Adiabatic quantum programming: minor embedding with hard faults

Abstract

Adiabatic quantum programming defines the timedependent mapping of a quantum algorithm into an underlying hardware or logical fabric. An essential step is embedding problem-specific information into the quantum logical fabric. We present algorithms for embedding arbitrary instances of the adiabatic quantum optimization algorithm into a square lattice of specialized unit cells. These methods extend with fabric growth while scaling linearly in time and quadratically in footprint. We also provide methods for handling hard faults in the logical fabric without invoking approximations to the original problem, and illustrate their versatility through numerical studies of embeddabilty versus fault rates in square lattices of complete bipartite unit cells. The studies show these algorithms are more resilient to faulty fabrics than naive embedding approaches, a feature which should prove useful in benchmarking the adiabatic quantum optimization algorithm on existing faulty hardware.

Identifying the minor set cover of dense connected bipartite graphs via random matching edge sets

“…For the Chimera(n, n, c) hardware graph,the MSC identifies K nc+1 as the largest complete graph which can be minor embedded into the virtual hardware K nc,nc . This agrees with results found by Choi [7] and results given in [11] based on treewidth arguments.…”

“…In general, graph minor embedding requires each logical vertex to be mapped into a connected subtree of the hardware graph, and Choi's TRIAD algorithm yielded a deterministic method for embedding a complete graph into the Chimera lattice. There are many approaches for determining if a graph can be embedded into another graph, including: identifying useful graph qualities [8][9][10], establishing a set of known embeddings * hamiltonke@ornl.gov as a "lookup table" [11,12] or other heuristic methods [13,14].…”

“…Their approach is based on employing Djikstra's algorithm to find the shortest-path between randomly mapped logical vertices. The CMR algorithm has proven useful for embedding arbitrary input graphs in current hardware because it can find smaller embeddings than using either the TRIAD algorithm [7] or maximal minor embedding [11]. However, the method is not guaranteed to succeed and has a worst case complexity that scales as O(n 9 ) with the input graph order n (though average case behavior appears to be O(n 3 )).…”

“…Our method builds upon two recent embedding concepts: "maximal minor embedding" developed in [11], which characterizes finding the most efficient embedding with respect to the minimal number of hardware nodes used in the embedded graph; and recent ideas developed by Goodrich and collaborators [20] that uses the complete bipartite (biclique) minor of the Chimera graph as a virtual hardware (as shown in Fig. 2).…”

“…Klymko et al [11] established tight bounds on the largest complete graph that can be minor embedded in a Chimera graph as well as demonstrating methods for embedding into faulty hardware graphs and introduced the concept of "maximal minor" embedding. We rename the set of "maximal minors" as the minor set cover (MSC): this is the set of minors for a given graph G where any subgraph or minor of G will either be a member of the MSC or is a subgraph contained in one of the members.…”