From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz

The next few years will be exciting as prototype universal quantum processors emerge, enabling implementation of a wider variety of algorithms. Of particular interest are quantum heuristics, which require experimentation on quantum hardware for their evaluation, and which have the potential to significantly expand the breadth of quantum computing applications. A leading candidate is Farhi et al.'s Quantum Approximate Optimization Algorithm, which alternates between applying a cost-function-based Hamiltonian and a mixing Hamiltonian. Here, we extend this framework to allow alternation between more general families of operators. The essence of this extension, the Quantum Alternating Operator Ansatz, is the consideration of general parametrized families of unitaries rather than only those corresponding to the time-evolution under a fixed local Hamiltonian for a time specified by the parameter. This ansatz supports the representation of a larger, and potentially more useful, set of states than the original formulation, with potential long-term impact on a broad array of application areas. For cases that call for mixing only within a desired subspace, refocusing on unitaries rather than Hamiltonians enables more efficiently implementable mixers than was possible in the original framework. Such mixers are particularly useful for optimization problems with hard constraints that must always be satisfied, defining a feasible subspace, and soft constraints whose violation we wish to minimize. More efficient implementation enables earlier experimental exploration of an alternating operator approach to a wide variety of approximate optimization, exact optimization, and sampling problems. Here, we introduce the Quantum Alternating Operator Ansatz, lay out design criteria for mixing operators, detail mappings for eight problems, and provide brief descriptions of mappings for diverse problems.Comment: 51 pages, 2 figures. Revised to match journal pape

Originality and Creativity in Copyright Law

Copyright law can be broadly viewed as a system that seeks an appropriate balance between the rights of authors and publishers with the rights of users and consumers. The case of Feist Publications Inc vs Rural Telephone Service Co is discussed

Joining Extractions of Regular Expressions

Regular expressions with capture variables, also known as "regex formulas," extract relations of spans (interval positions) from text. These relations can be further manipulated via Relational Algebra as studied in the context of document spanners, Fagin et al.'s formal framework for information extraction. We investigate the complexity of querying text by Conjunctive Queries (CQs) and Unions of CQs (UCQs) on top of regex formulas. We show that the lower bounds (NP-completeness and W[1]-hardness) from the relational world also hold in our setting; in particular, hardness hits already single-character text! Yet, the upper bounds from the relational world do not carry over. Unlike the relational world, acyclic CQs, and even gamma-acyclic CQs, are hard to compute. The source of hardness is that it may be intractable to instantiate the relation defined by a regex formula, simply because it has an exponential number of tuples. Yet, we are able to establish general upper bounds. In particular, UCQs can be evaluated with polynomial delay, provided that every CQ has a bounded number of atoms (while unions and projection can be arbitrary). Furthermore, UCQ evaluation is solvable with FPT (Fixed-Parameter Tractable) delay when the parameter is the size of the UCQ

On Training Neurons with Bounded Compilations

Knowledge compilation offers a formal approach to explaining and verifying the behavior of machine learning systems, such as neural networks. Unfortunately, compiling even an individual neuron into a tractable representation such as an Ordered Binary Decision Diagram (OBDD), is an NP-hard problem. In this thesis, we consider the problem of training a neuron from data, subject to the constraint that it has a compact representation as an OBDD. Our approach is based on the observation that a neuron can be compiled into an OBDD in polytime if (1) the neuron has integer weights, and (2) its aggregate weight is bounded. Unfortunately, we first show that it is also NP-hard to train a neuron, subject to these two constraints. On the other hand, we show that if we train a neuron generatively, rather than discriminatively, a neuron with bounded aggregate weight can be trained in pseudo-polynomial time. Hence, we propose the first efficient algorithm for training a neuron that is guaranteed to have a compact representation as an OBDD. Empirically, we show that our approach can train neurons with higher accuracy and more compact OBDDs

ExoMol: molecular line lists for exoplanet and other atmospheres

The discovery of extrasolar planets is one of the major scientific advances of the last two decades. Hundreds of planets have now been detected and astronomers are beginning to characterise their composition and physical characteristics. To do this requires a huge quantity of spectroscopic data most of which is not available from laboratory studies. The ExoMol project will offer a comprehensive solution to this problem by providing spectroscopic data on all the molecular transitions of importance in the atmospheres of exoplanets. These data will be widely applicable to other problems and will be used for studies on cool stars, brown dwarfs and circumstellar environments. This paper lays out the scientific foundations of this project and reviews previous work in this area. A mixture of first principles and empirically-tuned quantum mechanical methods will be used to compute comprehensive and very large rotation-vibration and rotation-vibration-electronic (rovibronic) line lists. Methodologies will be developed for treating larger molecules such as methane and nitric acid. ExoMol will rely on these developments and the use of state-of-the-art computing.Comment: MNRAS (in press

Creeping flow solution of the Leidenfrost phenomenon

Creeping flow solution of Leidenfrost phenomenon by use of Navier-Stokes, continuity, and energy equation

The ExoMol Atlas of Molecular Opacities

The ExoMol project is dedicated to providing molecular line lists for exoplanet and other hot atmospheres. The ExoMol procedure uses a mixture of ab initio calculations and available laboratory data. The actual line lists are generated using variational nuclear motion calculations. These line lists form the input for opacity models for cool stars and brown dwarfs as well as for radiative transport models involving exoplanets. This paper is a collection of molecular opacities for 52 molecules (130 isotopologues) at two reference temperatures, 300 K and 2000 K, using line lists from the ExoMol database. So far, ExoMol line lists have been generated for about 30 key molecular species. Other line lists are taken from external sources or from our work predating the ExoMol project. An overview of the line lists generated by ExoMol thus far is presented and used to evaluate further molecular data needs. Other line lists are also considered. The requirement for completeness within a line list is emphasized and needs for further line lists discussed

Lifted Successor Generation using Query Optimization Techniques

The standard PDDL language for classical planning uses sev eral first-order features, such as schematic actions. Yet, most classical planners ground this first-order representation into a propositional one as a preprocessing step. While this simpli fies the design of other parts of the planner, in several bench- marks the grounding process causes an exponential blowup that puts otherwise solvable tasks out of reach of the planners. In this work, we take a step towards planning with lifted representations . We tackle the successor generation task, a key operation in forward-search planning, directly on the lifted representation using well-known techniques from database theory . We show how computing the variable substitutions that make an action schema applicable in a given state is essentially a query evaluation problem. Interestingly, a large number of the action schemas in the standard benchmarks result in acyclic conjunctive queries, for which query evaluation is tractable. Our empirical results show that our approach is competitive with the standard (grounded) successor generation techniques in a few domains and outperforms them on benchmarks where grounding is challenging or infeasible