# Domain wall encoding of discrete variables for quantum annealing and QAOA

## TL;DR

This paper introduces a domain wall encoding method for discrete variables in quantum annealing and QAOA, which uses only two-body Ising interactions and may offer advantages in strength and performance over traditional encodings.

## Contribution

The paper presents a novel domain wall encoding technique for discrete variables in quantum optimization, enabling two-body interactions and improved embedding performance.

## Key findings

- Domain wall encoding uses only ferromagnetic couplings, which can be stronger than anti-ferromagnetic.
- Comparable or better performance than one hot encoding on synthetic scheduling and coloring problems.
- Potential advantages in specific problem structures and graph embeddings like Pegasus.

## Abstract

In this paper I propose a new method of encoding discrete variables into Ising model qubits for quantum optimization. The new method is based on the physics of domain walls in one dimensional Ising spin chains. I find that these encodings and the encoding of arbitrary two variable interactions is possible with only two body Ising terms. Following on from similar results for the `one hot' method of encoding discrete variables [Hadfield et. al. Algorithms 12.2 (2019): 34] I also demonstrate that it is possible to construct two body mixer terms which do not leave the logical subspace, an important consideration for optimising using the quantum alternating operator ansatz (QAOA). I additionally discuss how, since the couplings in the domain wall encoding only need to be ferromagnetic and therefore could in principle be much stronger than anti-ferromagnetic couplers, application specific quantum annealers for discrete problems based on this construction may be beneficial. Finally, I compare embedding for synthetic scheduling and colouring problems with the domain wall and one hot encodings on two graphs which are relevant for quantum annealing, the chimera graph and the Pegasus graph. For every case I examine I find a similar or better performance from the domain wall encoding as compared to one hot, but this advantage is highly dependent on the structure of the problem. For encoding some problems, I find an advantage similar to the one found by embedding in a Pegasus graph compared to embedding in a chimera graph.