Optimization with constraints is a typical problem in quantum physics and quantum information science that becomes especially challenging for high-dimensional systems and complex architectures like tensor networks. Here we use ideas of Riemannian geometry to perform optimization on the manifolds of unitary and isometric matrices as well as the cone of positive-definite matrices. Combining this approach with the up-to-date computational methods of automatic differentiation, we demonstrate the efficacy of the Riemannian optimization in the study of the low-energy spectrum and eigenstates of multipartite Hamiltonians, variational search of a tensor network in the form of the multiscale entanglement-renormalization ansatz, preparation of arbitrary states (including highly entangled ones) in the circuit implementation of quantum computation, decomposition of quantum gates, and tomography of quantum states. Universality of the developed approach together with the provided open source software enable one to apply the Riemannian optimization to complex quantum architectures well beyond the listed problems, for instance, to the optimal control of noisy quantum systems.

Both COVID-19 and novel pandemics challenge those of us within the modeling community, specifically in establishing suitable relations between lifecycles, scales, and existing methods. Herein we demonstrate transitions between models in space/time, individual-to-community, county-to-city, along with models for the trace beginning with exposure, then to symptom manifest, then to community transmission. To that end, we leverage publicly available data to compose a chain of Graphical Models (GMs) for predicting infection rates across communities, space, and time. We'll anchor our GMs against the more expensive yet state-of-the-art Agent-Based Models (ABMs). Insight obtained from designing novel GMs calibrated to ABMs furnishes reduced, yet reliable surrogates for the end-to-end public health challenge of community contact tracing and transmission. Further, this novel research transcends and synergizes information integration and informatics, leading to an advance in the science of GMs. Cognizance into the data lifecycle using properly coarse-grained modeling will broaden the toolkit available to public health specialists, and hopefully empower governments and health agencies, here and abroad, in addressing the profound challenges in disease and vaccination campaigns confronting us by COVID and future pandemics. In this proof of principle study, focusing on the GM methodology development, we show, first, how static GM of the Ising model type (characterised by pair-wise interaction between nodes related to traffic and communications between nodes representing communities, or census tracts within a given city, and with local infection bias) emerge from a dynamic GM of the Independent Cascade type, introduced and studied in Computer and Networks sciences mainly in the context of the spread of social influences. Second, we formulate the problem of inference in epidemiology as inference problems in the Ising model setting. Specifically, we pose the challenge of computing Conditional A-posteriori Level of Infection (CALI), which provides a quantitative answer to the questions: What is the probability that a given node in the GM (given census tract within the city) becomes infected in the result of injection of the infection at another node, e.g. due to arrival of a super-spreader agent or occurrence of the super-spreader event in the area. To answer the question exactly is not feasible for any realistic size (larger than 30-50 nodes) model. We therefore adopt and develop approximate inference techniques, of the variational and variable elimination types, developed in the GM literature. To demonstrate utility of the methodology, which seems new for the public health application, we build a 123-node model of Seattle, as well as its 10-node and 20-node coarse-grained variants, and then conduct the proof of principles experimental studies. The experiments on the coarse-grained models have helped us to validate the approximate inference by juxtaposing it to the exact inference. The experiments also lead to discovery of interesting and most probably universal phenomena. In particular, we observe (a) a strong sensitivity of CALI to the location of the initial infection, and (b) strong alignment of the resulting infection probability (values of CALI) observed at different nodes in the regimes of moderate interaction between the nodes. We then speculate how these, and other observations drawn from the synthetic experiments, can be extended to a more realistic, data driven setting of actual operation importance. We conclude the manuscript with an extensive discussion of how the methodology should be developed further, both at the level of devising realistic GMs from observational data (and also enhancing it with microscopic ABM modeling and simulations) and also regarding utilisation of the GM inference methodology for more complex problems of the pandemic mitigation and control.

Low-rank methods for semi-definite programming (SDP) have gained a lot of interest recently, especially in machine learning applications. Their analysis often involves determinant-based or Schattennorm penalties, which are difficult to implement in practice due to high computational efforts. In this paper, we propose Entropy-Penalized Semi-Definite Programming (EP-SDP) 1 , which provides a unified framework for a broad class of penalty functions used in practice to promote a low-rank solution. We show that EP-SDP problems admit an efficient numerical algorithm, having (almost) linear time complexity of the gradient computation; this makes it useful for many machine learning and optimization problems. We illustrate the practical efficiency of our approach on several combinatorial optimization and machine learning problems.Contribution. Our contributions can be summarized as follows. We show

Hard-to-predict bursts of COVID-19 pandemic revealed significance of statistical modeling which would resolve spatio-temporal correlations over geographical areas, for example spread of the infection over a city with census tract granularity. In this manuscript, we provide algorithmic answers to the following two inter-related public health challenges of immense social impact which have not been adequately addressed (1) Inference Challenge assuming that there are N census blocks (nodes) in the city, and given an initial infection at any set of nodes, e.g. any N of possible single node infections, any $$N(N-1)/2$$ N ( N - 1 ) / 2 of possible two node infections, etc, what is the probability for a subset of census blocks to become infected by the time the spread of the infection burst is stabilized? (2) Prevention Challenge What is the minimal control action one can take to minimize the infected part of the stabilized state footprint? To answer the challenges, we build a Graphical Model of pandemic of the attractive Ising (pair-wise, binary) type, where each node represents a census tract and each edge factor represents the strength of the pairwise interaction between a pair of nodes, e.g. representing the inter-node travel, road closure and related, and each local bias/field represents the community level of immunization, acceptance of the social distance and mask wearing practice, etc. Resolving the Inference Challenge requires finding the Maximum-A-Posteriory (MAP), i.e. most probable, state of the Ising Model constrained to the set of initially infected nodes. (An infected node is in the $$+ \, 1$$ + 1 state and a node which remained safe is in the $$- \, 1$$ - 1 state.) We show that almost all attractive Ising Models on dense graphs result in either of the two possibilities (modes) for the MAP state: either all nodes which were not infected initially became infected, or all the initially uninfected nodes remain uninfected (susceptible). This bi-modal solution of the Inference Challenge allows us to re-state the Prevention Challenge as the following tractable convex programming: for the bare Ising Model with pair-wise and bias factors representing the system without prevention measures, such that the MAP state is fully infected for at least one of the initial infection patterns, find the closest, for example in $$l_1$$ l 1 , $$l_2$$ l 2 or any other convexity-preserving norm, therefore prevention-optimal, set of factors resulting in all the MAP states of the Ising model, with the optimal prevention measures applied, to become safe. We have illustrated efficiency of the scheme on a quasi-realistic model of Seattle. Our experiments have also revealed useful features, such as sparsity of the prevention solution in the case of the $$l_1$$ l 1 norm, and also somehow unexpected features, such as localization of the sparse prevention solution at pair-wise links which are NOT these which are most utilized/traveled.

We propose a method for low-rank semidefinite programming in application to the semidefinite relaxation of unconstrained binary quadratic problems. The method improves an existing solution of the semidefinite programming relaxation to achieve a lower rank solution. This procedure is computationally efficient as it does not require projecting on the cone of positive-semidefinite matrices. Its performance in terms of objective improvement and rank reduction is tested over multiple graphs of large-scale Gset graph collection and over binary optimization problems from the Biq Mac collection.