Minor-embedding in adiabatic quantum computation: II. Minor-universal graph design

In Choi (Quantum Inf Process, 7:193---209, 2008), we introduced the notion of minor-embedding in adiabatic quantum optimization. A minor-embedding of a graph G in a quantum hardware graph U is a subgraph of U such that G can be obtained from it by contracting edges. In this paper, we describe the intertwined adiabatic quantum architecture design problem, which is to construct a hardware graph U that satisfies all known physical constraints and, at the same time, permits an efficient minor-embedding algorithm. We illustrate an optimal complete-graph-minor hardware graph. Given a family $${\mathcal{F}}$$ of graphs, a (host) graph U is called $${\mathcal{F}}$$ -minor-universal if for each graph G in $${\mathcal{F}, U}$$ contains a minor-embedding of G. The problem for designing a $${{\mathcal{F}}}$$ -minor-universal hardware graph U sparse in which $${{\mathcal{F}}}$$ consists of a family of sparse graphs (e.g., bounded degree graphs) is open.