A Lower Bound for the Sturm-Liouville Eigenvalue Problem on a Quantum Computer

We study the complexity of approximating the smallest eigenvalue of a univariate Sturm-Liouville problem on a quantum computer. This general problem includes the special case of solving a one-dimensional Schroedinger equation with a given potential for the ground state energy. The Sturm-Liouville problem depends on a function q, which, in the case of the Schroedinger equation, can be identified with the potential function V. Recently Papageorgiou and Wozniakowski proved that quantum computers achieve an exponential reduction in the number of queries over the number needed in the classical worst-case and randomized settings for smooth functions q. Their method uses the (discretized) unitary propagator and arbitrary powers of it as a query ("power queries"). They showed that the Sturm-Liouville equation can be solved with O(log(1/e)) power queries, while the number of queries in the worst-case and randomized settings on a classical computer is polynomial in 1/e. This proves that a quantum computer with power queries achieves an exponential reduction in the number of queries compared to a classical computer. In this paper we show that the number of queries in Papageorgiou's and Wozniakowski's algorithm is asymptotically optimal. In particular we prove a matching lower bound of log(1/e) power queries, therefore showing that log(1/e) power queries are sufficient and necessary. Our proof is based on a frequency analysis technique, which examines the probability distribution of the final state of a quantum algorithm and the dependence of its Fourier transform on the input.Comment: 23 pages, 2 figures; Major changes in Theorem 3 to previous version. To be published in the Journal of Complexit

Approximation of Various Quantum Query Types

Query complexity measures the amount of information an algorithm needs about a problem to compute a solution. On a quantum computer there are different realizations of a query and we will show that these are not always equivalent. Our definition of equivalence is based on the ability to simulate (or approximate) one query type by another. We show that a bit query can always approximate a phase query with just two queries, while there exist problems for which the number of phase queries which are necessary to approximate a bit query must grow exponentially with the precision of the bit query. This result follows from the query complexity bounds for the evaluation problem, for which we establish a strong lower bound for the number of phase queries by exploiting a relation between quantum algorithms and trigonometric polynomials.Comment: New version. To be published in the Journal of Complexity in August. Extended by an additional sectio

Epidemiology of Streptococcus dysgalactiae subsp. equisimilis in Tropical Communities, Northern Australia

This subspecies is common in communities with high rates of streptococcal disease, and its epidemiology differs from that of Streptococcus pyogenes

The Patent Litigation Explosion

This Article provides the first look at patent litigation hazards for public firms during the 1980s and 1990s. Litigation is more likely when prospective plaintiffs acquire more patents, when firms are larger and technologically close and when prospective defendants spend more on research and development ( R&D ). The latter suggests inadvertent infringement may be more important than piracy. Public firms face dramatically increased hazards of litigation as plaintiffs and even more rapidly increasing hazards as defendants, especially for small public firms. The increase cannot be explained by patenting rates, R&D, firm value or industry composition. Legal changes are the most likely explanatio

What\u27s Wrong with the Patent System? Fuzzy Boundaries and the Patent Tax

The annual number of patent lawsuits filed in the U.S. has roughly tripled from 1970 to 2004. The number of suits was more or less steady in the 1970s, climbed slowly in the 1980s, and exploded in the 1990s. Why? The usual answers point to (1) the growth of the “new economy” and the concomitant explosion of patenting, (2) the failure of the Patent Office to reject patents on old or obvious inventions, or (3) the rise of the patent troll. There is an element of truth in all these answers, but even collectively they do a poor job explaining the patent litigation explosion. The comprehensive empirical research, presented in our forthcoming book [1], identifies fuzzy boundaries of the patent property right as likely the main cause of the explosion. Our research also shows that as the problem of fuzzy boundaries has grown worse, the patent system has turned from a source of net subsidy to R&D to a net tax. Patents now discourage investment in innovation. The burden of litigation and the harm caused by fuzzy boundaries falls unevenly across fields of technology. Technologies that rely heavily on software are vexed by elevated patent litigation costs. This bodes poorly for cyberinfrastructure which will depend heavily on software innovation, and will probably attract significant unwanted attention from patent owners. We provide evidence below that software patents have more severe boundary problems and generate greater litigation costs than most other patents. Software patents tend to perform badly because the associated property rights are often expressed quite abstractly. The problem of mapping words to technology is difficult for any kind of technology, but it is especially difficult for software inventions because of the abstract nature of the technology. The problem has been made worse because when the courts have considered software inventions they have relaxed patent law doctrines that work to limit abstraction in other areas of technology. As a result, patent–based property rights to software inventions are not tethered to a specific device or to a specific physical or chemical process. Ironically, verbal descriptions corresponding to precise mathematical representations may be ambiguous; this is because of the inherent abstraction of the mathematical representations

Incremental Contributions of FbaA and Other Impetigo-Associated Surface Proteins to Fitness and Virulence of a Classical Group A Streptococcal Skin Strain

Group A streptococci (GAS) are highly prevalent human pathogens whose primary ecological niche is the superficial epithelial layers of the throat and/or skin. Many GAS strains having a strong tendency to cause pharyngitis are distinct from strains that tend to cause impetigo; thus, genetic differences between them may confer host tissue-specific virulence. In this study, the FbaA surface protein gene is found to be present in most skin specialist strains, but largely absent from a genetically-related subset of pharyngitis isolates. Using an DeltafbaA mutant constructed in the impetigo strain Alab49, loss of FbaA resulted in a slight but significant decrease in GAS fitness in a humanized mouse model for impetigo; the DeltafbaA mutant also exhibited decreased survival in whole human blood due to phagocytosis. Using assays with highly sensitive outcome measures, Alab49DeltafbaA was compared to other isogenic mutants lacking virulence genes known to be disproportionately associated with classical skin strains. FbaA and PAM (i.e., M53 protein) have additive effects in promoting GAS survival in whole blood. The pilus adhesin tip protein Cpa promotes Alab49 survival in whole blood, and appears to fully account for the antiphagocytic effect attributable to pili. That numerous skin strain-associated virulence factors make slight but significant contributions to virulence underscores the incremental contributions to fitness of individual surface protein genes and the multifactorial nature of GAS-host interactions

Differences in SpeB Protease Activity Among Group A Streptococci Associated With Superficial, Invasive, and Autoimmune Disease

The secreted cysteine proteinase SpeB is an important virulence factor of group A streptococci (GAS), whereby SpeB activity varies widely among strains. To establish the degree to which SpeB activity correlates with disease, GAS organisms were recovered from patients with pharyngitis, impetigo, invasive disease or acute rheumatic fever (ARF), and selected for analysis using rigorous sampling criteria; \u3e300 GAS isolates were tested for SpeB activity by casein digestion assays, and each GAS isolate was scored as a SpeB-producer or non-producer. Highly significant statistical differences (p \u3c 0.01) in SpeB production are observed between GAS recovered from patients with ARF (41.5% SpeB-non-producers) compared to pharyngitis (20.5%), invasive disease (16.7%), and impetigo (5.5%). SpeB activity differences between pharyngitis and impetigo isolates are also significant, whereas pharyngitis versus invasive isolates show no significant difference. The disproportionately greater number of SpeB-non-producers among ARF-associated isolates may indicate an altered transcriptional program for many rheumatogenic strains and/or a protective role for SpeB in GAS-triggered autoimmunity