From Quantum Chemistry to Verifiable Quantum Execution: SoftQuantus QCOS and the VQE Algorithm

From Quantum Chemistry to Verifiable Quantum Execution: SoftQuantus QCOS and the VQE Algorithm A Technical Research Study on Molecular Hydrogen Simulation, Hybrid Quantum-Classical Computing, and Cryptographically Auditable Scientific Execution SOFTQUANTUS | Quantum Computing Research & Engineering Research basis: Oscar Vicente, Scientific Researcher Research date: October 13, 2026 Technical collaboration and partnerships: roytman@softquantus.com Website: https://softquantus.com Abstract Quantum computing has the potential to transform computational chemistry by enabling new methods for studying molecular electronic structures. However, transforming quantum algorithms into reliable scientific services requires more than access to quantum processing units. Researchers must address circuit preparation, Hamiltonian decomposition, hardware compatibility, statistical uncertainty, classical optimization, and computational provenance. SoftQuantus is investigating these challenges through QCOS (Quantum Computer Operating System), a hybrid quantum-classical execution platform designed to connect scientific algorithms with simulators and physical quantum computing infrastructures. This article presents the technical methodology of a Variational Quantum Eigensolver (VQE) research demonstration focused on calculating the ground-state energy of molecular hydrogen (H₂). The research, developed by Oscar Vicente as part of SoftQuantus's Prototype & Technical Demonstration milestone, specifies an explicit implementation based on PySCF, Qiskit-compatible quantum primitives, OpenQASM 2.0 circuit serialization, classical optimization, and QCOS runtime integration. A defining objective of the study is to associate quantum computation with structured telemetry and cryptographically verifiable execution evidence. The work provides a reproducible experimental design for evaluating the portability, accuracy, and operational characteristics of hybrid quantum workloads. The benchmark results presented in the research specification remain experimental targets until confirmed by completed execution records. Keywords: Quantum Computing, QCOS, Variational Quantum Eigensolver, VQE, Quantum Chemistry, Molecular Hydrogen, Qiskit, OpenQASM, Hybrid Computing, Computational Provenance, Scientific Reproducibility.

The Scientific Problem: Molecular Ground-State Energy Accurately determining the electronic ground-state energy of a molecular system is a fundamental problem in computational chemistry. The electronic structure of a molecule is described by a many-electron Hamiltonian incorporating kinetic energy, electron–nucleus interactions, and electron–electron repulsion. Traditional electronic structure methods, including Full Configuration Interaction (FCI), can provide highly accurate solutions within a specified basis set. However, the computational resources required for exact treatments grow rapidly with the number of molecular orbitals. Quantum algorithms offer an alternative representation of electronic structure problems by mapping fermionic operators onto qubit operators. SoftQuantus's research focuses on molecular hydrogen (H₂), a well-established reference system for studying quantum chemistry algorithms. The proposed experimental configuration uses a hydrogen–hydrogen bond distance of approximately 0.735 Å and the STO-3G minimal basis set. This controlled problem provides a basis for comparing quantum simulation outputs with classical reference calculations.

Variational Quantum Eigensolver: The Hybrid Mathematical Model The Variational Quantum Eigensolver combines parameterized quantum circuits with classical numerical optimization. Its purpose is to approximate the minimum eigenvalue of a Hamiltonian by minimizing the expectation value of that Hamiltonian over a family of trial quantum states. The central objective is: E(θ) = ⟨ψ(θ)|H|ψ(θ)⟩ where H is the molecular Hamiltonian, |ψ(θ)⟩ is the parameterized quantum state, and θ represents the tunable parameters of the circuit. The optimization seeks: E = minθ E(θ)* Under the variational principle, an ideal expectation value obtained from a normalized trial state cannot fall below the exact ground-state energy of the same Hamiltonian. Practical experiments require additional attention to finite-shot sampling, noise, measurement estimation, and optimizer convergence. In the SoftQuantus research specification, this iterative computation is divided between a quantum execution layer and a classical numerical optimizer. 2.1 Hamiltonian decomposition The molecular Hamiltonian is converted from its fermionic representation into a qubit operator through the Jordan–Wigner transformation. The resulting operator is represented as a linear combination of Pauli strings: Hq = Σₖ cₖPₖ Here, cₖ denotes a real-valued coefficient and Pₖ is a tensor product of Pauli operators from the set {I, X, Y, Z}. After applying an appropriate symmetry-reduction procedure, the research proposes a compact representation using two active qubits. A representative reduced Hamiltonian contains terms of the form: Hq = c₀II + c₁IZ + c₂ZI + c₃ZZ + c₄XX The coefficients depend on the molecular geometry, basis, mapping conventions, and reduction procedure. Their numerical values must be generated consistently with the underlying electronic structure calculation. 2.2 Parameterized quantum ansatz The research evaluates two possible circuit constructions: a hardware-efficient parameterized ansatz and a simplified Unitary Coupled Cluster approach. The quantum state is prepared through a parameterized unitary transformation: |ψ(θ)⟩ = U(θ)|ψref⟩ The ansatz determines the family of quantum states that can be explored by the optimizer. A suitable circuit should balance expressibility, physical relevance, computational depth, and compatibility with the selected execution environment. 2.3 Explicit Pauli-basis measurements Instead of relying exclusively on high-level VQE wrappers, the research specifies explicit measurement circuits for individual Pauli components. Z observables are measured directly in the computational basis. X observables require a Hadamard rotation before measurement. Y observables use an S† rotation followed by a Hadamard rotation. Identity terms are evaluated classically. For a Pauli measurement whose computational outcomes are mapped to parity eigenvalues, the expectation-value estimator is: ⟨Pₖ⟩ = (N_even − N_odd) / N_total The molecular energy is then reconstructed from the weighted expectation values: E(θ) = Σₖ cₖ⟨Pₖ⟩θ + E_nuclear where E_nuclear is the nuclear repulsion contribution when it is not already included in the Hamiltonian coefficients. This explicit decomposition enables individual quantum measurements to be inspected and compared independently. QCOS: An Execution Layer for Scientific Quantum Workloads A central component of the research is the integration of the VQE algorithm with the QCOS runtime. The proposed workflow serializes quantum circuits into OpenQASM 2.0 and submits them through the QCOS execution interface: POST {BASE_URL}/runtime/jobs The architecture separates scientific algorithm development from the computational environment responsible for running the circuits. The specified workflow follows this sequence: Molecular Structure → Electronic Hamiltonian → Qubit Mapping → Parameterized Ansatz → Pauli Measurement Circuits → OpenQASM 2.0 → QCOS Runtime → Simulator or QPU → Measurement Results → Classical Optimization → Final Energy and Execution Evidence This abstraction is designed to support portability between simulation environments and compatible physical quantum hardware. Physical QPU execution remains a validation objective of the demonstration; it should not be interpreted as a completed multi-backend benchmark. Classical Optimization and Quantum Execution The VQE optimization process requires repeated evaluations of the molecular energy. For each iteration, the classical optimizer selects a parameter vector, the quantum circuit is reconstructed, the execution backend produces measurement results, and the energy estimator returns a scalar objective value. The research proposes COBYLA for initial simulator experiments and SPSA for experiments involving noisy quantum hardware. COBYLA is a derivative-free optimization method suitable for low-dimensional optimization problems. SPSA estimates parameter updates using stochastic perturbations and can be useful when measurement noise makes conventional gradient estimation expensive. The research does not assume that one optimizer is universally superior. Convergence, sample complexity, and robustness must be evaluated experimentally.

Cryptographically Verifiable Quantum Research An important objective of QCOS is to move beyond the generation of numerical outputs toward structured execution evidence. The research identifies three evidence categories required from the execution system: Execution identity: A unique job identifier that associates the workload with its execution record. Runtime telemetry: Measurements such as wall-clock execution time, used to evaluate operational overhead. Cryptographic audit evidence: Bundle hashes and Merkle-root information that allow recorded artifacts to be checked for integrity. These mechanisms are intended to support traceability between a scientific workload, its execution configuration, and the resulting evidence package. Cryptographic verification can establish that an archived artifact matches an expected integrity record. It does not, by itself, establish the correctness of the molecular model or prove that a physical quantum device produced the reported measurements. This distinction is essential for scientifically defensible evidence. Proposed Benchmark: Molecular Hydrogen Ground-State Energy The research specifies an experimental comparison between an exact classical reference, a QCOS-integrated Aer simulation, and a physical QPU implementation. Metric Exact reference QCOS Aer simulation target Physical QPU Ground-state energy −1.1373 Ha Approximately −1.135 Ha To be measured Absolute energy error Reference: 0 Below 3 mHa Backend-dependent Circuit depth Not applicable Target: 3 After hardware mapping Shots per Pauli term Exact expectation 1,024 2,048 or 4,096 Wall-clock latency Not applicable Preliminary target: ~3.5 ms Queue and execution dependent Provenance Reference calculation QCOS execution evidence QPU execution evidence Research status: These figures are the proposed reference values and performance targets from the October 13, 2026 technical specification. They are not reported here as independently verified benchmark results. The experimental study is designed to evaluate energy-estimation accuracy, circuit compilation effects, statistical variability, execution latency, and evidence completeness. A meaningful simulator-versus-QPU comparison must retain the compiled circuit characteristics and execution configuration for each backend. Otherwise, observed differences may reflect compilation changes rather than the physical hardware alone.

Scientific Services Enabled by the QCOS Research Architecture The H₂ VQE study illustrates several scientific service categories that SoftQuantus can develop and deliver through QCOS engineering engagements. Quantum chemistry algorithm engineering: Formulation of molecular Hamiltonians, active-space selection, qubit mappings, ansatz development, and VQE implementation. Quantum simulation and numerical benchmarking: Comparisons against exact classical references, evaluation of finite-shot error, optimizer studies, and uncertainty analysis. Hybrid quantum-classical workflow integration: Connections between scientific Python environments, OpenQASM-compatible circuit execution, classical optimizers, and QCOS APIs. Physical quantum hardware feasibility studies: Adaptation of representative quantum circuits to supported hardware, collection of backend-specific metrics, and evaluation of actual execution constraints. Scientific reproducibility and verification: Preparation of experiment metadata, telemetry, result records, integrity evidence, and auditable research reports. These services can support university laboratories, computational chemistry teams, materials researchers, pharmaceutical R&D organizations, HPC centers, and quantum software developers. The H₂ experiment establishes a methodological starting point. Applications to larger molecular systems require separate validation of accuracy, cost, algorithmic complexity, and hardware feasibility. Research Milestone and Required Validation The October 2026 specification defines the project as a Prototype & Technical Demonstration, identified as Milestone 2. Its required deliverables comprise a self-contained executable Jupyter Notebook, isolated configuration with no hardcoded API keys, an explicitly implemented VQE optimization loop, validated QCOS runtime integration, and an exported audit trail containing the relevant cryptographic evidence. Completion of this milestone requires actual execution records and a reproducible benchmark report. The specification alone does not establish that every deliverable has been completed. Future research should extend the experimental methodology to repeated trials, hardware noise sensitivity, compiler effects, reference-model comparisons, and execution provenance across additional computational environments.

Conclusion The SoftQuantus VQE research demonstrates a practical architectural direction for quantum computing infrastructure. The scientific goal is not merely to calculate the energy of a hydrogen molecule. It is to establish a structured process through which a molecular problem can be transformed into executable quantum circuits, evaluated by a hybrid optimizer, compared with classical references, and associated with verifiable computational evidence. QCOS is being developed as the software execution layer supporting this process. The value of such infrastructure lies in separating quantum algorithm development from execution complexity while preserving the information required for reproducibility and scientific evaluation. As quantum hardware and hybrid computing environments evolve, this methodology can provide a foundation for increasingly complex research workloads. SoftQuantus: From Quantum Algorithms to Reproducible, Verifiable Scientific Computing.

Research Attribution Scientific research: Oscar Vicente Original study: Hands-on Quantum Computing with SoftQuantus: From Qiskit to QCOS to Real Quantum Execution Date: October 13, 2026 Research program: SoftQuantus — QCOS Prototype & Technical Demonstration, Milestone 2 Research Collaboration and Technical Partnerships SoftQuantus welcomes technical collaboration with universities, quantum computing research groups, computational chemistry specialists, HPC laboratories, and industrial R&D organizations. Contact: roytman@softquantus.com Company: SoftQuantus innovative OÜ Website: https://softquantus.com Open-source research: https://github.com/softquantus Scientific AI ecosystem: https://huggingface.co/softquantus Tags: #computer science #quantum software #VQE