Quantum Algorithms for Clustering and Covert Factoring

Abstract

This thesis is composed of two projects – a quantum algorithm for clustering based on CERN’s event reconstruction algorithm, and a scheme to hide Shor’s algorithm in Hamiltonian Simulation and Ground State Estimation circuits.

Clustering algorithms are at the basis of several technological applications, and are fueling the development of rapidly evolving fields such as machine learning. In the recent past, however, it has become apparent that they face challenges stemming from datasets spanning several spatial dimensions. In fact, the best-performing clustering algorithms scale linearly in the number of points, but quadratically with respect to the local density of points. In this work, we introduce qLUE, a quantum clustering algorithm that scales linearly in both the number of points and their density. qLUE is inspired by CLUE, CERN's algorithm developed to address the challenging time and memory budgets of Event Reconstruction (ER) in future High-Energy Physics experiments. As such, qLUE marries decades of development with the quadratic speedup provided by quantum computers. We numerically test qLUE in several scenarios, demonstrating its effectiveness and proving it to be a promising route to handle complex data analysis tasks – especially in high-dimensional datasets with high densities of points. The code we developed for these simulations is available at: https://github.com/godspeed5/QLUE.

The advent of large-scale quantum computers promises transformative advances across various fields including optimization, materials science, and cryptography. However, this also poses a threat to traditional cryptography, due to Shor’s algorithm, which efficiently factors large integers. The existence of this algorithm undermines widely-used cryptographic protocols based on integer factorization and discrete logarithms. Even with Post-Quantum Cryptography, attacks of the “save now, decrypt later” type can compromise the security of critical systems. Keeping this mind, we would like to develop quantum systems that are designed specifically for benign applications such as Hamiltonian Simulation or Ground State Estimation – which could be of importance to the pharmaceutical industry. However, it cannot be taken for granted that even such a system is secure from malicious users attempting to run Shor’s Algorithm. In this note we propose the idea of using known circuit-to-Hamiltonian mappings to hide Shor’s algorithm in Hamiltonian simulation and Ground state estimation circuits. We provide the resource estimates for these mappings, and also propose some methods to potentially reduce these overhead costs.