Systematic and deterministic graph minor embedding for Cartesian products of graphs

  • Arman ZaribafiyanEmail author
  • Dominic J. J. Marchand
  • Seyed Saeed Changiz Rezaei


The limited connectivity of current and next-generation quantum annealers motivates the need for efficient graph minor embedding methods. These methods allow non-native problems to be adapted to the target annealer’s architecture. The overhead of the widely used heuristic techniques is quickly proving to be a significant bottleneck for solving real-world applications. To alleviate this difficulty, we propose a systematic and deterministic embedding method, exploiting the structures of both the specific problem and the quantum annealer. We focus on the specific case of the Cartesian product of two complete graphs, a regular structure that occurs in many problems. We decompose the embedding problem by first embedding one of the factors of the Cartesian product in a repeatable pattern. The resulting simplified problem comprises the placement and connecting together of these copies to reach a valid solution. Aside from the obvious advantage of a systematic and deterministic approach with respect to speed and efficiency, the embeddings produced are easily scaled for larger processors and show desirable properties for the number of qubits used and the chain length distribution. We conclude by briefly addressing the problem of circumventing inoperable qubits by presenting possible extensions of our method.


Graph minor embedding Cartesian product Quantum annealing Chip architecture 



The authors are grateful to Marko Bucyk for editing the manuscript, to Brad Woods, Natalie Mullin, Abbas Mehrabian, and Robyn Foerster for useful discussions and input, and to Aidan Roy for providing useful information on the treewidth of Chimera graphs. This work was supported by Mitacs (Grant No. IT03226).

Compliance with ethical standards

Conflict of interest

The authors are employees of 1QBit, a company that has quantum software as one of its areas of focus. D-Wave Systems is a minority investor in 1QBit.


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Arman Zaribafiyan
    • 1
    Email author
  • Dominic J. J. Marchand
    • 1
  • Seyed Saeed Changiz Rezaei
    • 1
  1. 1.1QB Information Technologies (1QBit)VancouverCanada

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