AlgoInvar: Simulation-Free Benchmarking of Quantum Hardware via Exact Algorithmic Invariants in VQE and QAOA

Overview As quantum processors scale into the utility regime (≥ 100 qubits), evaluating hardware fidelity on algorithmically relevant workloads becomes severely constrained. Prevailing methodologies such as Randomized Benchmarking (RB) characterize synthetic Clifford averages, while Cross-Entropy Benchmarking (XEB) is fundamentally bottlenecked by the exponential classical simulation cost of calculating statevectors beyond 50 qubits. In this work, we introduce AlgoInvar, an in situ, simulation-free, parameter-free benchmarking framework that provides an algorithm-specific complementary diagnostic layer by anchoring hardware evaluation to exact algebraic conservation laws and continuous physical symmetries in Variational Quantum Eigensolver (VQE) fermionic chemistry and Quantum Approximate Optimization Algorithm (QAOA) combinatorial optimization circuits. Core Mathematical Invariants Fermionic VQE Particle-Number Conservation QAOA Global Z2\mathbb{Z}_2Z2 Spin Parity Invariance Empirical Validation on 156-Qubit IBM Heron r2 Processors We demonstrate an initial physical proof-of-principle deployment on connected sub-chains of two independent 156-qubit IBM Heron r2 utility-era processors (ibm_kingston and ibm_fez): Execution Efficiency: Single-job batched executions with 8,192 shots per circuit (40,960 total physical shots per system) executed in exactly 13.0 seconds of QPU runtime. In-Sector Retention & Leakage Directional Asymmetry Diagnostic Classical Computational Scaling Controlled Simulation Suite & Active Error Mitigation 4-Channel Parameterized Noise Sweeps: Evaluated under Kraus operator models for longitudinal thermal relaxation (T1), pure phase dephasing (T2∗), coherent angle over-rotations, and asymmetric readout confusion. Active Molecular H2\mathrm{H}_2H2 VQE Symmetry Mitigation: Post-selection onto the physical 𝑤 = 2 sector suppresses runaway dissipation under composite noise, reducing energy error from 520.6 mHa down to 369.6 mHa (a 29.0% reduction). Open Science & Code Repository GitHub Repository: https://github.com/Quantum-Computing-Researcher/AlgoInvar Author: Rehan Arqam ([email protected]) Affiliation: Independent Researcher, Islamabad, 44000, Pakistan.