Energy Landscape Plummeting in Variational Quantum Eigensolver: Subspace Optimization, Non-iterative Corrections and Generator-informed Initialization for Improved Quantum Efficiency

TL;DR

This paper tackles the resource-related bottlenecks of variational quantum eigenproblem solvers (VQE) on NISQ devices by introducing AD(X)-ASC, a framework that decouples the variational parameter space into a low-dimensional principal subspace and a high-dimensional auxiliary subspace via a temporal hierarchy and an adiabatic approximation. It reconstructs the effects of auxiliary parameters through a principal-to-auxiliary mapping and incorporates them as one-step posteriori auxiliary subspace corrections (ASC) in the cost function, avoiding extra quantum hardware or iterative optimization. The authors demonstrate two practical implementations for selecting the principal subspace: ADAPT-VQE and MP2-based MP2S-VQE, showing up to 1–2 orders of magnitude improvements in energy minima on PES for molecular systems, both in noiseless and noisy simulations. Additionally, they introduce a generator-informed initialization to accelerate convergence, further reducing quantum-measurement costs. Overall, AD(X)-ASC constitutes a general, resource-efficient strategy to mitigate local traps and barren plateaus in VQE, with significant practical implications for scalable quantum chemistry on near-term devices.

Abstract

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 on to 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 minima 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. Additionally, 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 evidences of alleviating the effects of local traps, facilitating convergence toward a more optimal minimum.

Figures (5)

• Figure 1: Energy optimization landscape for AD(ADAPT-VQE)-ASC with recycled initialization: The x-axis shows the number of parameters in the ansatz and the left y-axis corresponds to the logarithm of the difference in energy with FCI (the global minimum) for stretched geometries of (a) $H_4$ linear chain ($R_{H-H}=1.75\text{\AA}$), (b) $BeH_2$ ($R_{Be-H}=2.0\text{\AA}$) and (c) $LiH$ ($R_{Li-H}=4.0\text{\AA}$). The right y-axis shows the number of expectation evaluations that includes both gradient calculations for operator selection and function evaluations for optimization. The initial energy trajectory is for conventional ADAPT-VQE. Once the convergence criteria is met with the threshold $\epsilon=10^{-3}$, ADAPT-VQE exits the selection and optimization cycle and one-step ASC is performed at the last step. It shows the signature plummeting effect of ASC where energy reaches a more optimal minima for all the cases shown here. The corresponding measurement overhead for ASC is also shown in the grey shaded area along the right y-axis. The chemical accuracy is taken to be $1.6mE_h$ energy difference from FCI.
• Figure 2: Comparison between ADAPT-VQE and AD(ADAPT-VQE)-ASC for different molecular geometries: The three columns are for linear $H_4$ chain, $BeH_2$ and $LiH$ molecule. Here y axes in all the subplots correspond to internuclear distances. The y-axes in (a), (b), (c) show the energy difference from FCI and (d), (e), (f) correspond to the associated number of CNOT gates. The solid and dotted lines correspond to ADAPT-VQE and AD(ADAPT-VQE)-ASC respectively with a specific threshold $\epsilon$. Note that AD(ADAPT-VQE)-ASC does not require additional CNOT gates in the circuit, so number of gates for it remains the same with ADAPT-VQE.
• Figure 3: Comparison between MP2S-VQE and AD(MP2S-VQE)-ASC for different molecular geometries: The axes information are same as Fig. \ref{['ad(adapt)-asc_CCSD(TQ)_pot_en_surf']} with MP2S-VQE performed with only one threshold $\bar{\epsilon}=10^{-4}$. The dUCCSD energy and CNOT count values are also shown here for comparison. Note that the CNOT count for AD(MP2S-VQE)-ASC is same as MP2S-VQE.
• Figure 4: Noisy simulation: Simulation with depolarising noise channel for $H_4$ at $1.5\text{\AA}$ with principal space selection done by MP2 screening at different thresholds $\bar{\epsilon}$. Energy in $E_h$ is plotted in y-axis and x-axis corresponds to number of iterations. The corresponding noiseless simulations are also shown for better comprehension of the disruptive effects of noise.
• Figure 5: Comparison between generator-informed, recycled and HF initialization for ADAPT-VQE: Energy error (in logarithm scale) for ADAPT-VQE ($\epsilon=10^{-5}$) with respect to FCI is plotted as a function of the number of cost-function evaluations for three different molecules. The numerical studies suggest generator-informed initialization converges much faster compared to HF or recycled initializations. For both $BeH_2$ and $LiH$, the generator-informed initialization shows almost 50% reduction in number of function evaluations compared to recycled initialization. The ASC (dotted part of the green curve) leads to better minima with almost no additional quantum resources.