Depth-Based Matrix Classification for the HHL Quantum Algorithm

TL;DR

This work tackles the practicality of the HHL quantum algorithm for solving linear systems by framing a depth-based binary classification problem: given a system matrix $A$, is the HHL circuit depth manageable under a predefined bound? It uses a data-driven pipeline with 58k+ random Hermitian matrices and an MLP that leverages 104 structural, statistical, and (estimated) condition-number features to predict well-suited versus poorly suited instances, labeling data by circuit depth relative to a cutoff. The findings show that the condition number is a dominant predictor of depth, and exact conditioning yields high classification accuracy, while removing it reduces performance. Approaches that tailor training distributions to target data (e.g., iris-like matrices) substantially improve the classifier’s ability to estimate the share of problems that can benefit from HHL, suggesting a practical, data-driven tool for problem selection in quantum linear algebra workloads.

Abstract

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.

Figures (7)

• Figure 1: Quantum Circuit Depth vs. Ideal Matrix Size
• Figure 2: Harrow-Hassidim-Lloyd circuit overview.
• Figure 3: Quantum phase estimation (not inverse) circuit overview.
• Figure 4: Distribution of HHL circuit depths for $4\times 4$ matrices. Each point represents a sample corresponding to one matrix in the dataset. Purple points have the highest depth, and light blue have the lowest.
• Figure 5: Average HHL circuit depth vs. random matrix size for three $\kappa$ bins and the ideal matrices. The number of layers in the exact quantum circuit grows exponentially with the input size, at faster rates for higher condition numbers. The horizontal blue lines depict the potential depth cutoffs for suitable/non-suitable problems.