Realization of a scalable Shor algorithm

TLDR

Quantum computers can outperform classical algorithms, a fact noted by Feynman and later formalized by Shor’s efficient factoring algorithm, which has spurred extensive research and experimental implementations demonstrating the feasibility of quantum factoring. The authors aim to achieve a fully scalable Shor algorithm, following Kitaev’s proposal, to address general scalability challenges. They factor the number fifteen using an ion‑trap system that controls seven computational qubits and four cache qubits while implementing modular multipliers. The scalable implementation achieves success probabilities exceeding 90 %.

Abstract

Quantum computers are able to outperform classical algorithms. This was long recognized by the visionary Richard Feynman who pointed out in the 1980s that quantum mechanical problems were better solved with quantum machines. It was only in 1994 that Peter Shor came up with an algorithm that is able to calculate the prime factors of a large number vastly more efficiently than known possible with a classical computer. This paradigmatic algorithm stimulated the flourishing research in quantum information processing and the quest for an actual implementation of a quantum computer. Over the last fifteen years, using skillful optimizations, several instances of a Shor algorithm have been implemented on various platforms and clearly proved the feasibility of quantum factoring. For general scalability, though, a different approach has to be pursued. Here, we report the realization of a fully scalable Shor algorithm as proposed by Kitaev. For this, we demonstrate factoring the number fifteen by effectively employing and controlling seven qubits and four "cache-qubits", together with the implementation of generalized arithmetic operations, known as modular multipliers. The scalable algorithm has been realized with an ion-trap quantum computer exhibiting success probabilities in excess of 90%.