Avoiding local minima in variational quantum eigensolvers with the natural gradient optimizer

We compare the BFGS optimizer, ADAM and Natural Gradient Descent (NatGrad) in the context of Variational Quantum Eigensolvers (VQEs). We systematically analyze their performance on the QAOA ansatz for the Transverse Field Ising Model (TFIM) as well as on overparametrized circuits with the ability to break the symmetry of the Hamiltonian. The BFGS algorithm is frequently unable to find a global minimum for systems beyond about 20 spins and ADAM easily gets trapped in local minima. On the other hand, NatGrad shows stable performance on all considered system sizes, albeit at a significantly higher cost per epoch. In sharp contrast to most classical gradient based learning, the performance of all optimizers is found to decrease upon seemingly benign overparametrization of the ansatz class, with BFGS and ADAM failing more often and more severely than NatGrad. Additional tests for the Heisenberg XXZ model corroborate the accuracy problems of BFGS in high dimensions, but they reveal some shortcomings of NatGrad as well. Our results suggest that great care needs to be taken in the choice of gradient based optimizers and the parametrization for VQEs.Comment: 16 pages, 6 figures, 1 tabl

Backpropagation scaling in parameterised quantum circuits

The discovery of the backpropagation algorithm ranks among one of the most important moments in the history of machine learning, and has made possible the training of large-scale neural networks through its ability to compute gradients at roughly the same computational cost as model evaluation. Despite its importance, a similar backpropagation-like scaling for gradient evaluation of parameterised quantum circuits has remained elusive. Currently, the best known method requires sampling from a number of circuits that scales with the number of circuit parameters, making training of large-scale quantum circuits prohibitively expensive in practice. Here we address this problem by introducing a class of structured circuits that admit gradient estimation with significantly fewer circuits. In the simplest case -- for which the parameters feed into commuting quantum gates -- these circuits allow for fast estimation of the gradient, higher order partial derivatives and the Fisher information matrix, and are not expected to suffer from the problem of vanishing gradients. Moreover, classes of parameterised circuits exist for which the scaling of gradient estimation is in line with classical backpropagation, and can thus be trained at scale. In a toy classification problem on 16 qubits, such circuits show competitive performance with other methods, while reducing the training cost by about two orders of magnitude.Comment: Comments welcom

Longevity of immediate rehabilitation with direct metal-wire reinforced composite fixed partial dentures.

OBJECTIVES This study aimed to analyze the longevity of direct metal-wire reinforced composite fixed partial dentures (MRC-FPD) and factors influencing their survival and success. METHODS Within one private practice 513 MRC-FPD were directly applied. The preparation of a proximal cavity in abutment teeth was not limited. MRC-FPD were reinforced by one to three metal-wires. At the last follow-up MRC-FPD were considered successful, if they were still in function without any need of therapy. MRC-FPD were considered as survived, if they were repaired or replaced. Multi-level Cox proportional hazard models were used to evaluate the association between clinical factors and time. RESULTS Mean follow-up period (range) was 59(2-249) months. Seventy-three bridges did not survive (cumulative survival rate(CSR):86%) and further 129 bridges had received a restorative follow-up treatment (CSR:61%). AFR was 2.2% for survival and 8.6% for success. In multivariate analysis MRC-FPD with> 1 wire showed a up to 2.3x higher failure rate than MRC-FPD with one wire(p ≤ 0.023). Dentist's experience in designing MRC-FDP (p ≤ 0.017), patient's caries risk (p ≤ 0.040) and bruxism (p = 0.033) significantly influenced the failure rate: the more experience, the lower caries risk and bruxism, the lower the failure rate. SIGNIFICANCE For directly prepared metal-wire reinforced composite bridges high survival and moderate success rates were observed. MRC-FPD might, thus, be an immediate, short- and medium-term solution for replacing missing teeth. However, several factors on the levels of practice (dentist's experience in designing MRC-FDP), patient (bruxism, caries risk) and restoration (number of wires) were identified as significant predictors for the failure rate. The study was registered in the German Clinical Trials Register (DRKS-ID: DRKS00021576)

Root canal morphology of the mandibular second premolar: a systematic review and meta- analysis

Background: The aim of this paper was to systematically review the root canal configuration (RCC) and morphology literature of the mandibular second premolar (Mn2P). Methods: Systematic research of five electronic databases was performed to identify published literature concerning the root canal configuration (RCC) of the Mn2P up through July 2020. Studies were selected according to predefined search terms and keywords inclusion criteria: “root canal configuration”, “root canal system”, “root canal morphology”, “mandibular second premolar”, “mandibular premolars”, “morphology” and “anatomy”. Further possible studies were identified by cross-referencing and screening the bibliographies of the selected articles. Results: From 1622 retrieved studies, 44 studies investigating the internal morphology of 17,839 Mn2Ps were included. Most examined Mn2Ps were single-rooted (89.5–100%); two-rooted (0.1–8%) and three-rooted (0.1–3.5%) Mn2Ps at lower frequency. Most frequent RCCs reported were 1–1–1/1 (55.3–99.6%) followed by 1–1–2/2 (0.5–57%) and 2–2–2/2 (0.6–18%). The meta-analysis of seven studies demonstrated that a significantly higher number of RCC type 1–2–1/1 (OR [95%CI] = 2.05 [1.27, 3.33]) and 2–2–2/2 (OR [95%CI] = 2.32 [0.65, 8.63]) were observed in male than in female patients. Conclusions: Different RCC research methods have been reported. Whereas clearing and radiographs were com- monly used in the past, CBCT has been prevalent in recent years. A globally high frequency of a 1–1–1/1 RCC in the Mn2P has been reported. Nevertheless, the probability that different, more complicated RCCs can appear in Mn2Ps should not be underestimated and, thus, should be taken into consideration when making decisions during an endo- dontic treatment

Masking-efficacy and caries arrestment after resin infiltration or fluoridation of initial caries lesions in adolescents during orthodontic treatment - a randomised controlled trial.

OBJECTIVES The aim of this randomised, controlled, split-mouth trial was to assess the masking results in initial caries lesions (ICL) that were either resin infiltrated or fluoridated during treatment with fixed orthodontic appliances. METHODS Adolescent patients (age range:12-18years) with fixed orthodontic appliances who had developed ICL [ICDAS 1 or 2 (International Caries Detection and Assessment System)] during orthodontic treatment were consecutively recruited and randomly assigned to either resin infiltration with up to 3 etching procedures (Group:Inf) or to 3-monthly application of a fluoride varnish (Group:FV). Both interventions were performed according to the manufacturer's recommendations. Primary and secondary outcomes (ΔE, ICDAS, DIAGNOdent) included the evaluation of the appearance of the ICL before (T0), 1 week after (T1) treatment and at the last appointment before debonding (T2). RESULTS Fifteen patients (8females, 7males) with 57ICL were included. Mean (SD) observation time at the last appointment before debonding was 0.5 (0.3) years. At T0 FV and Inf did not differ significantly in ΔE (median ΔE0,FV(25th/75th percentiles):11.6 (8.7/20.3): ΔE0,Inf:15.1 (11.4/19.5); pT0=0.135), ICDAS (pT0=0.920) and DD (pT0=0.367). At T1 and T2 ΔE values (pT1<0.001,pT2<0.001), ICDAS scores (pT1<0.001,pT2<0.001) and DIAGNOdent values (pT1=<0.001,pT2=<0.001) for Inf were significantly reduced whereas ΔE values (pT1=0.382,pT2=0.072) and ICDAS scores (pT1=0.268,pT2<0.001) for FV remained unchanged. CONCLUSIONS Resin infiltration effectively masked ICL during treatment with fixed orthodontic appliances both immediately after application and at the last appointment before debonding. Furthermore, the visual appearance of fluoridated lesions was not as satisfactory as that of the infiltrated ones at both T1 and T2. CLINICAL SIGNIFICANCE Resin infiltration effectively masked ICL during treatment with fixed orthodontic appliances both immediately after application and at the last appointment before debonding. Furthermore, the visual appearance of fluoridated lesions was not as satisfactory as that of the infiltrated ones immediately after first application as well as half a year after application. TRIAL REGISTRATION German Clinical Trials Register (DRKS-ID: DRKS00011797)

Training quantum embedding kernels on near-term quantum computers

Kernel methods are a cornerstone of classical machine learning. The idea of using quantum computers to compute kernels has recently attracted attention. Quantum embedding kernels (QEKs), constructed by embedding data into the Hilbert space of a quantum computer, are a particular quantum kernel technique that is particularly suitable for noisy intermediate-scale quantum devices. Unfortunately, kernel methods face three major problems: Constructing the kernel matrix has quadratic computational complexity in the number of training samples, choosing the right kernel function is nontrivial, and the effects of noise are unknown. In this work, we addressed the latter two. In particular, we introduced the notion of trainable QEKs, based on the idea of classical model optimization methods. To train the parameters of the QEK, we proposed the use of kernel-target alignment. We verified the feasibility of this method, and showed that for our experimental setup we could reduce the training error significantly. Furthermore, we investigated the effects of device and finite sampling noise, and we evaluated various mitigation techniques numerically on classical hardware. We took the best performing strategy and evaluated it on data from a real quantum processing unit. We found that using this mitigation strategy demonstrated an increased kernel matrix quality

Evaluation of the masking efficacy of caries infiltration in post-orthodontic initial caries lesions: 1-year follow-up.

OBJECTIVES This study aimed to qualitatively and quantitatively assess the masking efficacy and color stability of resin infiltration on post-orthodontic ICL after 1 year. MATERIALS AND METHODS In 17 adolescents, 112 ICL (ICDAS-1: n = 1; ICDAS-2: n = 111) in 112 teeth were treated by resin infiltration (Icon, DMG) 3 to 12 months after bracket removal. The etching procedure was performed up to 3 times. Standardized digital images were taken before treatment (T0), 7 days (T7) and 12 months (T365) after treatment. Outcomes included the evaluation of the color differences between infiltrated and healthy enamel at T0, T7, and T365 by quantitative (colorimetric analysis (ΔE), ICDAS scores) and qualitative methods (5-point Likert scale (deteriorated (1), unchanged (2), improved, but not satisfying (3), improved and no further treatment required (4), completely masked (5)).) Differences between time points were analyzed by using Friedman test (ΔΕ) and chi-square tests (ICDAS). RESULTS The median color difference (25th/75th percentiles) between carious and healthy enamel at baseline (ΔΕ0) was 10.2(7.7/13.6). A significant decrease was observed 7 days after treatment (ΔΕ7 = 3.1(1.8/5.0); p < 0.001; ICDAS; p < 0.001). No significant changes based on ΔΕ (p = 1.000), and ICDAS grade (p = 0.305) were observed between T7 and T365 (ΔΕ12 = 3.4 (1.8/4.9)). Furthermore, at T365 four experienced dentists classified 55% and 39% of the lesions as "improved and no further treatment required" and "completely masked," respectively (Fleiss kappa: T365 = 0.851 (almost perfect)). CONCLUSION Resin infiltration efficaciously masked post-orthodontic ICL 7 days and 12 months after treatment. These results for most of the teeth could not only be observed by quantitative but also by qualitative analysis. CLINICAL RELEVANCE Resin infiltration efficaciously masks post-orthodontic initial carious lesions. The optical improvement can be observed directly after treatment and remains stable for at least 12 months