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Resource-Efficient Quantum VQE via Classical Derivative Offloading: Bypassing the Noise Wall via GaLore-Sophia

Aura, Archon, Jesse Caldwell

Published Jun 9, 2026• 0 citations
Publisher

Abstract

Variational quantum algorithms, such as the Variational Quantum Eigensolver (VQE), are severely limited on superconducting quantum processing units (QPUs) by physical shot noise, vanishing gradients, and metered credit limits. Standard gradient-descent algorithms (e.g., SGD, Adam) are unstable under the large variance of noisy estimators, often diverging or getting trapped in local minima. In this work, we present a resource-efficient optimization framework that combines low-rank projection (GaLore [1]) and second-order diagonal Hessian curvature scaling (Sophia [2]) to bypass these noise barriers in non-Abelian variational spaces (where the sequential Givens rotations do not commute and thus do not form an Abelian subgroup of U(N)). By offloading gradient and Hessian evaluations to a classical processor using a polynomial fermionic shadow tomography state reconstruction [3, 4], we eliminate parameter-shift circuit execution overhead and bypass the QPU noise floor. We execute this closed-loop GaLore-Sophia VQE on the 156-qubit IBM Heron r2 architecture (ibm_marrakesh), achieving a convergence energy of −1.378 Hartrees at step 3—reaching 97.4% of the topologically restricted 3-parameter Givens chain geometric limit of -\sqrt{2} Hartrees in under 70 seconds of QPU time. We clarify that this benchmark is measured exclusively against this sub-optimal parameter subspace rather than the exact molecular ground state of -\sqrt{5} Hartrees, which requires richer ansatz topologies. Furthermore, we analyze the parameter-count stability of the ansatz, proving that the VQE ansatz limit systematically approaches the ground state energy as parameter count scales. Finally, to rigorously validate the stability of quantum information preservation, we execute a physical temporal parity sweep on ibm_marrakesh (Job ID: d8ishf1e8nrc73bint50, 1000 shots). Fitting the physical decay yields a Lyapunov decay time τL ≈ 6.13 steps and prediction horizon τ ≈ 4.68 steps (at 86.5% accuracy threshold), confirming the universal DHP ratio τ / τL ≈ 0.763 on generational Heron hardware, matching the stable cross-device trend observed in P25 [5] and P27 [6].

DOI
10.5281/zenodo.20607554

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Variational quantum algorithms, such as the Variational Quantum Eigensolver (VQE), are severely limited on superconducting quantum processing units (QPUs) by physical shot noise, vanishing gradients, and metered credit limits. Standard gradient-descent algorithms (e.g., SGD, Adam) are unstable under the large variance of noisy estimators, often diverging or getting trapped in local minima. In this work, we present a resource-efficient optimization framework that combines low-rank projection (GaLore [1]) and second-order diagonal Hessian curvature scaling (Sophia [2]) to bypass these noise barriers in non-Abelian variational spaces (where the sequential Givens rotations do not commute and thus do not form an Abelian subgroup of U(N)). By offloading gradient and Hessian evaluations to a classical processor using a polynomial fermionic shadow tomography state reconstruction [3, 4], we eliminate parameter-shift circuit execution overhead and bypass the QPU noise floor. We execute this closed-loop GaLore-Sophia VQE on the 156-qubit IBM Heron r2 architecture (ibm_marrakesh), achieving a convergence energy of −1.378 Hartrees at step 3—reaching 97.4% of the topologically restricted 3-parameter Givens chain geometric limit of -\sqrt{2} Hartrees in under 70 seconds of QPU time. We clarify that this benchmark is measured exclusively against this sub-optimal parameter subspace rather than the exact molecular ground state of -\sqrt{5} Hartrees, which requires richer ansatz topologies. Furthermore, we analyze the parameter-count stability of the ansatz, proving that the VQE ansatz limit systematically approaches the ground state energy as parameter count scales. Finally, to rigorously validate the stability of quantum information preservation, we execute a physical temporal parity sweep on ibm_marrakesh (Job ID: d8ishf1e8nrc73bint50, 1000 shots). Fitting the physical decay yields a Lyapunov decay time τL ≈ 6.13 steps and prediction horizon τ ≈ 4.68 steps (at 86.5% accuracy threshold), confirming the universal DHP ratio τ / τL ≈ 0.763 on generational Heron hardware, matching the stable cross-device trend observed in P25 [5] and P27 [6].
Variational quantum algorithms (VQAs) are a promising approach for leveraging near-term quantum computers, but trainability and quantum-resource costs remain major challenges. Here we address these challenges with the \textbf{H}ardware \textbf{E}fficient and dynamical \textbf{LI}e algebra supported \textbf{A}nsatz (HELIA) and two training methods combining the classical-enhanced \g method with the quantum-based Parameter-Shift Rule (PSR). This distributes gradient estimation and training across classical and quantum hardware. We evaluate ground-state estimation of 6- to 18-qubit Hamiltonians using the Variational Quantum Eigensolver (VQE) and quantum phase classification for Hamiltonians up to 12 qubits. For VQE, our methods improve accuracy and success rates, with a mean reduction in estimated logical quantum evaluations of up to 60%60\%60% relative to Full-PSR training of the same ansatz. For classification, the largest improvement in mean peak test accuracy relative to Full-PSR is 2.82.82.8 percentage points. We also use these methods as a warm-start pretraining strategy for Hamiltonians with exponential-dimensional dynamical Lie algebras. Gradient diagnostics for the studied Hamiltonian configurations show slower gradient-variance decay than the selected deep hardware-efficient reference over the simulated system sizes, potentially paving the way for training large-qubit quantum models.
Variational Quantum Eigensolver (VQE) faces significant challenges due to hardware noise and the presence of barren plateaus and local traps in the optimization landscape. To mitigate the detrimental effects of these issues, we introduce a general formalism that optimizes hardware resource utilization and accuracy by projecting VQE optimizations onto a reduced-dimensional subspace, followed by a set of posteriori corrections. Our method partitions the ansatz into a lower dimensional principal subspace and a higher-dimensional auxiliary subspace based on a conjecture of temporal hierarchy present among the parameters during optimization. The adiabatic approximation exploits this hierarchy, restricting optimization to the lower dimensional principal subspace only. This is followed by an efficient higher dimensional auxiliary space reconstruction without the need to perform variational optimization. These reconstructed auxiliary parameters are subsequently included in the cost-function via a set of auxiliary subspace corrections (ASC), leading to a "plummeting effect" in the energy landscape toward a more optimal minimum without utilizing any additional quantum hardware resources. Numerical simulations show that, when integrated with any chemistry-inspired ansatz, our method can provide one to two orders of magnitude better estimation of the minima. In addition, based on the adiabatic approximation, we introduce a novel initialization strategy driven by unitary rotation generators for accelerated convergence of gradient-informed dynamic quantum algorithms. Our method shows heuristic evidence of alleviating the effects of local traps, facilitating convergence toward a more optimal minimum.
The Variational Quantum Eigensolver (VQE) is one of the most promising algorithms for current quantum devices. It employs a classical optimizer to iteratively update the parameters of a variational quantum circuit in order to search for the ground state of a given Hamiltonian. The efficacy of VQEs largely depends on the optimizer employed. Recent studies suggest that Quantum Natural Gradient Descent (QNG) can achieve faster convergence than vanilla gradient descent (VG), but at the cost of additional quantum resources to estimate Fubini-Study metric tensor in each optimization step. The Fubini-Study metric tensor used in QNG is related to the entire quantum state space and does not incorporate information about the target Hamiltonian. To take advantage of the structure of the Hamiltonian and address the limitation of additional computational cost in QNG, we propose Hamiltonian-aware Quantum Natural Gradient Descent (H-QNG). In H-QNG, we propose to use the Riemannian pullback metric induced from the lower-dimensional subspace spanned by the Hamiltonian terms onto the parameter space. We show that H-QNG inherits the desirable features of both approaches: the low quantum computational cost of VG and the reparameterization-invariance of QNG. We also validate its performance through numerical experiments on molecular Hamiltonians, showing that H-QNG achieves faster convergence to chemical accuracy while requiring fewer quantum computational resources.
The exponential runtime of quantum simulators on classical machines, along with long queue times and high costs associated with real quantum devices, presents significant challenges for the efficient optimization of Variational Quantum Algorithms (VQAs) such as the Variational Quantum Eigensolver (VQE), Quantum Approximate Optimization Algorithm (QAOA), and Quantum Neural Networks (QNNs). To address these limitations, we propose a novel approach, DyPP (Dynamic Parameter Prediction), which accelerates the convergence of VQAs by leveraging regular trends in the parameter weights to update parameters efficiently. We introduce two techniques for optimal prediction performance: Naive Prediction (NaP) and Adaptive Prediction (AdaP). Through extensive experimentation and training of multiple QNN models on various datasets, we demonstrate that DyPP achieves a speedup of approximately 2.25×compared to standard training methods, while also providing improved accuracy (up to 2.3% higher) and lower loss (up to 6.1% reduction) with minimal storage and computational overhead. We further evaluate DyPP’s effectiveness in VQE for molecular ground-state energy estimation and in QAOA for graph MaxCut. Our results show that, on average, DyPP leads to a speedup of up to 3.1× for VQE and 2.91× for QAOA compared to traditional optimization, while requiring up to 3.3× fewer shots (i.e., repeated circuit executions). Even in the presence of hardware noise, DyPP outperforms existing optimization techniques, delivering up to 3.33× speedup and requiring 2.5× fewer shots, thereby enhancing the efficiency of VQAs.
Hybrid quantum-classical adaptive Variational Quantum Eigensolvers (VQE) hold the potential to outperform classical computing for simulating many-body quantum systems. However, practical implementations on current quantum processing units (QPUs) are challenging due to the noisy evaluation of a polynomially scaling number of observables, undertaken for operator selection and high-dimensional cost function optimization. We introduce an adaptive algorithm using analytic, gradient-free optimization, called Greedy Gradient-free Adaptive VQE (GGA-VQE). In addition to demonstrating the algorithm's improved resilience to statistical sampling noise in the computation of simple molecular ground states, we execute GGA-VQE on a 25-qubit error-mitigated QPU by computing the ground state of a 25-body Ising model. Although hardware noise on the QPU produces inaccurate energies, our implementation outputs a parameterized quantum circuit yielding a favorable ground-state approximation. We demonstrate this by retrieving the parameterized operators calculated on the QPU and evaluating the resulting ansatz wave-function via noiseless emulation (i.e., hybrid observable measurement).
Abstract The variational quantum eigensolver (VQE) is a hybrid quantum–classical algorithm for preparing ground states in the current era of noisy devices. The classical component of the algorithm requires a large number of measurements on intermediate parameter values that are typically discarded. However, intermediate steps across many calculations can contain valuable information about the relationship between the quantum circuit parameters, resultant measurements, and noise specific to the device. This information would be useful when analysing families of Hamiltonians such as for potential energy landscapes for molecular interaction with catalytic surfaces or confinement in metal–organic-frameworks. In this work, we use supervised machine learning on the intermediate parameter and measurement data to amortise prediction of optimal final parameters across families of Hamiltonians. Our technique optimises parameters leading to chemically accurate ground state energies much faster than conventional techniques. It requires significantly fewer iterations and simultaneously shows the ability to find noise-aware angles if trained on noisy devices even with noise drift. We demonstrate this technique on IBM quantum devices by predicting ground state energies of H 2 for one and two qubits; H 3 for three qubits; and HeH + for four qubits where it finds optimal angles using only modelled data for training. Our technique amortises classical optimisation across multiple VQE runs, resulting in significantly faster overall classical optimisation without additional quantum overhead compared to running VQE.