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A practical heuristic for finding graph minors
TL;DR: A heuristic algorithm for finding a graph H as a minor of a graph G that is practical for sparse $G$ and $H$ with hundreds of vertices is presented.
Abstract: We present a heuristic algorithm for finding a graph $H$ as a minor of a graph $G$ that is practical for sparse $G$ and $H$ with hundreds of vertices We also explain the practical importance of finding graph minors in mapping quadratic pseudo-boolean optimization problems onto an adiabatic quantum annealer
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TL;DR: This is one of the first studies of a quantum annealer’s performance on parametrized families of hard problems from a practical domain, exploring two different general mappings of planning problems to quadratic unconstrained binary optimization (QUBO) problems, and applying them to two parametrizated families of plans.
Abstract: We report on a case study in programming an early quantum annealer to attack optimization problems related to operational planning. While a number of studies have looked at the performance of quantum annealers on problems native to their architecture, and others have examined performance of select problems stemming from an application area, ours is one of the first studies of a quantum annealer's performance on parametrized families of hard problems from a practical domain. We explore two different general mappings of planning problems to quadratic unconstrained binary optimization (QUBO) problems, and apply them to two parametrized families of planning problems, navigation-type and scheduling-type. We also examine two more compact, but problem-type specific, mappings to QUBO, one for the navigation-type planning problems and one for the scheduling-type planning problems. We study embedding properties and parameter setting and examine their effect on the efficiency with which the quantum annealer solves these problems. From these results, we derive insights useful for the programming and design of future quantum annealers: problem choice, the mapping used, the properties of the embedding, and the annealing profile all matter, each significantly affecting the performance.
197 citations
Cites methods from "A practical heuristic for finding g..."
...From a QUBO instance generated as described in Section V, we generate the embedded instance by running DWave’s heuristic embedding software [10] on the original QUBO instance....
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...For example, D-Wave recommends both minimizing the size of the embedding and the maximum component size in an embedding [10]....
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TL;DR: A quantum annealing solver for the renowned job-shop scheduling problem (JSP) is presented in detail and the results from the processor are compared against state-of-the-art global-optimum solvers.
Abstract: A quantum annealing solver for the renowned job-shop scheduling problem (JSP) is presented in detail. After formulating the problem as a time-indexed quadratic unconstrained binary optimization problem, several pre-processing and graph embedding strategies are employed to compile optimally parametrized families of the JSP for scheduling instances of up to six jobs and six machines on the D-Wave Systems Vesuvius processor. Problem simplifications and partitioning algorithms, including variable pruning and running strategies that consider tailored binary searches, are discussed and the results from the processor are compared against state-of-the-art global-optimum solvers.
164 citations
Additional excerpts
...Mathematically, finding the optimal tiling that uses the least amount of qubits is NP-hard [24], and the standard approach is to employ heuristic algorithms [25]....
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TL;DR: A combinatorial class of native clique minors in Chimera graphs with vertex images of uniform, near minimal size are defined and a polynomial-time algorithm is provided that finds a maximumnative clique minor in a given induced subgraph of a Chimera graph.
Abstract: The current generation of D-Wave quantum annealing processor is designed to minimize the energy of an Ising spin configuration whose pairwise interactions lie on the edges of a Chimera graph $${\mathcal {C}}_{M,N,L}$$CM,N,L. In order to solve an Ising spin problem with arbitrary pairwise interaction structure, the corresponding graph must be minor-embedded into a Chimera graph. We define a combinatorial class of native clique minors in Chimera graphs with vertex images of uniform, near minimal size and provide a polynomial-time algorithm that finds a maximum native clique minor in a given induced subgraph of a Chimera graph. These minors allow improvement over recent work and have immediate practical applications in the field of quantum annealing.
157 citations
Cites methods from "A practical heuristic for finding g..."
...The method is studied in greater detail elsewhere [4,13,11,9,3]....
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12 Nov 2017
TL;DR: Results for graph partitioning using quantum and hybrid classical-quantum approaches are shown to be comparable to current "state of the art" methods and sometimes better.
Abstract: Graph partitioning (GP) applications are ubiquitous throughout mathematics, computer science, chemistry, physics, bio-science, machine learning, and complex systems. Post Moore's era supercomputing has provided us an opportunity to explore new approaches for traditional graph algorithms on quantum computing architectures. In this work, we explore graph partitioning using quantum annealing on the D-Wave 2X machine. Motivated by a recently proposed graph-based electronic structure theory applied to quantum molecular dynamics (QMD) simulations, graph partitioning is used for reducing the calculation of the density matrix into smaller subsystems rendering the calculation more computationally efficient. Unconstrained graph partitioning as community clustering based on the modularity metric can be naturally mapped into the Hamiltonian of the quantum annealer. On the other hand, when constraints are imposed for partitioning into equal parts and minimizing the number of cut edges between parts, a quadratic unconstrained binary optimization (QUBO) reformulation is required. This reformulation may employ the graph complement to fit the problem in the Chimera graph of the quantum annealer. Partitioning into 2 parts and k parts concurrently for arbitrary k are demonstrated with benchmark graphs, random graphs, and small material system density matrix based graphs. Results for graph partitioning using quantum and hybrid classical-quantum approaches are shown to be comparable to current "state of the art" methods and sometimes better.
153 citations
Cites methods from "A practical heuristic for finding g..."
...Graph theory algorithms are used to determine the embedding of a problem graph on the D-Wave system [5, 9, 18]....
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TL;DR: In this article, the Sherrington-Kirkpatrick model with random $1$ couplings is programmed on the D-Wave Two annealer featuring 509 qubits interacting on a Chimera-type graph.
Abstract: The Sherrington-Kirkpatrick model with random $\pm1$ couplings is programmed on the D-Wave Two annealer featuring 509 qubits interacting on a Chimera-type graph. The performance of the optimizer compares and correlates to simulated annealing. When considering the effect of the static noise, which degrades the performance of the annealer, one can estimate an improvement on the comparative scaling of the two methods in favor of the D-Wave machine. The optimal choice of parameters of the embedding on the Chimera graph is shown to be associated to the emergence of the spin-glass critical temperature of the embedded problem.
143 citations
References
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TL;DR: How heuristic information from the problem domain can be incorporated into a formal mathematical theory of graph searching is described and an optimality property of a class of search strategies is demonstrated.
Abstract: Although the problem of determining the minimum cost path through a graph arises naturally in a number of interesting applications, there has been no underlying theory to guide the development of efficient search procedures. Moreover, there is no adequate conceptual framework within which the various ad hoc search strategies proposed to date can be compared. This paper describes how heuristic information from the problem domain can be incorporated into a formal mathematical theory of graph searching and demonstrates an optimality property of a class of search strategies.
10,366 citations
TL;DR: This programmable artificial spin network bridges the gap between the theoretical study of ideal isolated spin networks and the experimental investigation of bulk magnetic samples, and may provide a practical physical means to implement a quantum algorithm, possibly allowing more-effective approaches to solving certain classes of hard combinatorial optimization problems.
Abstract: Many interesting but practically intractable problems can be reduced to that of finding the ground state of a system of interacting spins. It is believed that the ground state of some naturally occurring spin systems can be effectively attained through a process called quantum annealing. Johnson et al. use quantum annealing to find the ground state of an artificial Ising spin system comprised of an array of eight superconducting flux qubits with programmable spin–spin couplings. With an increased number of spins, the system may provide a practical physical means to implement quantum algorithms, possibly enabling more effective approaches towards solving certain classes of hard combinatorial optimization problems. Many interesting but practically intractable problems can be reduced to that of finding the ground state of a system of interacting spins; however, finding such a ground state remains computationally difficult1. It is believed that the ground state of some naturally occurring spin systems can be effectively attained through a process called quantum annealing2,3. If it could be harnessed, quantum annealing might improve on known methods for solving certain types of problem4,5. However, physical investigation of quantum annealing has been largely confined to microscopic spins in condensed-matter systems6,7,8,9,10,11,12. Here we use quantum annealing to find the ground state of an artificial Ising spin system comprising an array of eight superconducting flux quantum bits with programmable spin–spin couplings. We observe a clear signature of quantum annealing, distinguishable from classical thermal annealing through the temperature dependence of the time at which the system dynamics freezes. Our implementation can be configured in situ to realize a wide variety of different spin networks, each of which can be monitored as it moves towards a low-energy configuration13,14. This programmable artificial spin network bridges the gap between the theoretical study of ideal isolated spin networks and the experimental investigation of bulk magnetic samples. Moreover, with an increased number of spins, such a system may provide a practical physical means to implement a quantum algorithm, possibly allowing more-effective approaches to solving certain classes of hard combinatorial optimization problems.
1,593 citations
"A practical heuristic for finding g..." refers methods in this paper
...In this paper we focus instead on heuristic techniques: we find a minor with some probability, without attempting an exhaustive search and without attempting to prove minor-exclusion in the case of failure....
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TL;DR: An algorithm, which for fixed k ≥ 0 has running time O (| V(G) | 3 ), to solve the following problem: given a graph G and k pairs of vertices of G, decide if there are k mutually vertex-disjoint paths of G joining the pairs.
1,438 citations
TL;DR: In a spin glass with Ising spins, the problems of computing the magnetic partition function and finding a ground state are studied and are shown to belong to the class of NP-hard problems, both in the two-dimensional case within a magnetic field, and in the three-dimensional cases.
Abstract: In a spin glass with Ising spins, the problems of computing the magnetic partition function and finding a ground state are studied. In a finite two-dimensional lattice these problems can be solved by algorithms that require a number of steps bounded by a polynomial function of the size of the lattice. In contrast to this fact, the same problems are shown to belong to the class of NP-hard problems, both in the two-dimensional case within a magnetic field, and in the three-dimensional case. NP-hardness of a problem suggests that it is very unlikely that a polynomial algorithm could exist to solve it.
1,205 citations
"A practical heuristic for finding g..." refers background in this paper
...Currently the best known algorithm has running time O(2(2k+1) log k|H|2k22|H|2 |H|), where k is the branchwidth of G [1]....
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TL;DR: In this article, a quantum algorithm for solving instances of the satisfiability problem, based on adiabatic evolution, is given, where the evolution of the quantum state is governed by a time-dependent Hamiltonian that interpolates between an initial Hamiltonian and a final Hamiltonian, whose ground state encodes the satisfying assignment.
Abstract: We give a quantum algorithm for solving instances of the satisfiability problem, based on adiabatic evolution. The evolution of the quantum state is governed by a time-dependent Hamiltonian that interpolates between an initial Hamiltonian, whose ground state is easy to construct, and a final Hamiltonian, whose ground state encodes the satisfying assignment. To ensure that the system evolves to the desired final ground state, the evolution time must be big enough. The time required depends on the minimum energy difference between the two lowest states of the interpolating Hamiltonian. We are unable to estimate this gap in general. We give some special symmetric cases of the satisfiability problem where the symmetry allows us to estimate the gap and we show that, in these cases, our algorithm runs in polynomial time.
996 citations
"A practical heuristic for finding g..." refers background in this paper
...As a result, none of the known exact algorithms for minor-embedding are practical for more than tens of vertices....
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