Quantum Lower Bounds by Polynomials

We examine the number of queries to input variables that a quantum algorithm requires to compute Boolean functions on {0,1} N in the black-box model. We show that the exponential quantum speed-up obtained for partial functions (i.e., problems involving a promise on the input) by Deutsch and Jozsa, Simon, and Shor cannot be obtained for any total function: if a quantum algorithm computes some total Boolean function f with small error probability using T black-box queries, then there is a classical deterministic algorithm that computes f exactly with O ( Ts 6 ) queries. We also give asymptotically tight characterizations of T for all symmetric f in the exact, zero-error, and bounded-error settings. Finally, we give new precise bounds for AND, OR, and PARITY. Our results are a quantum extension of the so-called polynomial method, which has been successfully applied in classical complexity theory, and also a quantum extension of results by Nisan about a polynomial relationship between randomized and deterministic decision tree complexity. </jats:p

Region of interest analysis: by selecting regions with denuded areas can we detect greater amounts of change?

SummaryIntroductionBased on recent analyses, the measures of short-term responsiveness of magnetic resonance imaging (MRI) derived cartilage morphometry may not be as large as earlier studies had suggested. We examined if by selecting regions of interest with denuded cartilage, the remaining cartilage within this region of interest was susceptible to greater rates of cartilage loss.MethodsSubjects included for this analysis are a subset of the approximately 4700 participants in the Osteoarthritis Initiative (OAI) Study. Bilateral radiographs and 3T MRI (Siemens Trio) of the knees and clinical data are obtained at baseline and annually in all participants. Hundred and fifty subjects from the OAI progression subcohort all of whom had both frequent symptoms and, in the same knee, radiographic osteoarthritis (ROA defined as definite tibio-femoral osteophytes on X-ray) based on a screening reading done at the OAI clinics. One knee from each subject was selected for analysis. Using sagittal 3D DESSwe MR images from the baseline and 12-month follow-up visit, a segmentation algorithm was applied to the cartilage plates of the index knee to compute the cartilage volume, normalized cartilage volume (volume normalized to bone surface interface area), and percent denuded area (Total Cartilage Bone Interface area denuded of cartilage). Summary statistics of the changes (absolute and percentage) from baseline at 1 year and the standardized response mean (SRM), i.e., mean change divided by the standard deviation (SD) of that change were calculated. Analyses are stratified into three groups according to baseline assessment of denuded area: those with no denuded area in the region of interest at baseline, and then two groups (intermediate denuded area (≤median) and severe (≥median) denuded area) of equal sample size.ResultsOn average the subjects were 60.9 years of age and obese with a mean body mass index (BMI) of 30.3kg/m2. For the combined central medial femur and tibia the mean volume change for the whole sample was −48.2 (SD 159.8) mm3, which gives an SRM of −0.30. In the subsample of knees with no denuded area the SRM was −0.25, in the knees with intermediate denuded area the SRM was −0.30, and in knees with severe denuded area the SRM was −1.00. For normalized volume of the central medial femur in the subsample of knees with no denuded area the SRM was −0.22, in the knees with intermediate denuded area the SRM was −0.26, and in knees with severe denuded area (n=23) the SRM was −0.71. The magnitude of the SRMs was generally smaller in participants with no denuded area. In contrast, the SRMs in participants with denuded area were larger.ConclusionBy selecting participants with the presence of cartilage regions with denuded area the ability to demonstrate change in cartilage loss in that specific location is markedly improved compared to persons without a full thickness lesion in that cartilage plate. This option for screening during recruitment in clinical trials could facilitate the detection of participants at greater risk of subsequent cartilage loss