The current state of quantum processing units (QPU) is characterized by Noisy Intermediate-Scale Quantum (NISQ) devices. Among the methods to utilize NISQ devices, the Variational Quantum Eigensolver (VQE) is prominent for finding ground state energy of quantum systems. However, VQE parameter optimization is NP-Hard, posing significant challenges. To address this, we introduce the CoolMomentum method, a global optimization approach based on Langevin dynamics with Simulated Annealing. We compared CoolMomentum with ADAM, SPSA, and NFT in VQE using a random weighted Max-Cut problem for up to 20 qubits. Our results confirm that CoolMomentum consistently outperforms the other methods. ADAM and SPSA tend to get trapped in local minima or exhibit infeasible optimization durations, whereas NFT, despite its fast convergence, also suffers from local minima. In contrast, CoolMomentum demonstrates higher accuracy and robustness across various numbers of qubits, highlighting its potential as a superior optimization strategy for VQE.
The current state of quantum processing units (QPU) is characterized by Noisy Intermediate-Scale Quantum (NISQ) devices. Among the methods to utilize NISQ devices, the Variational Quantum Eigensolver (VQE) is prominent for finding ground state energy of quantum systems. However, VQE parameter optimization is NP-Hard, posing significant challenges. To address this, we introduce the CoolMomentum method, a global optimization approach based on Langevin dynamics with Simulated Annealing. We compared CoolMomentum with ADAM, SPSA, and NFT in VQE using a random weighted Max-Cut problem for up to 20 qubits. Our results confirm that CoolMomentum consistently outperforms the other methods. ADAM and SPSA tend to get trapped in local minima or exhibit infeasible optimization durations, whereas NFT, despite its fast convergence, also suffers from local minima. In contrast, CoolMomentum demonstrates higher accuracy and robustness across various numbers of qubits, highlighting its potential as a superior optimization strategy for VQE.
Abstract We compare the use of adaptive moment estimation (ADAM), simultaneous perturbation stochastic approximation (SPSA), Nakanishi–Fujii–Todo method (NFT), and CoolMomentum in a variational quantum eigensolver. Using a random weighted max-cut problem, we numerically analyze these methods and confirm that CoolMomentum performs better than the other methods. ADAM and SPSA tend to get trapped in local minima or exhibit infeasible optimization durations. Although NFT exhibits fast convergence, it tends to suffer from local minima similar to ADAM and SPSA. Contrarily, CoolMomentum shows a higher accuracy under noiseless and realistic hardware noise conditions.
Variational quantum algorithms have promising applications in noisy intermediate-scale quantum (NISQ) devices. These algorithms rely on a classical optimization outer loop that minimizes a parameterized quantum circuit function. The optimization in variational quantum eigensolver (VQE) is NP-hard, meaning that finding the optimal solution is infeasible in the worst-case scenario. One way to address this challenge is through parallel optimization of parameters using multiple-parameterized quantum circuits. However, this approach is unsuitable for cloud-based quantum processing unit utilization due to the increased number of quantum circuit executions. Although NISQ devices have limitations in terms of gate depth, their size has been growing in recent years. Therefore, implementing multiple-parameterized quantum circuits in NISQ devices can suppress the increase in the number of executions. In this study, we propose a parallel-VQE, which leverages the parallel execution of parameterized quantum circuits to perform parallel parameter optimization in VQE, achieving convergence to solutions closer to the ground state. We validate the effectiveness of parallel-VQE in solving the random weighted max-cut problem using numerical simulations and a real quantum device. We present the results of running up to six circuits in parallel (120 qubits) and demonstrate the advantages of using multiple units to improve computational accuracy. This study provides a potential method for solving eigenvalue problems and combinatorial optimization problems for future quantum devices.
Abstract In this study, we study the Variational Quantum Eigensolver (VQE) application for the Ising model as a test bed model, in which we pivotally delved into several optimization methods, both classical and quantum, and analyzed the quantum advantage that each of these methods offered, and then we proposed a new combinatorial optimization scheme, deemed as QN-SPSA+PSR which combines calculating approximately Fubini-study metric (QN-SPSA) and the exact evaluation of gradient by Parameter-Shift Rule (PSR). The QN-SPSA+PSR method integrates the QN-SPSA computational efficiency with the precise gradient computation of the PSR, improving both stability and convergence speed while maintaining low computational consumption. Our results provide a new potential quantum supremacy in the VQAs's optimization subroutine, even in Quantum Machine Learning's optimization section, and enhance viable paths toward efficient quantum simulations on Noisy Intermediate-Scale Quantum Computing (NISQ) devices. Additionally, we also conducted a detailed study of quantum circuit ansatz structures in order to find the one that would work best with the Ising model and NISQ, in which we utilized the properties of the investigated model.
Abstract The variational quantum eigensolver (VQE) is one of the most promising and widely used algorithms for exploiting the capabilities of current Noisy Intermediate-Scale Quantum (NISQ) devices. However, VQE algorithms suffer from a plethora of issues, such as barren plateaus, local minima, quantum hardware noise, and limited qubit connectivity, thus posing challenges for their successful deployment on hardware and simulators. In this work, we propose a VQE optimization strategy that builds upon recent advances in the literature, and exhibits very shallow circuit depths when applied to the specific system of interest, namely a model Hamiltonian representing a cuprate superconductor. These features make our approach a favorable candidate for generating good ground state approximations on current NISQ devices. Our findings illustrate the potential of VQE algorithmic development for leveraging the full capabilities of NISQ devices.
In this study, we investigated the Variational Quantum Eigensolver (VQE) application for the Ising model as a testbed model, in which we thoroughly delved into several optimizers, both classical and quantum, and analyzed the extent to which each of these methods would offer a benefit. We then investigated a new combinatorial optimization scheme, termed QN-SPSA+PSR, in which the Fubini-Study metric is approximated within the Quantum Natural Gradient (QN) framework, with its inner gradient estimated by the Simultaneous Perturbation Stochastic Approximation (SPSA), while the outer gradient of the cost function is evaluated exactly by the Parameter-Shift Rule (PSR). The QN-SPSA+PSR method integrates the QN-SPSA computational efficiency with the precise gradient computation of the PSR, improving the stability of QN-SPSA-based and convergence speed per parameter update while maintaining low computational consumption. Our results provide a potential performance improvement in the VQAs' optimization subroutine, even in Quantum Machine Learning's optimization section, and enhance viable paths toward efficient quantum simulations on Noisy Intermediate-Scale Quantum Computing (NISQ) devices. Additionally, we also conducted a detailed study of quantum circuit ansatz structures in order to find the one that would work best with the Ising model and NISQ, in which we utilized the properties of the investigated model.
The Variational Quantum Eigensolver (VQE), a quantum-classical hybrid algorithm, holds significant potential for advancements in quantum chemistry and optimization tasks. Our study focuses on accelerating the VQE in combinatorial optimization tasks such as Max-Cut, k-Clique, and k-SAT, while addressing the challenges of scalability and measurement overhead in quantum devices. Notably, while solving one problem, it allows us to concurrently compute essential information like expectation values and gradients for multiple other problems. This capability is made feasible because combinatorial optimization problems often share the same measurement basis. Drawing inspiration from this insight, we introduce a method that efficiently solves multiple problems by appropriately selecting their initial points. Subsequently, we propose two selection strategies for the method. Employing these strategies can, to some extent, help avoid the issue of becoming trapped in local optima, thus leading to improved accuracy. Our tests on 5-node and 10-node graphs for Max-Cut and k-Clique show that, in some cases, our method outperforms random initialization in terms of both accuracy and number of iterations.