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arXiv:2410.01935 · Physical Review A

Variational quantum eigensolver approach to prime factorization on IBM's noisy intermediate scale quantum computer

Mona Sobhani, Yahui Chai, Tobias Hartung, Karl Jansen

Published Apr 8, 2025• 5 citations
arXivHTMLPDF

Abstract

This paper presents a hybrid quantum-classical approach to prime factorization. The proposed algorithm is based on the variational quantum eigensolver (VQE), which employs a classical optimizer to find the ground state of a given Hamiltonian. A numerical study is presented, evaluating the performance of the proposed method across various instances on both IBM's real quantum computer and its classical simulator. The results demonstrate that the method is capable of successfully factorizing numbers up to 253 on a real quantum computer and up to 1 048 561 on a classical simulator. These findings show the potential of the approach for practical applications on near-term quantum computers.

DOI
10.1103/physreva.111.042413

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This paper presents a hybrid quantum-classical approach to prime factorization. The proposed algorithm is based on the variational quantum eigensolver (VQE), which employs a classical optimizer to find the ground state of a given Hamiltonian. A numerical study is presented, evaluating the performance of the proposed method across various instances on both IBM's real quantum computer and its classical simulator. The results demonstrate that the method is capable of successfully factorizing numbers up to 253 on a real quantum computer and up to 1 048 561 on a classical simulator. These findings show the potential of the approach for practical applications on near-term quantum computers.
We describe the contextual subspace variational quantum eigensolver (CS-VQE), a hybrid quantum-classical algorithm for approximating the ground state energy of a Hamiltonian. The approximation to the ground state energy is obtained as the sum of two contributions. The first contribution comes from a noncontextual approximation to the Hamiltonian, and is computed classically. The second contribution is obtained by using the variational quantum eigensolver (VQE) technique to compute a contextual correction on a quantum processor. In general the VQE computation of the contextual correction uses fewer qubits and measurements than the VQE computation of the original problem. Varying the number of qubits used for the contextual correction adjusts the quality of the approximation. We simulate CS-VQE on tapered Hamiltonians for small molecules, and find that the number of qubits required to reach chemical accuracy can be reduced by more than a factor of two. The number of terms required to compute the contextual correction can be reduced by more than a factor of ten, without the use of other measurement reduction schemes. This indicates that CS-VQE is a promising approach for eigenvalue computations on noisy intermediate-scale quantum devices.
The variational quantum eigensolver (VQE) is a promising algorithm to demonstrate quantum advantage on near-term noisy-intermediate-scale quantum (NISQ) computers. One central problem of VQE is the effect of noise, especially physical noise, on realistic quantum computers. We systematically study the effect of noise for the VQE algorithm by performing numerical simulations with various local noise models, including amplitude damping, dephasing, and depolarizing noise. We show that the ground state energy will deviate from the exact value as the noise probability increase, and typically, the noise will accumulate as the circuit depth increase. The results suggest that the noisy quantum system can remain entanglement at the noise level of NISQ devices by comparing the VQE solution with the mean-field solution for the many-body ground state problem. We build a noise model to capture the noise in a real quantum computer, and the corresponding numerical simulation is consistent with experimental results on IBM Quantum computers through cloud. Our work sheds new light on the practical research of noisy VQE, and the deep understanding of the noise effect of VQE will also help develop error mitigation techniques on near-term quantum computers.
Currently available quantum processors are noisy intermediate-scale quantum (NISQ) devices. The variational quantum eigensolver (VQE) is an algorithm that is closer to near-term applicability due to lower quantum hardware requirements. In VQE, trial states with variational parameters are prepared by quantum computers, and the optimal parameters are determined by a classical optimizer. This optimization is known to be an NP-hard problem. In this work, we make a comparison between various gradient-free optimizers in terms of approximation ratio and function evaluations. As a result, we find that COBYLA method is the best to find the approximated solution with a lower number of executions of quantum computers.
Variational Quantum Eigensolvers (VQEs) are a powerful class of hybrid quantum-classical algorithms designed to approximate the ground state of a quantum system described by its Hamiltonian. VQEs hold promise for various applications, including lattice field theory. However, the inherent noise of Noisy Intermediate-Scale Quantum (NISQ) devices poses a significant challenge for running VQEs as these algorithms are particularly susceptible to noise, e.g., measurement shot noise and hardware noise. In a recent work, it was proposed to enhance the classical optimization of VQEs with Gaussian Processes (GPs) and Bayesian Optimization, as these machine-learning techniques are well-suited for handling noisy data. In these proceedings, we provide additional insights into this new algorithm and present further numerical experiments. In particular, we examine the impact of hardware noise and error mitigation on the algorithm's performance. We validate the algorithm using classical simulations of quantum hardware, including hardware noise benchmarks, which have not been considered in previous works. Our numerical experiments demonstrate that GP-enhanced algorithms can outperform state-of-the-art baselines, laying the foundation for future research on deploying these techniques to real quantum hardware and lattice field theory setups.
Variational quantum eigensolver (VQE), a kind of quantum-classical hybrid method, has latent powers to leverage noisy quantum devices by effectively managing a limited number of qubits with finite coherent times. We implemented VQE with real quantum processors through IBM Quantum Experience and applied it to combinatorial optimizations. Notably, a short-depth quantum circuit consisting of a lower number of quantum gate than that in the quantum approximate optimization algorithm can search the ground states of diagonal Hamiltonians encoding the problems. Since the simplest case of the quantum circuit for VQE requires only single-qubit gates, it can be realized using current quantum computers.
The variational quantum eigensolver (VQE) is a hybrid quantum-classical algorithm for finding the minimum eigenvalue of a Hamiltonian that involves the optimization of a parameterized quantum circuit. Since the resulting optimization problem is in general nonconvex, the method can converge to suboptimal parameter values that do not yield the minimum eigenvalue. In this work, we address this shortcoming by adopting the concept of variational adiabatic quantum computing (VAQC) as a procedure to improve VQE. In VAQC, the ground state of a continuously parameterized Hamiltonian is approximated via a parameterized quantum circuit. We discuss some basic theory of VAQC to motivate the development of a hybrid quantum-classical homotopy continuation method. The proposed method has parallels with a predictor-corrector method for numerical integration of differential equations. While there are theoretical limitations to the procedure, we see in practice that VAQC can successfully find good initial circuit parameters to initialize VQE. We demonstrate this with two examples from quantum chemistry. Through these examples, we provide empirical evidence that VAQC, combined with other techniques (an adaptive termination criteria for the classical optimizer and a variance-based resampling method for the expectation evaluation), can provide more accurate solutions than “plain” VQE, for the same amount of effort.