Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Discrete quadratic model QUBO solution landscapes

Tristan Zaborniak, Ulrike Stege

TL;DR

The effects of choice of encoding and penalty strength on the structure of QUBO DQM solution landscapes and their optimization are investigated, focusing specifically on one-hot and domain-wall encodings.

Abstract

Many computational problems involve optimization over discrete variables with quadratic interactions. Known as discrete quadratic models (DQMs), these problems in general are NP-hard. Accordingly, there is increasing interest in encoding DQMs as quadratic unconstrained binary optimization (QUBO) models to allow their solution by quantum and quantum-inspired hardware with architectures and solution methods designed specifically for such problem types. However, converting DQMs to QUBO models often introduces invalid solutions to the solution space of the QUBO models. These solutions must be penalized by introducing appropriate constraints to the QUBO objective function that are weighted by a tunable penalty parameter to ensure that the global optimum is valid. However, selecting the strength of this parameter is non-trivial, given its influence on solution landscape structure. Here, we investigate the effects of choice of encoding and penalty strength on the structure of QUBO DQM solution landscapes and their optimization, focusing specifically on one-hot and domain-wall encodings.

Discrete quadratic model QUBO solution landscapes

TL;DR

The effects of choice of encoding and penalty strength on the structure of QUBO DQM solution landscapes and their optimization are investigated, focusing specifically on one-hot and domain-wall encodings.

Abstract

Many computational problems involve optimization over discrete variables with quadratic interactions. Known as discrete quadratic models (DQMs), these problems in general are NP-hard. Accordingly, there is increasing interest in encoding DQMs as quadratic unconstrained binary optimization (QUBO) models to allow their solution by quantum and quantum-inspired hardware with architectures and solution methods designed specifically for such problem types. However, converting DQMs to QUBO models often introduces invalid solutions to the solution space of the QUBO models. These solutions must be penalized by introducing appropriate constraints to the QUBO objective function that are weighted by a tunable penalty parameter to ensure that the global optimum is valid. However, selecting the strength of this parameter is non-trivial, given its influence on solution landscape structure. Here, we investigate the effects of choice of encoding and penalty strength on the structure of QUBO DQM solution landscapes and their optimization, focusing specifically on one-hot and domain-wall encodings.
Paper Structure (9 sections, 9 theorems, 39 equations, 1 figure, 1 table)

This paper contains 9 sections, 9 theorems, 39 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. DQMs encoded as QUBO models
  3. One-hot encoding
  4. Domain-wall encoding
  5. Solution landscape features of DQMs encoded as QUBO models
  6. One-hot encoding
  7. Domain-wall encoding
  8. Discussion
  9. Conclusions

Key Result

Lemma 1

Let $f_{OH}(x)=c(x)+\gamma_{OH} p(x)$ be a one-hot QUBO-encoded DQM function and $x_a$ and $x_b$ be neighboring solutions such that $|x_b-x_a|=1$. Then $p(x_b)-p(x_a)\neq 0$.

Figures (1)

  • Figure 1: Valid and invalid solution energies as a function of $\gamma$ for a $k=2$, $l=3$ DQM, expressed as (a) a one-hot QUBO-encoded DQM, and (b) a domain-wall QUBO-encoded DQM. Notice that the solution energies of valid solutions are constant, while invalid solutions linearly increase in $\gamma$. Different invalid solutions exhibit different slopes and intercepts such that their rank order changes with $\gamma$, and it may be seen that steeper slopes are present in the one-hot QUBO-encoded DQM versus the domain-wall QUBO-encoded DQM. $\gamma^*$ is the penalty parameter which above which the optimal valid solution, $x^*$, occupies the global minimum. $\gamma^{\prime\prime}$ is the penalty parameter below which no valid solution occupies a local minimum.

Theorems & Definitions (18)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Corollary 1.1
  • proof
  • Theorem 2
  • proof
  • Corollary 2.1
  • proof
  • ...and 8 more