A Gradient-Cost Multiobjective Alternate Framework for Variational Quantum Eigensolver with Variable Ansatz

Variational quantum eigensolver (VQE), aiming at determining the ground state energy of a quantum system described by a Hamiltonian on noisy intermediate scale quantum (NISQ) devices. However, the accuracy and trainability of the current VQE algorithm are significantly influenced due to the barren plateau (BP), the non-negligible gate error and limited coherence time in NISQ devices. To tackle these issues, a gradient-cost multiobjective alternate framework with variable ansatz is proposed. A theoretical framework is first proposed for VQE with variable ansatz (VA-VQE) via alternately solving a multiobjective optimization problem defined by cost function values and gradient magnitudes and the original VQE problem. Then, a novel implementation method based on the double epsilon-greedy strategy with the candidate tree and modified multiobjective genetic algorithm is proposed. As a result, the local optima are avoided both in ansatz and parameter perspectives, the BP phenomenon is alleviated, and the stability of output ansatz is enhanced. The experimental results indicate that this framework shows considerably average improvement of the error and stability by 59.6% and 78.8% compared with the state-of-the-art cost-criterion-based VA-VQE implementation, and 54.1% and 73.5% compared with the gradient-criterion-based counterpart, respectively, with lower quantum costs.