Enhanced Digitized Adiabatic Quantum Factorization Algorithm Using Null-Space Encoding

TL;DR

This paper introduces a modified QAOA-based quantum factorization method that simplifies the Hamiltonian to two-body interactions, achieving high fidelity and efficiency on near-term quantum devices for small integers.

Contribution

It presents a novel Hamiltonian simplification for quantum factorization algorithms, reducing experimental complexity and improving performance in NISQ-era quantum computers.

Findings

Comparable or higher fidelities than standard protocols

Fewer quantum resources required

Faster convergence for small problem instances

Abstract

Integer factorization is a computational problem of fundamental importance in cybersecurity and secure communications, as its difficulty form the basis of modern public-key cryptography. While Shor's algorithm can solve this problem efficiently on a universal quantum computer, near-term devices require alternative approaches. The Adiabatic Factorization Algorithm and its digitized counterparts offer a promising NISQ-era pathway but suffer from high-order many-body interactions that are difficult to implement. In this work, we propose a modified QAOA-based factorization protocol that simplifies the interacting Hamiltonian to include only two-body terms, significantly reducing its experimental complexity. Numerical simulations show that this method achieves comparable or higher fidelities than the standard protocol, while requiring fewer quantum resources and converging more rapidly for problem instances up to eight qubits. We analyze the characteristic fidelity behavior introduced by the Hamiltonian modification. Additionally, we report on simulations with alternative cost-function definitions that frequently yielded improved performance.