Demonstration of Shor's factoring algorithm for N=21 on IBM quantum processors

TL;DR

This paper demonstrates a quantum implementation of Shor's algorithm for factoring 21 on IBM quantum processors, showcasing the use of approximate gates and entanglement verification with limited qubits.

Contribution

The work introduces a compiled quantum phase estimation routine with approximate Toffoli gates enabling successful factorization of 21 on a 5-qubit IBM quantum device.

Findings

Successful factorization of N=21 using 5 qubits

Verification of entanglement between control and work registers

Use of approximate gates preserves correctness in noisy systems

Abstract

We report a proof-of-concept demonstration of a quantum order-finding algorithm for factoring the integer 21. Our demonstration involves the use of a compiled version of the quantum phase estimation routine, and builds upon a previous demonstration by Mart\'in-L\'{o}pez et al. in Nature Photonics 6, 773 (2012). We go beyond this work by using a configuration of approximate Toffoli gates with residual phase shifts, which preserves the functional correctness and allows us to achieve a complete factoring of N=21. We implemented the algorithm on IBM quantum processors using only 5 qubits and successfully verified the presence of entanglement between the control and work register qubits, which is a necessary condition for the algorithm's speedup in general. The techniques we employ may be useful in carrying out Shor's algorithm for larger integers, or other algorithms in systems with a limited number of noisy qubits.