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Experimental demonstration of Shor’s algorithm for factoring N=21 on Tianyan-176 superconducting processor via Fredkin gate synthesis

Guihui Li, Yadong Yang, Haibo Lin, Jing Zhang, Shijian Zheng, Xiancun Zhou

Published Jan 1, 2026• 0 citations
Publisher

Abstract

We present an experimental demonstration of factoring 21 using a compiled version of Shor’s algorithm on the Tianyan-176 superconducting processor. Our compilation approach—Fredkin gate synthesis—is primarily derived from the symmetry of the permutation operations in the modular exponentiation function (MEF) modules. The effectiveness of the approximate Toffoli gate substitution is also theoretically validated. Additionally, we propose several novel compilation strategies for implementing a lower-depth, topology-adapted inverse quantum Fourier transform (QFT). The 6-qubit demonstration achieves a state-of-the-art implementation with only 19 two-qubit gates on a grid-based qubit topology, without introducing any SWAP gates. Owing to the permutation symmetry, the demonstration is readily extensible to larger-scale instances and other factoring cases by introducing only a minimal number of SWAP gates, leveraging the commutation properties between the controlled MEF and inverse QFT modules.

DOI
10.1587/elex.23.20260366

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We present an experimental demonstration of factoring 21 using a compiled version of Shor’s algorithm on the Tianyan-176 superconducting processor. Our compilation approach—Fredkin gate synthesis—is primarily derived from the symmetry of the permutation operations in the modular exponentiation function (MEF) modules. The effectiveness of the approximate Toffoli gate substitution is also theoretically validated. Additionally, we propose several novel compilation strategies for implementing a lower-depth, topology-adapted inverse quantum Fourier transform (QFT). The 6-qubit demonstration achieves a state-of-the-art implementation with only 19 two-qubit gates on a grid-based qubit topology, without introducing any SWAP gates. Owing to the permutation symmetry, the demonstration is readily extensible to larger-scale instances and other factoring cases by introducing only a minimal number of SWAP gates, leveraging the commutation properties between the controlled MEF and inverse QFT modules.
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. 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 [Formula: see text]. We implemented the algorithm on IBM quantum processors using only five 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.
We report an experimental demonstration of a complied version of Shor's algorithm using four photonic qubits. We choose the simplest instance of this algorithm, that is, factorization of N=15 in the case that the period r=2 and exploit a simplified linear optical network to coherently implement the quantum circuits of the modular exponential execution and semiclassical quantum Fourier transformation. During this computation, genuine multiparticle entanglement is observed which well supports its quantum nature. This experiment represents an essential step toward full realization of Shor's algorithm and scalable linear optics quantum computation.
To factor an integer N into a product of two other numbers N1 and N2, Shor’s algorithm uses the Quantum Fourier Transform on n qubits, denoted QFTn, to determine the period r of a Modular Exponentiation Function (MEF), which is later used to compute a factor of N. In this work, we present Shor’s algorithm for factoring an integer of n qubits and illustrate an example with n=4. We implement our algorithm using Qiskit and IBM quantum computers, which also can be adapted on other quantum computing platforms with minor adjustments. We discuss the IBM quantum computer limitations for running our circuit and show the impact of device imperfections on the factored result.
We study the results of a compiled version of Shor's factoring algorithm on the ibmqx5 superconducting chip, for the particular case of N=15N=15N=15, 21, and 35. The semiclassical quantum Fourier transform is used to implement the algorithm with only a small number of physical qubits, and the circuits are designed to reduce the number of gates to the minimum. We use the square of the statistical overlap to give a quantitative measure of the similarity between the experimentally obtained distribution of phases and the predicted theoretical distribution of phases for different values of the period. This allows us to assign a period to the experimental data without the use of the continued fraction algorithm. A quantitative estimate of the error in our assignment of the period is then given by the overlap coefficient.
Shor's algorithm proved a significant milestone in quantum computing since it promises an exponential speedup over conventional algorithms for integer factorization, a problem critical to modern cryptography systems. This work covers implementation steps of Shor’s algorithm and analyzes Quantum Fourier Transform (QFT) usage for extracting periodicity from quantum superpositions. Firstly, theoretical foundations of algorithm are illustrated concentrating on QFT construction and operation within the setting of the quantum circuit. Thereafter, each implementation step such as qubit optimization, modular exponentiation, and circuit design is discussed. Our findings validate the theoretical effectiveness of the QFT in solving the period-finding subroutine, while also highlighting the practical difficulties and scaling constraints presented by existing quantum hardware. The paper ends with a summary of possible advancements and future paths for error mitigation strategies and quantum algorithm design.
Quantum algorithms are computational routines that exploit quantum mechanics to solve problems exponentially faster than the best classical algorithms. Shor's quantum factoring algorithm is a key example and the prime motivator in the international effort to realise a quantum computer. However, due to the large number of resources required, to date, there have been only four small scale demonstrations. Here we address this resource demand and demonstrate a scalable version of Shor's algorithm in which then qubit control register is replaced by a single qubit that is recycled n times: the total number of qubits is one third of that required in the standard protocol. Encoding the work register in higher-dimensional states, we implement a two-photon compiled algorithm to factor N = 21. Significantly, the algorithmic output exhibits structure that is distinguishable from noise, in contrast to previous demonstrations.