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

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.