# Embedding Equality Constraints of Optimization Problems into a Quantum Annealer

## Abstract

Quantum annealers such as D-Wave machines are designed to propose solutions for quadratic unconstrained binary optimization (QUBO) problems by mapping them onto the quantum processing unit, which tries to find a solution by measuring the parameters of a minimum-energy state of the quantum system. While many NP-hard problems can be easily formulated as binary quadratic optimization problems, such formulations almost always contain one or more constraints, which are not allowed in a QUBO. Embedding such constraints as quadratic penalties is the standard approach for addressing this issue, but it has drawbacks such as the introduction of large coefficients and using too many additional qubits. In this paper, we propose an alternative approach for implementing constraints based on a combinatorial design and solving mixed-integer linear programming (MILP) problems in order to find better embeddings of constraints of the type Σ x _{i} = k for binary variables x _{i}. Our approach is scalable to any number of variables and uses a linear number of ancillary variables for a fixed k.

- Authors:

- The State University of New Jersey, Piscataway, NJ (United States)
- Los Alamos National Laboratory; Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Publication Date:

- Research Org.:
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA); USDOE Laboratory Directed Research and Development (LDRD) Program

- OSTI Identifier:
- 1544690

- Report Number(s):
- LA-UR-19-20224

Journal ID: ISSN 1999-4893; ALGOCH

- Grant/Contract Number:
- 89233218CNA000001

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Algorithms

- Additional Journal Information:
- Journal Volume: 12; Journal Issue: 4; Journal ID: ISSN 1999-4893

- Publisher:
- MDPI

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; quantum annealing; D-Wave; QUBO; constrained optimization; mixed-integer programming

### Citation Formats

```
Vyskocil, Tomas, and Djidjev, Hristo Nikolov. Embedding Equality Constraints of Optimization Problems into a Quantum Annealer. United States: N. p., 2019.
Web. doi:10.3390/a12040077.
```

```
Vyskocil, Tomas, & Djidjev, Hristo Nikolov. Embedding Equality Constraints of Optimization Problems into a Quantum Annealer. United States. doi:10.3390/a12040077.
```

```
Vyskocil, Tomas, and Djidjev, Hristo Nikolov. Wed .
"Embedding Equality Constraints of Optimization Problems into a Quantum Annealer". United States. doi:10.3390/a12040077. https://www.osti.gov/servlets/purl/1544690.
```

```
@article{osti_1544690,
```

title = {Embedding Equality Constraints of Optimization Problems into a Quantum Annealer},

author = {Vyskocil, Tomas and Djidjev, Hristo Nikolov},

abstractNote = {Quantum annealers such as D-Wave machines are designed to propose solutions for quadratic unconstrained binary optimization (QUBO) problems by mapping them onto the quantum processing unit, which tries to find a solution by measuring the parameters of a minimum-energy state of the quantum system. While many NP-hard problems can be easily formulated as binary quadratic optimization problems, such formulations almost always contain one or more constraints, which are not allowed in a QUBO. Embedding such constraints as quadratic penalties is the standard approach for addressing this issue, but it has drawbacks such as the introduction of large coefficients and using too many additional qubits. In this paper, we propose an alternative approach for implementing constraints based on a combinatorial design and solving mixed-integer linear programming (MILP) problems in order to find better embeddings of constraints of the type Σ xi = k for binary variables xi. Our approach is scalable to any number of variables and uses a linear number of ancillary variables for a fixed k.},

doi = {10.3390/a12040077},

journal = {Algorithms},

number = 4,

volume = 12,

place = {United States},

year = {2019},

month = {4}

}

Works referenced in this record:

##
Computationally Related Problems

journal, December 1974

- Sahni, Sartaj
- SIAM Journal on Computing, Vol. 3, Issue 4

##
Quantum annealing with manufactured spins

journal, May 2011

- Johnson, M. W.; Amin, M. H. S.; Gildert, S.
- Nature, Vol. 473, Issue 7346

##
Architectural Considerations in the Design of a Superconducting Quantum Annealing Processor

journal, August 2014

- Bunyk, P. I.; Hoskinson, Emile M.; Johnson, Mark W.
- IEEE Transactions on Applied Superconductivity, Vol. 24, Issue 4

##
Ising formulations of many NP problems

journal, January 2014

- Lucas, Andrew
- Frontiers in Physics, Vol. 2

##
Discrete optimization using quantum annealing on sparse Ising models

journal, September 2014

- Bian, Zhengbing; Chudak, Fabian; Israel, Robert
- Frontiers in Physics, Vol. 2

##
Satisfiability modulo theories: introduction and applications

journal, September 2011

- De Moura, Leonardo; Bjørner, Nikolaj
- Communications of the ACM, Vol. 54, Issue 9

##
Asymptotic Analysis for Penalty and Barrier Methods in Convex and Linear Programming

journal, February 1997

- Auslender, A.; Cominetti, R.; Haddou, M.
- Mathematics of Operations Research, Vol. 22, Issue 1

##
The unconstrained binary quadratic programming problem: a survey

journal, April 2014

- Kochenberger, Gary; Hao, Jin-Kao; Glover, Fred
- Journal of Combinatorial Optimization, Vol. 28, Issue 1

##
Minor-embedding in adiabatic quantum computation: I. The parameter setting problem

journal, September 2008

- Choi, Vicky
- Quantum Information Processing, Vol. 7, Issue 5

##
Adiabatic quantum programming: minor embedding with hard faults

journal, November 2013

- Klymko, Christine; Sullivan, Blair D.; Humble, Travis S.
- Quantum Information Processing, Vol. 13, Issue 3

##
Fast clique minor generation in Chimera qubit connectivity graphs

journal, October 2015

- Boothby, Tomas; King, Andrew D.; Roy, Aidan
- Quantum Information Processing, Vol. 15, Issue 1

##
A case study in programming a quantum annealer for hard operational planning problems

journal, December 2014

- Rieffel, Eleanor G.; Venturelli, Davide; O’Gorman, Bryan
- Quantum Information Processing, Vol. 14, Issue 1

##
Mapping Constrained Optimization Problems to Quantum Annealing with Application to Fault Diagnosis

journal, July 2016

- Bian, Zhengbing; Chudak, Fabian; Israel, Robert Brian
- Frontiers in ICT, Vol. 3

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

journal, February 2017

- Hamilton, Kathleen E.; Humble, Travis S.
- Quantum Information Processing, Vol. 16, Issue 4

##
Optimizing adiabatic quantum program compilation using a graph-theoretic framework

journal, April 2018

- Goodrich, Timothy D.; Sullivan, Blair D.; Humble, Travis S.
- Quantum Information Processing, Vol. 17, Issue 5

##
Efficiently embedding QUBO problems on adiabatic quantum computers

journal, March 2019

- Date, Prasanna; Patton, Robert; Schuman, Catherine
- Quantum Information Processing, Vol. 18, Issue 4