3 research outputs found
Advanced unembedding techniques for quantum annealers
The D-Wave quantum annealers make it possible to obtain high quality
solutions of NP-hard problems by mapping a problem in a QUBO (quadratic
unconstrained binary optimization) or Ising form to the physical qubit
connectivity structure on the D-Wave chip. However, the latter is restricted in
that only a fraction of all pairwise couplers between physical qubits exists.
Modeling the connectivity structure of a given problem instance thus
necessitates the computation of a minor embedding of the variables in the
problem specification onto the logical qubits, which consist of several
physical qubits "chained" together to act as a logical one. After annealing, it
is however not guaranteed that all chained qubits get the same value (-1 or +1
for an Ising model, and 0 or 1 for a QUBO), and several approaches exist to
assign a final value to each logical qubit (a process called "unembedding"). In
this work, we present tailored unembedding techniques for four important
NP-hard problems: the Maximum Clique, Maximum Cut, Minimum Vertex Cover, and
Graph Partitioning problems. Our techniques are simple and yet make use of
structural properties of the problem being solved. Using Erd\H{o}s-R\'enyi
random graphs as inputs, we compare our unembedding techniques to three popular
ones (majority vote, random weighting, and minimize energy). We demonstrate
that our proposed algorithms outperform the currently available ones in that
they yield solutions of better quality, while being computationally equally
efficient
Embedding of Complete Graphs in Broken Chimera Graphs
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 as well as 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.Comment: 26 pages, 9 figures, 2 table