Shor's factorization algorithm with a single control qubit and imperfections

TL;DR

This study numerically investigates the robustness of Shor's quantum factoring algorithm with a single control qubit against static imperfections, demonstrating its operational limits and scaling behavior on realistic quantum computers.

Contribution

It provides a detailed numerical analysis of the algorithm's resilience to static imperfections, extending understanding of its practical feasibility with many qubits.

Findings

Algorithm remains operational up to a critical coupling strength that decreases polynomially with log2 N.

Numerical results agree with analytical estimates for the scaling of imperfections.

Demonstrates feasibility of factoring large numbers with realistic quantum hardware.

Abstract

We formulate and numerically simulate the single control qubit Shor algorithm for the case of static imperfections induced by residual couplings between qubits. This allows us to study the accuracy of Shor's algorithm with respect to these imperfections using numerical simulations of realistic quantum computations with up to $n_q=18$ computational qubits allowing to factor numbers up to N=205193. We confirm that the algorithm remains operational up to a critical coupling strength $\epsilon_c$ which drops only polynomially with $\log_2 N$. The obtained numerical dependence of $\epsilon_c$ on $\log_2 N$ is in a good agreement with the analytical estimates that allows to obtain the scaling for functionality of Shor's algorithm on realistic quantum computers with a large number of qubits.