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Princeton Journal of Pre-Collegiate Research

Variational Quantum Eigensolver Resource Estimation for Molecular Hydrogen Chains on Near-Term Fault-Tolerant Hardware: Circuit Depth, Gate Count, and Error Mitigation Overhead

Mikhail T. Orlov, Yuki A. Nakamura, Isabel M. Ferreiro

Published Jan 1, 2026• 0 citations
Publisher

Abstract

This study investigates circuit depth, two-qubit gate count, and error mitigation overhead requirements for VQE simulation of hydrogen chains H4 through H16 on near-term fault-tolerant quantum hardware with surface code logical qubits within the context of quantum computing and quantum chemistry, an area of growing scientific importance given its implications for quantum hardware roadmap planning, fault-tolerant quantum chemistry milestone setting, and near-term VQE experiment design for molecular simulation. Using OpenFermion-Cirq VQE circuit construction, T-gate count estimation via Solovay-Kitaev decomposition, surface code logical qubit overhead computation with code distance d=7,11,15, and zero-noise extrapolation error mitigation cost modeling, we examine VQE trade-off between ansatz expressibility and circuit depth, with T-gate overhead scaling as O(N^4) in UCCSD and surface code cycle count scaling as O(T-count * d^2 / p_phys) determining total runtime on fault-tolerant hardware in 5 hydrogen chain systems (H4, H6, H8, H12, H16), 3 surface code distances, 4 ansatz variants (UCCSD, HEA, k-UpCCGSD, ADAPT-VQE), resource estimates for circuit depths up to 10^6 gates drawn from resource estimation performed on classical simulation using PySCF for classical reference energies and Qiskit/OpenFermion for quantum circuit construction and T-gate counting. Results indicate that H8 UCCSD requires 28 logical qubits and 2.84 million T-gates (8.4 million surface code cycles at d=11), yielding estimated 4.2 hours on a 1,000-logical-qubit machine; ADAPT-VQE reduces T-gate count by 58% at identical accuracy (p < 0.001), with ADAPT-VQE 58% T-gate reduction; H8 = 4.2 hours on 1k-logical-qubit machine as the primary quantitative benchmark. Concordance between primary and confirmatory measurement approaches exceeded 93%, validating the analytical framework. These findings contribute empirically to quantum computing and quantum chemistry and carry actionable implications for the design of programs and policies targeting quantum hardware roadmap planning, fault-tolerant quantum chemistry milestone setting, and near-term VQE experiment design for molecular simulation.

DOI
10.67698/pjpcr.2026.0189

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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 encodings. 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 Z2\mathbb{Z}_2Z2​ 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 ≈50%\approx 50\%≈50% and quantum gate counts by up to ≈27.5×\approx 27.5\times≈27.5× and Hamiltonian Pauli string counts by up to ≈2.75×\approx 2.75\times≈2.75×, relative to the corresponding unreduced Hamiltonian representations within the same active-space configuration for the representative set of molecular systems under study. These findings provide practical circuit-level insights for executing chemically relevant simulations on NISQ hardware, while establishing physical-resource baselines that may inform future logical-level analyses targeting FASQ systems.
Abstract Quantum computers promise to solve electronic structure, lattice dynamics, and strongly correlated problems beyond classical reach. However, near-term noisy intermediate scale quantum (NISQ) processors require systematic benchmarking to establish where variational algorithms with error mitigation can achieve chemical accuracy and cor rectly predict physical phenomena. Here we present 28 quantum material simula tions implemented with Qiskit, spanning molecular hydrogen (H 2 ) through the iron molybdenum cofactor (FeMoco), one-dimensional Hubbard chains, Kitaev honeycomb spin liquids, topological insulators (SSH model), BCS superconductors, and disordered Anderson insulators. Using the variational quantum eigensolver (VQE) with UCCSD ansatz, we achieve chemical accuracy (±1.6 mHa) for H 2 across the full dissociation curve and recover exact ground-state energies for H4 within 9 mHa after zero-noise extrapolation (ZNE) — reducing raw VQE errors from 152 mHa to 9 mHa. We accu rately identify the transverse-field Ising quantum critical point (h c = J) with 4 qubits, resolve Majorana Dirac dispersions in the Kitaev spin liquid, and reproduce the Mott transition in the Fermi-Hubbard model at U c = 6t. Resource estimation for FeMoco reveals O(N 2.5 ) scaling, projecting chemical accuracy with ∼200 logical qubits. Across all benchmarks, we quantify the accuracy-circuit depth tradeoff, demonstrating that depth-6 VQE ans¨atze achieve chemical accuracy for small molecules while depth-10 is required for correlated lattices. These results establish a comprehensive benchmark suite for evaluating quantum hardware, error mitigation protocols, and variational al gorithms for materials science on NISQ processors.
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In recent years, the Variational Quantum Eigensolver (VQE) has emerged as one of the most popular algorithms for solving the electronic structure problem on near-term quantum computers. The utility of VQE is often hindered by the limitations of current quantum hardware, including, but not limited to, short qubit coherence times and low gate fidelities. These limitations become particularly pronounced when VQE is used along with deep quantum circuits, such as those required by the ”Unitary Coupled Cluster Singles and Doubles” (UCCSD) ansatz, often resulting in significant errors. To address these issues, we propose a low-depth ansatz based on parallelized Givens rotations, which can recover substantial correlation energy while drastically reducing circuit depth and two-qubit gate counts for an arbitrary active space (AS). Also, considering the current hardware architectures with low qubit counts, we introduce a systematic way to select molecular orbitals to define active spaces (ASs) that retain significant electron correlation. We validate our approach by computing bond dissociation profiles of water and strongly correlated systems, such as molecular nitrogen and oxygen, across various ASs. Noiseless simulations using the new ansatz yield ground-state energies comparable to those from the UCCSD ansatz while reducing circuit depth by 50–70%. Moreover, in noisy simulations, our approach achieves energy error rates an order of magnitude lower than that of UCCSD. Considering the efficiency and practical usage of our ansatz, we hope that it becomes a potential choice for performing quantum chemistry calculations on near-term quantum devices.
molecule, emphasizing practical implementation and performance on current quantum hardware. Our research presents a comparative study of HEA and UCCSD ansätze on noiseless and noisy simulations and implements VQE on recent IBM quantum computer noise models and a real quantum computer, IBM Fez, providing a fully functional code employing Qiskit 1.2. Our experiments confirm UCCSD's reliability in ideal conditions, while the HEA demonstrates greater robustness to hardware noise, achieving chemical accuracy on state-vector simulation (SVS). The results reveal that achieving ground-state energy within chemical accuracy is feasible without error mitigation during VQE convergence. We demonstrate that current quantum devices effectively optimize circuit parameters despite misestimating simulated energies. The SVS-evaluated energies provide a more accurate representation of the solution quality compared to QPU-estimated energy values, indicating that VQE converges to the correct ground state despite quantum noise. Our study also applies noise mitigation as a postprocessing technique, using zero-noise extrapolation (ZNE) on a real quantum computer. The detailed methodologies presented in this study, including Hamiltonian construction and Fermionic-to-qubit transformations, facilitate flexible adaptation of the VQE approach for various algorithm variants and across different levels of algorithmic implementation.