Resource Estimation for VQE on Small Molecules: Impact of Fermion Mappings and Hamiltonian Reductions

TL;DR

The paper tackles the challenge of estimating quantum resources for VQE-based chemistry on small molecules by integrating a four-stage workflow that tracks how fermion-to-qubit mappings (JW, BK, Pa) and Hamiltonian reduction techniques (frozen-core, Z2 tapering) impact qubit counts, circuit depth, and gate counts. The approach combines theory (second quantization and FTQMs) with an end-to-end pipeline (Hamiltonian Modeling, Qubit Mapping, Ansatz Preparation, Quantum Circuit Preparation) implemented in a PySCF/Qiskit-based framework. Key findings show that appropriate mappings and symmetry-based reductions can dramatically compress resources, reducing qubits by up to about $50\%$ and two-qubit gate counts by up to roughly $45\times$ for a representative molecule set, with configuration-dependent advantages across JW, BK, and Pa. These results offer practical guidance for running chemically relevant VQE simulations on NISQ and emerging FASQ hardware, informing algorithm-hardware co-design and resource-aware benchmarking.

Abstract

Accurate determination of ground-state energies for molecules remains a challenge in quantum chemistry and a cornerstone for progress in fields such as drug discovery and materials design. The Variational Quantum Eigensolver (VQE) represents a leading hybrid quantum-classical paradigm for addressing this challenge; however, its widespread realization is limited by noise and the restricted scalability of current quantum hardware. Achieving efficient simulations on Noisy Intermediate-Scale Quantum (NISQ) devices and forthcoming Fault-Tolerant Application-Scalable Quantum (FASQ) systems demands a detailed understanding of how computational resources scale with molecular complexity and fermion-to-qubit encoding schemes. In this study, resource requirements for VQE implementations employing the Unitary Coupled Cluster Singles and Doubles (UCCSD) ansatz are systematically analyzed. The molecular Hamiltonian is formulated in second quantization and mapped to qubit operators through the Jordan-Wigner (JW), Bravyi-Kitaev (BK), and Parity (Pa) transformations. Hamiltonian reduction strategies, including $\mathbb{Z}_2$ tapering and frozen-core approximations, are examined to assess their effect on quantum resource scaling. The analysis reveals that appropriate transformations, when combined with symmetry-based reductions, can substantially reduce qubit counts by up to $\approx 50\%$ and quantum gate counts by up to $\approx 45\times$ for the representative set of molecular systems under study. This provides practical insights for executing chemically relevant simulations on NISQ and FASQ hardware.

Figures (6)

• Figure 1: Schematic representation of a diatomic molecular system described by the non-relativistic molecular Hamiltonian in Eq. (\ref{['eq:molecular_hamiltonian']}). The nuclei $A$ and $B$ (shown in red) and electrons $i$ and $j$ (shown in blue) are located by their respective position vectors $\vec{R}_A, \vec{R}_B, \vec{r}_i,$ and $\vec{r}_j$ with respect to the origin. The inter-particle vectors $\vec{r}_{iA}, \vec{r}_{jA},$ and $\vec{r}_{ij}$ are represented as solid lines, while the internuclear vector $\vec{R}_{AB} = \vec{R}_A - \vec{R}_B$ is shown as a dotted line to indicate that, under the Born--Oppenheimer approximation, the internuclear separation is treated as a fixed parameter.
• Figure 2: Workflow for resource estimation in the VQE framework, showing the sequence from molecular input to Hamiltonian modeling, qubit mapping, ansatz construction, and circuit compilation, along with key outputs at each stage.
• Figure 3: 3D molecular structures: (a) Methane ($CH_{4}$) and (b) Fluoromethane ($CH_{3}F$).
• Figure 4: Scaling behaviour of qubit and gate requirements across molecular systems using JW, BK, Pa mappings and Hamiltonian reduction configurations. (a)-(d) correspond to the first set of molecular systems $(LiH, HF, BeH_2, \text{and } H_2O)$. Here, T denotes qubit tapering via $\mathbb{Z}_2$ symmetries, and F denotes the FC approximation. Symbols indicate the applied configurations: (–) both T and F are false, (T) only tapering is applied, (F) only frozen-core is applied, and (TF) both techniques are applied. Error bars are omitted as the resource-estimation procedure is deterministic.
• Figure 4: (continued) Scaling behaviour of qubit and gate requirements across molecular systems using JW, BK, Pa mappings and Hamiltonian reduction configurations. (e)-(h) correspond to the next set of molecular systems $(NH_3, CH_4, O_2,\text{and } N_2)$. Here, T denotes qubit tapering via $\mathbb{Z}_2$ symmetries, and F denotes the FC approximation. Symbols indicate the applied configurations: (–) both T and F are false, (T) only tapering is applied, (F) only frozen-core is applied, and (TF) both techniques are applied. Error bars are omitted as the resource-estimation procedure is deterministic.