Minor Embedding in Broken Chimera and Pegasus Graphs is NP-complete

TL;DR

This paper proves that the problem of embedding problems in broken Chimera and Pegasus graphs, used in D-Wave quantum computers, is NP-complete, highlighting fundamental computational hardness in quantum hardware mapping.

Contribution

The paper establishes the NP-completeness of the minor embedding problem for broken Chimera and Pegasus graphs, extending known complexity results to practical hardware scenarios.

Findings

Embedding in broken Chimera graphs is NP-complete.

Embedding in broken Pegasus graphs is NP-complete.

Hamiltonian cycle problem reduction demonstrates hardness.

Abstract

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 both types of existing hardware, represented by the Chimera and the Pegasus 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 Pegasus graphs, the proof is then further extended to the Pegasus graphs.