Benchmarking VQE Configurations: Architectures, Initializations, and Optimizers for Silicon Ground State Energy

TL;DR

This work benchmarks Variational Quantum Eigensolver configurations for the silicon atom, analyzing how initialization, ansatz architecture, and classical optimization jointly affect convergence and accuracy. By minimizing $E(\theta)=\langle \psi(\theta)|\hat{H}|\psi(\theta)\rangle$ with a fermion-to-qubit mapped Hamiltonian, the authors compare four ansatz (DexcG, PCU2, UCCSD, k-UpCCGSD) and three optimizers (GD, SPSA, ADAM) across multiple seed initializations. The study finds that zero initialization consistently improves stability, and that a chemically inspired ansatz with adaptive optimization (notably UCCSD with ADAM) yields the most accurate and robust ground-state energies for silicon, while PCU2 shows exceptional initialization robustness. These results provide practical guidance for configuring VQE in medium-scale quantum chemistry problems on NISQ devices, highlighting the value of co-designing initialization, ansatz, and optimizer choices to enable reliable simulations.

Abstract

Quantum computing presents a promising path toward precise quantum chemical simulations, particularly for systems that challenge classical methods. This work investigates the performance of the Variational Quantum Eigensolver (VQE) in estimating the ground-state energy of the silicon atom, a relatively heavy element that poses significant computational complexity. Within a hybrid quantum-classical optimization framework, we implement VQE using a range of ansatz, including Double Excitation Gates, ParticleConservingU2, UCCSD, and k-UpCCGSD, combined with various optimizers such as gradient descent, SPSA, and ADAM. The main contribution of this work lies in a systematic methodological exploration of how these configuration choices interact to influence VQE performance, establishing a structured benchmark for selecting optimal settings in quantum chemical simulations. Key findings show that parameter initialization plays a decisive role in the algorithm's stability, and that the combination of a chemically inspired ansatz with adaptive optimization yields superior convergence and precision compared to conventional approaches.

Figures (7)

• Figure 1: Motivational overview of the VQE framework for estimating the ground-state energy of the silicon atom. Panel (A) presents the physical system and its electronic Hamiltonian, defining the objective of obtaining the ground-state energy $E_0$. Panel (B) illustrates the hybrid quantum--classical VQE loop used to approximate $E_0$ through parameterized quantum circuits and iterative optimization. Panel (C) shows the convergence of the estimated energy toward $E_{\text{exact}}$. While the ultimate goal is precise energy estimation, the effectiveness of VQE critically depends on several design choices, including ansatz structure, optimizer configuration, and parameter initialization, motivating the comparative analysis conducted in this study.
• Figure 2: Schematic of the VQE workflow, starting with the preparation of parameterized quantum states, then measuring observables and feeding the results to a classical optimizer, which iteratively updates the parameters.
• Figure 3: An overview of our proposed methodology.
• Figure 4: Circuit structures of the ansatz architectures investigated in this study: (a) Double Excitation Gates Ansatz, (b) Particle-Conserving U2 Ansatz (composed of RX, RZ, and CNOT gates), (c) UCCSD Ansatz (including Fermionic Single and Double excitation gates), and (d) k-UpCCGSD Ansatz (implemented with three layers of the UCCSD ansatz).
• Figure 5: Energy convergence comparison for different parameter initialization strategies across the selctect ansatz: (a) Random, (b) Zero, (c) Half, and (d) One.