Depth-Based Matrix Classification for the HHL Quantum Algorithm

--Under the nearing error-corrected era of quantum computing, it is necessary to understand the suitability of certain post-NISQ algorithms for practical problems. One of the most promising, applicable and yet difficult to implement in practical terms is the Harrow, Hassidim and Lloyd (HHL) algorithm for linear systems of equations. An enormous number of problems can be expressed as linear systems of equations, from Machine Learning to fluid dynamics. However, in most cases, HHL will not be able to provide a practical, reasonable solution to these problems. This paper's goal inquires about whether problems can be labeled using Machine Learning classifiers as suitable or unsuitable for HHL implementation when some numerical information about the problem is known beforehand. This work demonstrates that training on significantly representative data distributions is critical to achieve good classifications of the problems based on the numerical properties of the matrix representing the system of equations. Accurate classification is possible through Multi-Layer Perceptrons, although with careful design of the training data distribution and classifier parameters. The HHL algorithm by Harrow, Hassidim and Lloyd is a well known quantum algorithm for quantum-mechanically constructing the solution of a linear systems of equations [1]. HHL is one of those quantum algorithms that will only make sense under quantum error-corrected implementation. Although its depth (number of gate layers) varies depending on certain conditions as it will be shown, HHL results in deep quantum circuits. As we approach this new era of quantum computing, it is necessary to gain understanding of the actual implementability of certain algorithms. The linear system of equations problem can be defined as, given a matrix A and a vector b, find a vector xsuch that A x= b. In quantum notation, this is expressed as A | x = | b, where A is a Hermitian operator -- a workaround exists when A is not Hermitian-- and b has to be encoded in a quantum state |b and, hence, it has to be normalized.