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Variational Quantum Factorization (VQF) using Different Variations of Quantum Approximate Optimization Algorithm (QAOA)

Janhavi Hatwar, Nakul Aggarwal, Debanjan Bhowmik

Published Jan 6, 2025• 1 citation
Publisher

Abstract

Variational quantum factoring (VQF), a variational quantum algorithm, has been proposed as an alternative to Shor’s algorithm for factoring numbers and has been experimentally implemented to factor numbers much larger than what Shor’s algorithm has factorized thus far. The focus of this work is comparing the performance of different variations of quantum approximate optimization algorithm (QAOA) in terms of implementing VQF. Hence, VQF has been implemented here on the Qiskit quantum simulation framework using standard QAOA, multi-angle QAOA (maQAOA), conditional value at risk (CVaR) QAOA, and CVar maQAOA. Using these methods, VQF has been implemented for factorizing bi-prime numbers 143 and 291311 using 4 and 6 qubits respectively, and for factorizing a tri-prime number, 175, using 3 qubits. For factoring of 143 and 175, maQAOA yields the highest success probability numbers (0.92 and 0.99 respectively), while for factoring 291311, CVar maQAOA with 0.75 of the total measurements contributing to the expectation value yields the highest success probability.

DOI
10.1109/comsnets63942.2025.10885675

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Variational quantum factoring (VQF), a variational quantum algorithm, has been proposed as an alternative to Shor’s algorithm for factoring numbers and has been experimentally implemented to factor numbers much larger than what Shor’s algorithm has factorized thus far. The focus of this work is comparing the performance of different variations of quantum approximate optimization algorithm (QAOA) in terms of implementing VQF. Hence, VQF has been implemented here on the Qiskit quantum simulation framework using standard QAOA, multi-angle QAOA (maQAOA), conditional value at risk (CVaR) QAOA, and CVar maQAOA. Using these methods, VQF has been implemented for factorizing bi-prime numbers 143 and 291311 using 4 and 6 qubits respectively, and for factorizing a tri-prime number, 175, using 3 qubits. For factoring of 143 and 175, maQAOA yields the highest success probability numbers (0.92 and 0.99 respectively), while for factoring 291311, CVar maQAOA with 0.75 of the total measurements contributing to the expectation value yields the highest success probability.
Abstract In the near-term, hybrid quantum-classical algorithms hold great potential for outperforming classical approaches. Understanding how these two computing paradigms work in tandem is critical for identifying areas where such hybrid algorithms could provide a quantum advantage. In this work, we study a QAOA-based quantum optimization approach by implementing the Variational Quantum Factoring (VQF) algorithm. We execute experimental demonstrations using a superconducting quantum processor, and investigate the trade off between quantum resources (number of qubits and circuit depth) and the probability that a given biprime is successfully factored. In our experiments, the integers 1099551473989, 3127, and 6557 are factored with 3, 4, and 5 qubits, respectively, using a QAOA ansatz with up to 8 layers and we are able to identify the optimal number of circuit layers for a given instance to maximize success probability. Furthermore, we demonstrate the impact of different noise sources on the performance of QAOA, and reveal the coherent error caused by the residual ZZ-coupling between qubits as a dominant source of error in a near-term superconducting quantum processor.
Variational algorithm using Quantum Approximate Optimization Algorithm (QAOA) can solve the prime factorization problem in near-term noisy quantum computers. Conventional Variational Quantum Factoring (VQF) requires a large number of 2-qubit gates (especially for factoring a large number) resulting in deep circuits. The output quality of the deep quantum circuit is degraded due to errors limiting the computational power of quantum computing. In this paper, we explore various transformations to optimize the QAOA circuit for integer factorization. We propose two criteria to select the optimal quantum circuit that can improve the noise resiliency of VQF.
The content of this poster is a description of the implementation of two well-known quantum integer factorization methods and, in addition, a proposal of a new, third way along with some examples of its use. For all three methods, the measured results of their circuits will be shown.The first method is Shor's algorithm, where a quantum circuit is used only to find the period of a specially assembled modular exponentiating function. The rest of the algorithm is preprocessing and postprocessing that can be done on a classical computer, so one could say that this is a hybrid algorithm. New results of the successful factorization of the number 119, performed by this algorithm on a real 7-qubit quantum computer, will be presented. It is worth mentioning here that the highest number successfully factorized by this algorithm on a real quantum computer so far was 35.The second method is variational quantum factorization (VQF). In this method, preprocessing builds a system of equations based on the classical binary multiplication procedure of two numbers, resulting in a factorized number. This system of equations is then used to form a Hamiltonian in such a way that its ground state corresponds to the solution of this equation system. Only now the hybrid quantum optimization comes into play to iteratively find this ground state. The advantage over Shor's algorithm is that this method does not require a fault-tolerant quantum computer, but on the other hand it approaches the solution iteratively using a classical computer, which reduces the potential quantum advantage.The new, third method employs Grover's search algorithm in an unusual way to create a quantum factorization circuit. A key part of this approach is the quantum implementation of a multiplying oracle. The main advantage of this method over the previous ones is that it can be used to assemble a relatively simple universal factorization circuit. Moreover, there is almost no preprocessing and postprocessing required. This method is still under development, but this poster will present the first results.
We study the performance and resource usage of the variational quantum factoring (VQF) algorithm for different instance sizes and optimization algorithms. Our simulations show better chance of finding the ground state when using VQE rather than QAOA for optimization. In gradient-based optimization we find that the time required for quantum circuit gradient estimation is a significant problem if VQF is to become competitive with classical factoring algorithms. Further, we compare entangled and non-entangled circuits in VQE optimization and fail to see significant evidence in favour of including entanglement in the VQE circuit.
Abstract: This paper explores hybrid quantum-classical optimization algorithms implemented across multiple quantum software development kits (SDKs): Qiskit and Cirq,. We evaluate the Variational Quantum Eigensolver (VQE) and Quantum Approximate Optimization Algorithm (QAOA) in terms of convergence rate, final cost value, and fidelity, using both simulators and real quantum hardware backends. Our results offer comparative insights into SDK performance and propose a standardized benchmarking approach for future quantum algorithm research. Keywords: Hybrid Quantum Algorithms, Qiskit, Cirq, VQE, QAOA, Quantum SDKs, Quantum Optimization, NISQ
Abstract Factoring integers is considered as a computationally hard problem for classical methods, whereas there exists polynomial-time Shor’s quantum algorithm for solving this task. However, requirements for running Shor’s algorithm for realistic tasks, which are beyond the capabilities of existing and upcoming generations of quantum computing devices, motivate to search for alternative approaches. In this work, we experimentally demonstrate factoring of the integer with a trapped ion quantum processor using the Schnorr approach and a modified version of the quantum approximate optimization algorithm (QAOA). The key difference of our approach in comparison with the recently proposed QAOA-based factoring method is the use of the fixed-point feature, which relies on the use of universal parameters. We present experimental results on factoring 1591 = 37 × 43 using 6 qubits as well as simulation results for 74425657 = 9521 × 7817 with 10 qubits and 35183361263263 = 4194191 × 8388593 with 15 qubits. Although we present all the necessary details for reproducing our results and analysis of the performance of the factoring method, the scalability of this approach in both the classical and quantum domains still requires further studies.