scholarly journals On the Representation of Boolean and Real Functions as Hamiltonians for Quantum Computing

2021 ◽  
Vol 2 (4) ◽  
pp. 1-21
Author(s):  
Stuart Hadfield
Keyword(s):  
Quantum Computing ◽  
Boolean Functions ◽  
Optimization Problems ◽  
Quantum Algorithms ◽  
Building Blocks ◽  
Constraint Satisfaction Problems ◽  
Boolean Formula ◽  
Combinatorial Optimization Problems ◽  
Compute Function ◽  
Real Functions

Mapping functions on bits to Hamiltonians acting on qubits has many applications in quantum computing. In particular, Hamiltonians representing Boolean functions are required for applications of quantum annealing or the quantum approximate optimization algorithm to combinatorial optimization problems. We show how such functions are naturally represented by Hamiltonians given as sums of Pauli Z operators (Ising spin operators) with the terms of the sum corresponding to the function’s Fourier expansion. For many classes of Boolean functions which are given by a compact description, such as a Boolean formula in conjunctive normal form that gives an instance of the satisfiability problem, it is #P-hard to compute its Hamiltonian representation, i.e., as hard as computing its number of satisfying assignments. On the other hand, no such difficulty exists generally for constructing Hamiltonians representing a real function such as a sum of local Boolean clauses each acting on a fixed number of bits as is common in constraint satisfaction problems. We show composition rules for explicitly constructing Hamiltonians representing a wide variety of Boolean and real functions by combining Hamiltonians representing simpler clauses as building blocks, which are particularly suitable for direct implementation as classical software. We further apply our results to the construction of controlled-unitary operators, and to the special case of operators that compute function values in an ancilla qubit register. Finally, we outline several additional applications and extensions of our results to quantum algorithms for optimization. A goal of this work is to provide a design toolkit for quantum optimization which may be utilized by experts and practitioners alike in the construction and analysis of new quantum algorithms, and at the same time to provide a unified framework for the various constructions appearing in the literature.

scholarly journals Graph Neural Networks for Maximum Constraint Satisfaction

2021 ◽  
Vol 3 ◽  
Author(s):  
Jan Tönshoff ◽  
Martin Ritzert ◽  
Hinrikus Wolf ◽  
Martin Grohe
Keyword(s):  
Constraint Satisfaction ◽  
Network Architecture ◽  
Optimization Problems ◽  
Independent Set ◽  
Constraint Satisfaction Problems ◽  
Maximum Independent Set ◽  
Combinatorial Optimization Problems ◽  
Maximum Cut ◽  
Binary Constraint ◽  
Graph Neural Networks

Many combinatorial optimization problems can be phrased in the language of constraint satisfaction problems. We introduce a graph neural network architecture for solving such optimization problems. The architecture is generic; it works for all binary constraint satisfaction problems. Training is unsupervised, and it is sufficient to train on relatively small instances; the resulting networks perform well on much larger instances (at least 10-times larger). We experimentally evaluate our approach for a variety of problems, including Maximum Cut and Maximum Independent Set. Despite being generic, we show that our approach matches or surpasses most greedy and semi-definite programming based algorithms and sometimes even outperforms state-of-the-art heuristics for the specific problems.

scholarly journals Global constraints in modeling and solving problems within the Constraint Programming paradigm.

2020 ◽  
Vol 11 (8-2020) ◽  
pp. 67-83
Author(s):  
Yu.A. Oleynik ◽  
◽  
A.A. Zuenko ◽  
Keyword(s):  
Constraint Programming ◽  
High Performance ◽  
Optimization Problems ◽  
Building Blocks ◽  
Global Constraints ◽  
Combinatorial Search ◽  
Combinatorial Optimization Problems ◽  
Programming Paradigm ◽  
Moment Constraint ◽  
The Moment

At the moment, constraint programming technology is a powerful tool for solving combinatorial search and combinatorial optimization problems. To use this technology, any task must be formulated as a task of satisfying constraints. The role of the concept of global constraints in modeling and solving applied problems within the framework of the constraint programming paradigm can hardly be overestimated. The procedures that implement the algorithms of filtering global constraints are the elementary “building blocks” from which the model of a specific applied problem is built. Algorithms for filtering global constraints, as a rule, are supported by the corresponding developed theories that allow organizing high-performance computing. The choice of a particular software library is primarily determined by the extent to which the set and method of implementing global constraints corresponds tothe level of modern research in this area. The main focus of this article is focused on an overview of global constraints that are implemented within the most popular constraint programming libraries: Choco, GeCode, JaCoP, MiniZinc.

scholarly journals Influence of Probability of Variation Operator on the Performance of Quantum-Inspired Evolutionary Algorithm for 0/1 Knapsack Problem

2010 ◽  
Vol 4 (1) ◽  
pp. 37-48
Author(s):  
Mozammel H.A. Khan
Keyword(s):  
Genetic Algorithm ◽  
Quantum Computing ◽  
Combinatorial Optimization ◽  
Evolutionary Algorithm ◽  
Knapsack Problem ◽  
Optimization Problems ◽  
Exploration And Exploitation ◽  
Good Balance ◽  
Combinatorial Optimization Problems ◽  
Variation Operator

Quantum-Inspired Evolutionary Algorithm (QEA) has been shown to be better performing than classical Genetic Algorithm based evolutionary techniques for combinatorial optimization problems like 0/1 knapsack problem. QEA uses quantum computing-inspired representation of solution called Q-bit individual consisting of Q-bits. The probability amplitudes of the Q-bits are changed by application of Q-gate operator, which is classical analogous of quantum rotation operator. The Q-gate operator is the only variation operator used in QEA, which along with some problem specific heuristic provides exploitation of the properties of the best solutions. In this paper, we analyzed the characteristics of the QEA for 0/1 knapsack problem and showed that a probability in the range 0.3 to 0.4 for the application of the Q-gate variation operator has the greatest likelihood of making a good balance between exploration and exploitation. Experimental results agree with the analytical finding.

A benchmark for quantum optimization: the traveling salesman

2021 ◽  
Vol 21 (7&8) ◽  
pp. 557-562
Author(s):  
Richard H. Warren
Keyword(s):  
Quantum Computing ◽  
Combinatorial Optimization ◽  
Optimization Problems ◽  
Traveling Salesman ◽  
Traveling Salesman Problems ◽  
Combinatorial Optimization Problems ◽  
Quantum Optimization

We present compelling reasons for symmetric traveling salesman problems (TSPs) to be the benchmark for quantum computing of combinatorial optimization problems for all types of quantum hardware. There are seven reasons for endorsing these TSPs to be the benchmark and no shortcomings.

Adiabatic Quantum Computing and Quantum Annealing

2020 ◽  
Author(s):  
Erica K. Grant ◽  
Travis S. Humble
Keyword(s):  
Quantum Computing ◽  
Quantum Computation ◽  
Large Scale ◽  
Optimization Problems ◽  
Quantum Annealing ◽  
Adiabatic Condition ◽  
Physical Systems ◽  
Adiabatic Quantum Computing ◽  
Combinatorial Optimization Problems ◽  
Computational Performance

Adiabatic quantum computing (AQC) is a model of computation that uses quantum mechanical processes operating under adiabatic conditions. As a form of universal quantum computation, AQC employs the principles of superposition, tunneling, and entanglement that manifest in quantum physical systems. The AQC model of quantum computing is distinguished by the use of dynamical evolution that is slow with respect to the time and energy scales of the underlying physical systems. This adiabatic condition enforces the promise that the quantum computational state will remain well-defined and controllable thus enabling the development of new algorithmic approaches. Several notable algorithms developed within the AQC model include methods for solving unstructured search and combinatorial optimization problems. In an idealized setting, the asymptotic complexity analyses of these algorithms indicate computational speed-ups may be possible relative to state-of-the-art conventional methods. However, the presence of non-ideal conditions, including non-adiabatic dynamics, residual thermal excitations, and physical noise complicate the assessment of the potential computational performance. A relaxation of the adiabatic condition is captured in the complementary computational heuristic of quantum annealing, which accommodates physical systems operating at finite temperature and in open environments. While quantum annealing (QA) provides a more accurate model for the behavior of actual quantum physical systems, the possibility of non-adiabatic effects obscures a clear separation with conventional computing complexity. A series of technological advances in the control of quantum physical systems have enabled experimental AQC and QA. Prominent examples include demonstrations using superconducting electronics, which encode quantum information in the magnetic flux induced by a weak current operating at cryogenic temperatures. A family of devices developed specifically for unconstrained optimization problems has been applied to solve problems in specific domains including logistics, finance, material science, machine learning, and numerical analysis. An accompanying infrastructure has also developed to support these experimental demonstrations and to enable access of a broader community of users. Although AQC is most commonly applied in superconducting technologies, alternative approaches include optically trapped neutral atoms and ion-trap systems. The significant progress in the understanding of AQC has revealed several open topics that continue to motivate research into this model of quantum computation. Foremost is the development of methods for fault-tolerant operation that will ensure the scalability of AQC for solving large-scale problems. In addition, unequivocal experimental demonstrations that differentiate the computational power of AQC and its variants from conventional computing approaches are needed. This will also require advances in the fabrication and control of quantum physical systems under the adiabatic restrictions.

scholarly journals Quantum-accelerated constraint programming

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 550
Author(s):  
Kyle E. C. Booth ◽  
Bryan O'Gorman ◽  
Jeffrey Marshall ◽  
Stuart Hadfield ◽  
Eleanor Rieffel
Keyword(s):  
Constraint Programming ◽  
Optimization Problems ◽  
Fault Tolerant ◽  
Quantum Algorithms ◽  
Global Constraints ◽  
Quantum Computers ◽  
Promising Candidate ◽  
Backtracking Search ◽  
Combinatorial Optimization Problems ◽  
Classical Quantum

Constraint programming (CP) is a paradigm used to model and solve constraint satisfaction and combinatorial optimization problems. In CP, problems are modeled with constraints that describe acceptable solutions and solved with backtracking tree search augmented with logical inference. In this paper, we show how quantum algorithms can accelerate CP, at both the levels of inference and search. Leveraging existing quantum algorithms, we introduce a quantum-accelerated filtering algorithm for the alldifferent global constraint and discuss its applicability to a broader family of global constraints with similar structure. We propose frameworks for the integration of quantum filtering algorithms within both classical and quantum backtracking search schemes, including a novel hybrid classical-quantum backtracking search method. This work suggests that CP is a promising candidate application for early fault-tolerant quantum computers and beyond.

scholarly journals Quantum SDP-Solvers: Better upper and lower bounds

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 230 ◽  
Author(s):  
Joran van Apeldoorn ◽  
András Gilyén ◽  
Sander Gribling ◽  
Ronald de Wolf
Keyword(s):  
Combinatorial Optimization ◽  
Lower Bounds ◽  
Optimization Problems ◽  
Quantum Algorithms ◽  
Upper And Lower Bounds ◽  
Smooth Functions ◽  
Worst Case ◽  
Semidefinite Programs ◽  
New Techniques ◽  
Combinatorial Optimization Problems

Brandão and Svore \cite{brandao2016QSDPSpeedup} recently gave quantum algorithms for approximately solving semidefinite programs, which in some regimes are faster than the best-possible classical algorithms in terms of the dimension n of the problem and the number m of constraints, but worse in terms of various other parameters. In this paper we improve their algorithms in several ways, getting better dependence on those other parameters. To this end we develop new techniques for quantum algorithms, for instance a general way to efficiently implement smooth functions of sparse Hamiltonians, and a generalized minimum-finding procedure.We also show limits on this approach to quantum SDP-solvers, for instance for combinatorial optimization problems that have a lot of symmetry. Finally, we prove some general lower bounds showing that in the worst case, the complexity of every quantum LP-solver (and hence also SDP-solver) has to scale linearly with mn when m≈n, which is the same as classical.

scholarly journals Learning for Graph Matching and Related Combinatorial Optimization Problems

2020 ◽  
Author(s):  
Junchi Yan ◽  
Shuang Yang ◽  
Edwin Hancock
Keyword(s):  
Machine Learning ◽  
Neural Networks ◽  
Combinatorial Optimization ◽  
Prediction Models ◽  
Graph Matching ◽  
Optimization Problems ◽  
Weighted Graph ◽  
Building Blocks ◽  
Transformative Change ◽  
Combinatorial Optimization Problems

This survey gives a selective review of recent development of machine learning (ML) for combinatorial optimization (CO), especially for graph matching. The synergy of these two well-developed areas (ML and CO) can potentially give transformative change to artificial intelligence, whose foundation relates to these two building blocks. For its representativeness and wide-applicability, this paper is more focused on the problem of weighted graph matching, especially from the learning perspective. For graph matching, we show that many learning techniques e.g. convolutional neural networks, graph neural networks, reinforcement learning can be effectively incorporated in the paradigm for extracting the node features, graph structure features, and even the matching engine. We further present outlook for the new settings for learning graph matching, and direction towards more integrated combinatorial optimization solvers with prediction models, and also the mutual embrace of traditional solver and machine learning components.

scholarly journals Hysteresis in Combinatorial Optimization Problems

2021 ◽  
Vol 34 (1) ◽  
Author(s):  
Yuling Guan ◽  
Ang Li ◽  
Sven Koenig ◽  
Stephan Haas ◽  
T. K. Satish Kumar
Keyword(s):  
Combinatorial Optimization ◽  
Statistical Mechanics ◽  
Optimization Problems ◽  
Physical Phenomenon ◽  
Magnetic Hysteresis ◽  
Constraint Satisfaction Problems ◽  
Combinatorial Optimization Problems ◽  
Ferromagnetic Metals ◽  
Spin Interactions ◽  
Iron Nickel

Hysteresis is a physical phenomenon reflected in macroscopic observables of materials that are subjected to external perturbations. For example, magnetic hysteresis is observed in ferromagnetic metals such as iron, nickel and cobalt in the presence of a changing external magnetic field. In this paper, we model hysteresis using combinatorial models of microscopic spin interactions, for which we invoke the top K solution framework for Ising models and their generalizations, called Weighted Constraint Satisfaction Problems (WCSPs). We show that the WCSP model with a simple "memory effect" can be used to understand hysteresis combinatorially and from the perspective of statistical mechanics. Compared to the basic Ising model, the WCSP framework allows accurate simulations of long-range and k-body interactions between the spins; and compared to other simulation frameworks, such as Monte Carlo methods, our WCSP framework has the advantage of using a principled statistical mechanics perspective. Our WCSP framework also allows us to understand hysteresis more generally in combinatorial optimization problems, with or without a connection to physically occurring phenomena.

Computation

2021 ◽  
pp. 168-222
Author(s):  
Jochen Rau
Keyword(s):  
Quantum Computing ◽  
Optimization Problems ◽  
Quantum Computer ◽  
Classical Case ◽  
Building Blocks ◽  
Quantum Circuit ◽  
Carry Over ◽  
Classical Computing ◽  
Classical Computer ◽  
Classical Optimization

This chapter introduces the basic building blocks of quantum computing and a variety of specific algorithms. It begins with a brief review of classical computing and discusses how its key elements – bits, gates, circuits – carry over to the quantum realm. It highlights crucial differences to the classical case, such as the impossibility of copying a qubit. The quantum circuit model is shown to be universal, and a peculiar variant of quantum computing, based on measurements only, is illustrated. That a quantum computer can perform some calculations more efficiently than a classical computer, at least in principle, is exemplified with the Deutsch-Jozsa algorithm. Other examples covered in this chapter are the variational quantum eigensolver, which can be applied to the study of molecules and classical optimization problems; quantum simulation; and entanglement-assisted metrology.

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