# Floating-Point Calculations on a Quantum Annealer: Division and Matrix Inversion

## TL;DR

This paper demonstrates how to formulate floating-point division and matrix inversion as QUBO problems suitable for quantum annealers, providing solutions for small matrices and scalable algorithms for larger systems.

## Contribution

It introduces a novel QUBO formulation for floating-point operations and matrix inversion, enabling their implementation on quantum annealers with scalable algorithms.

## Key findings

- Successfully implemented division and inversion on D-Wave quantum annealer for 2x2 and 3x3 matrices
- Proposed a scalable algorithm for large linear systems
- Provides full matrix solutions, unlike HHL which only gives expectation values

## Abstract

Systems of linear equations are employed almost universally across a wide range of disciplines, from physics and engineering to biology, chemistry and statistics. Traditional solution methods such as Gaussian elimination become very time consuming for large matrices, and more efficient computational methods are desired. In the twilight of Moore's Law, quantum computing is perhaps the most direct path out of the darkness. There are two complementary paradigms for quantum computing, namely, gated systems and quantum annealers. In this paper, we express floating point operations such as division and matrix inversion in terms of a quadratic unconstrained binary optimization (QUBO) problem, a formulation that is ideal for a quantum annealer. We first address floating point division, and then move on to matrix inversion. We provide a general algorithm for any number of dimensions, but we provide results from the D-Wave quantum anneler for $2\times 2$ and $3 \times 3$ general matrices. Our algorithm scales to very large numbers of linear equations. We should also mention that our algorithm provides the full solution the the matrix problem, while HHL provides only an expectation value.