonance techniques with a solution of chloroform molecules. Experimental results show the good agreement between theory and experiment.Comment: 11 pages,3 figure. Accepted by Phys. Rev. A. Scheduled Issue: 01 Mar 200

Nested quantum search and NP-complete problems

A quantum algorithm is known that solves an unstructured search problem in a number of iterations of order $\sqrt{d}$, where $d$ is the dimension of the search space, whereas any classical algorithm necessarily scales as $O(d)$. It is shown here that an improved quantum search algorithm can be devised that exploits the structure of a tree search problem by nesting this standard search algorithm. The number of iterations required to find the solution of an average instance of a constraint satisfaction problem scales as $\sqrt{d^\alpha}$, with a constant $\alpha<1$ depending on the nesting depth and the problem considered. When applying a single nesting level to a problem with constraints of size 2 such as the graph coloring problem, this constant $\alpha$ is estimated to be around 0.62 for average instances of maximum difficulty. This corresponds to a square-root speedup over a classical nested search algorithm, of which our presented algorithm is the quantum counterpart.Comment: 18 pages RevTeX, 3 Postscript figure

Targeting foreclosure interventions: an analysis of neighborhood characteristics associated with high foreclosure rates in two Minnesota counties

This study examines the statistical association of foreclosure sales with social, economic and housing variables measured at the Census tract level for two purposes of interest to foreclosure mitigation practitioners —- to assess whether it is feasible to identify in advance neighborhoods likely to have high rates of foreclosure, and to explore the socioeconomic traits of high-foreclosure neighborhoods so as to design appropriate mitigation programs. We collected data on foreclosure sales in 2002 from the sheriff’s departments of Hennepin and Ramsey counties, the two core counties that comprise the Minneapolis-St. Paul MSA. We find that several factors commonly associated with high foreclosure sale rates could have correctly identified, in advance, most neighborhoods with high rates of mortgage foreclosure. To guide the design of foreclosure mitigation programs, we also present evidence that foreclosure risks in our two counties were highest in neighborhoods with elevated credit risk indicators and a high proportion of homeowners who are recent minority buyers or young. We show that an accurate credit risk variable is among the best predictors of foreclosure and also critically affects our multivariate analysis of factors associated with foreclosure. To limit social losses associated with foreclosures, we conclude that consideration should be given to enhancing public access to data on mortgages, foreclosures, and foreclosure risk factors, especially the neighborhood distribution of credit scores.