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arXiv:2012.12720 · Quantum Information Processing

Embedding of complete graphs in broken Chimera graphs

Elisabeth Lobe, Lukas Schürmann, Tobias Stollenwerk

Published Jul 1, 2021• 19 citations
arXivHTMLPDF

Abstract

Abstract In order to solve real-world combinatorial optimization problems with a D-Wave quantum annealer, it is necessary to embed the problem at hand into the D-Wave hardware graph, namely Chimera or Pegasus. Most hard real-world problems exhibit a strong connectivity. For the worst-case scenario of a complete graph, there exists an efficient solution for the embedding into the ideal Chimera graph. However, since real machines almost always have broken qubits, it is necessary to find an embedding into the broken hardware graph. We present a new approach to the problem of embedding complete graphs into broken Chimera graphs. This problem can be formulated as an optimization problem, more precisely as a matching problem with additional linear constraints. Although being NP-hard in general, it is fixed-parameter tractable in the number of inaccessible vertices in the Chimera graph. We tested our exact approach on various instances of broken hardware graphs, both related to real hardware and randomly generated. For fixed runtime, we were able to embed larger complete graphs compared to previous, heuristic approaches. As an extension, we developed a fast heuristic algorithm which enables us to solve even larger instances. We compared the performance of our heuristic and exact approaches.

DOI
10.1007/s11128-021-03168-z

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Abstract In order to solve real-world combinatorial optimization problems with a D-Wave quantum annealer, it is necessary to embed the problem at hand into the D-Wave hardware graph, namely Chimera or Pegasus. Most hard real-world problems exhibit a strong connectivity. For the worst-case scenario of a complete graph, there exists an efficient solution for the embedding into the ideal Chimera graph. However, since real machines almost always have broken qubits, it is necessary to find an embedding into the broken hardware graph. We present a new approach to the problem of embedding complete graphs into broken Chimera graphs. This problem can be formulated as an optimization problem, more precisely as a matching problem with additional linear constraints. Although being NP-hard in general, it is fixed-parameter tractable in the number of inaccessible vertices in the Chimera graph. We tested our exact approach on various instances of broken hardware graphs, both related to real hardware and randomly generated. For fixed runtime, we were able to embed larger complete graphs compared to previous, heuristic approaches. As an extension, we developed a fast heuristic algorithm which enables us to solve even larger instances. We compared the performance of our heuristic and exact approaches.
The embedding is an essential step when calculating on the D-Wave machine. In this work, we show the hardness of the embedding problem for all types of currently existing hardware, represented by the Chimera and the derived Pegasus and Zephyr graphs, containing unavailable qubits. We construct certain broken Chimera graphs, where it is hard to find a Hamiltonian cycle. As the Hamiltonian cycle problem is a special case of the embedding problem, this proves the general complexity result for the Chimera graphs. By exploiting the subgraph relation between the Chimera and the derived graphs, the proof is then further extended to the Pegasus graphs and to the Zephyr graphs.
Quantum annealing is a heuristic algorithm that solves combinatorial optimization problems, and D-Wave Systems Inc. has developed hardware implementation of this algorithm. However, in general, we cannot embed all the logical variables of a large-scale problem, since the number of available qubits is limited. In order to handle a large problem, qbsolv has been proposed as a method for partitioning the original large problem into subproblems that are embeddable in the D-Wave quantum annealer, and it then iteratively optimizes the subproblems using the quantum annealer. Multiple logical variables in the subproblem are simultaneously updated in this iterative solver, and using this approach we expect to obtain better solutions than can be obtained by conventional local search algorithms. Although embedding of large subproblems is essential for improving the accuracy of solutions in this scheme, the size of the subproblems are small in qbsolv since the subproblems are basically embedded by using an embedding of a complete graph even for sparse problem graphs. This means that the resource of the D-Wave quantum annealer is not exploited efficiently. In this paper, we propose a fast algorithm for embedding larger subproblems, and we show that better solutions are obtained efficiently by embedding larger subproblems.
Quantum annealing (QA) can be used to quickly obtain near-optimal solutions for quadratic unconstrained binary optimization (QUBO) problems. In QA hardware, each decision variable of a QUBO should be mapped to one or more adjacent qubits in such a way that pairs of variables defining a quadratic term in the objective function are mapped to some pair of adjacent qubits. However, qubits have limited connectivity in existing QA hardware. This has spurred work on preprocessing algorithms for embedding the graph representing problem variables with quadratic terms into the hardware graph representing qubits adjacencies, such as the Chimera graph in hardware produced by D-Wave Systems. In this paper, we use integer linear programming to search for an embedding of the problem graph into certain classes of minors of the Chimera graph, which we call template embeddings. One of these classes corresponds to complete bipartite graphs, for which we show the limitation of the existing approach based on minimum odd cycle transversals (OCTs). One of the formulations presented is exact and thus can be used to certify the absence of a minor embedding using that template. On an extensive test set consisting of random graphs from five different classes of varying size and sparsity, we can embed more graphs than a state-of-the-art OCT-based approach, our approach scales better with the hardware size, and the runtime is generally orders of magnitude smaller. Summary of Contribution: Our work combines classical and quantum computing for operations research by showing that integer linear programming can be successfully used as a preprocessing step for adiabatic quantum optimization. We use it to determine how a quadratic unconstrained binary optimization problem can be solved by a quantum annealer in which the qubits are coupled as in a Chimera graph, such as in the quantum annealers currently produced by D-Wave Systems. The paper also provides a timely introduction to adiabatic quantum computing and related work on minor embeddings.
We propose two new algorithms – Spring-Based MinorMiner (SPMM) and Clique-Based MinorMiner (CLMM) – which take as input the connectivity graph of a Quadratic Unconstrained Binary Optimization (QUBO) problem and produce as output an embedding of the input graph on a host graph that models the topology of a quantum computing device. As host graphs, we take the Chimera graph and the Pegasus graph, which are the topology graphs of D-Wave’s 2000 qubit (first introduced in 2017) and 5000 qubit (expected 2020) quantum annealer devices, respectively. We evaluate our algorithms on a large set of random graph QUBO inputs (Erdős-Renyi Gn,pG_{n,p}Gn,p​, Barabasi-Albert and d-regular graphs) on both host topologies against other embedding algorithms. For the Pegasus topology, we find that CLMM outperforms all other algorithms at edge densities larger than 0.08, while SPMM wins at edge densities smaller than 0.08 for Erdős-Renyi graphs, with very similar transition densities for the other graph classes. Surprisingly, the standard D-Wave MinorMiner embedding algorithm – while also getting slightly outperformed by SPMM for sparse and very dense graphs on Chimera – does not manage to extend its overall good performance on Chimera to Pegasus as it fails to embed even medium-density graphs on 175–180 nodes which are known to have clique embeddings on Pegasus.
One of the main bottlenecks in solving combinatorial optimization problems with quantum annealers is the qubit connectivity in the hardware. A possible solution for larger connectivity is minor embedding. This technique makes the geometrical properties of the combinatorial optimization problem, encoded as a Hamiltonian, match the properties of the quantum annealing hardware. The embedding itself is a hard computational problem, and therefore heuristic algorithms are required. In this work we present fixed, modular, and scalable embeddings that can be used to embed any combinatorial optimization problem described as an Ising Hamiltonian. These embeddings are the result of an extension of the well-known parity mapping, which has been used in the past to map higher-order Ising Hamiltonians to quadratic Hamiltonians, which are suitable for existing quantum hardware. We show how our new embeddings can be mapped to existing quantum annealers and that the embedded Hamiltonian physical properties match the original Hamiltonian properties.
Here we consider using quantum annealing to solve Set Cover with Pairs (SCP), an NP-hard combinatorial optimization problem that plays an important role in networking, computational biology, and biochemistry. We show an explicit construction of Ising Hamiltonians whose ground states encode the solution of SCP instances. We numerically simulate the time-dependent Schrödinger equation in order to test the performance of quantum annealing for random instances and compare with that of simulated annealing. We also discuss explicit embedding strategies for realizing our Hamiltonian construction on the D-wave type restricted Ising Hamiltonian based on Chimera graphs. Our embedding on the Chimera graph preserves the structure of the original SCP instance and in particular, the embedding for general complete bipartite graphs and logical disjunctions may be of broader use than that the specific problem we deal with.