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

## Abstract

Using quantum annealing to solve an optimization problem requires minor embedding a logic graph into a known hardware graph. We introduce the minor set cover (MSC) of a known graph GG : a subset of graph minors which contain any remaining minor of the graph as a subgraph, in an effort to reduce the complexity of the minor embedding problem. Any graph that can be embedded into GG will be embeddable into a member of the MSC. Focusing on embedding into the hardware graph of commercially available quantum annealers, we establish the MSC for a particular known virtual hardware, which is a complete bipartite graph. Furthermore, we show that the complete bipartite graph K _{N,N} has a MSC of N minors, from which K _{N+1} is identified as the largest clique minor of K _{N,N}. In the case of determining the largest clique minor of hardware with faults we briefly discussed this open question.

- Authors:

- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). Computer Science Mathematics Division

- Publication Date:

- Research Org.:
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1361309

- Grant/Contract Number:
- AC05-00OR22725

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Quantum Information Processing

- Additional Journal Information:
- Journal Volume: 16; Journal Issue: 4; Journal ID: ISSN 1570-0755

- Publisher:
- Springer

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; Minor embedding; Adiabatic quantum computing; Quantum annealing; Clique minor; Graph theory

### Citation Formats

```
Hamilton, Kathleen E., and Humble, Travis S.
```*Identifying the minor set cover of dense connected bipartite graphs via random matching edge sets*. United States: N. p., 2017.
Web. doi:10.1007/s11128-016-1513-7.

```
Hamilton, Kathleen E., & Humble, Travis S.
```*Identifying the minor set cover of dense connected bipartite graphs via random matching edge sets*. United States. doi:10.1007/s11128-016-1513-7.

```
Hamilton, Kathleen E., and Humble, Travis S. Thu .
"Identifying the minor set cover of dense connected bipartite graphs via random matching edge sets". United States.
doi:10.1007/s11128-016-1513-7. https://www.osti.gov/servlets/purl/1361309.
```

```
@article{osti_1361309,
```

title = {Identifying the minor set cover of dense connected bipartite graphs via random matching edge sets},

author = {Hamilton, Kathleen E. and Humble, Travis S.},

abstractNote = {Using quantum annealing to solve an optimization problem requires minor embedding a logic graph into a known hardware graph. We introduce the minor set cover (MSC) of a known graph GG : a subset of graph minors which contain any remaining minor of the graph as a subgraph, in an effort to reduce the complexity of the minor embedding problem. Any graph that can be embedded into GG will be embeddable into a member of the MSC. Focusing on embedding into the hardware graph of commercially available quantum annealers, we establish the MSC for a particular known virtual hardware, which is a complete bipartite graph. Furthermore, we show that the complete bipartite graph KN,N has a MSC of N minors, from which KN+1 is identified as the largest clique minor of KN,N. In the case of determining the largest clique minor of hardware with faults we briefly discussed this open question.},

doi = {10.1007/s11128-016-1513-7},

journal = {Quantum Information Processing},

number = 4,

volume = 16,

place = {United States},

year = {Thu Feb 23 00:00:00 EST 2017},

month = {Thu Feb 23 00:00:00 EST 2017}

}

*Citation information provided by*

Web of Science

Web of Science