quant-phApr 12, 2026
SaveAdaptive H-EFT-VA: A Provably Safe Trajectory Through the Trainability-Expressibility Landscape of Variational Quantum Algorithms
Organizations: Department of Physics, International University of Africa, Khartoum, Sudan
Abstract
H-EFT-VA established a physics-informed solution to the Barren Plateau (BP) problem via a hierarchical EFT UV-cutoff, guaranteeing gradient variance in Omega(1/poly(N)). However, localization restricts the ansatz to a polynomial subspace, creating a reference-state gap for states distant from |0>^N. We introduce Adaptive H-EFT-VA (A-H-EFT) to navigate the trainability-expressibility tradeoff by expanding the reachable Hilbert space along a safe trajectory. Gradient variance is maintained in Omega(1/poly(N)) if sigma(t) <= 0.5/sqrt(LN) (Theorem 1). A Safe Expansion Corollary and Monotone Growth Lemma confirm expansion without discontinuous jumps. Benchmarking across 16 experiments (up to N=14) shows A-H-EFT achieves fidelity F=0.54, doubling static H-EFT-VA (F=0.27) and outperforming HEA (F~0.01), with gradient variance >= 0.5 throughout. For Heisenberg XXZ (Delta_ref=1), A-H-EFT identifies the negative ground state while static methods fail. Results are statistically significant (p < 10^-37). Robustness over three decades of hyperparameters enables deployment without search. This is the first rigorously bounded trajectory through the VQA landscape.
Figures & tables
Figure 1: Gradient variance scaling under adaptive training. Left (Phase I, static EFT initialization ): power-law decay from at to at , all depths nearly coincident, confirming depth-independent BP avoidance. Right (Phase II, ): gradient variance ranges from ( ) to ( ), remaining – times larger than HEA’s floor [ 9 ] . All values are mean s.e. over 50 seeds.
Figure 2: Critical cutoff sweep ( , , 50 seeds). vs. (log scale). The theoretical prediction (red dashed) marks the onset of gradient variance collapse, verified experimentally. The gradient variance is flat at for , peaks near at , then collapses to at . Error bars: s.e. over 50 seeds.
Figure 3: Gradient norm across Phase I Phase II ( , ). Mean s.d. over 10 independent runs. Phase I shows damped oscillatory convergence from to . Phase II (steps ) shows stable non-vanishing gradients with no spike, validating Corollary 1 experimentally.
Figure 4: Noiseless convergence: A-H-EFT vs. baselines ( seed, 200 steps). Nine panels for . A-H-EFT (blue) converges to deeper minima than static H-EFT-VA (green, dashed) at all system sizes. HEA (orange, dotted) stagnates near , unable to make progress past the barren plateau. The performance gap widens with and is depth-independent for A-H-EFT.
Figure 5: Final energy vs. system size ( , 200 steps). A-H-EFT (blue solid) and static H-EFT-VA (green dashed) both improve linearly with , but A-H-EFT maintains a consistent additional advantage of – energy units. HEA (orange dotted) fails entirely for .
Figure 6: Ground-state fidelity ( , ). Mean s.d. over 5 seeds. A-H-EFT (blue): rises from at to at . Static H-EFT-VA (green): plateau at , depth-independent. HEA (orange): near-zero fidelity with high variance, reflecting BP stagnation.
Figure 7: Reference-state gap vs. . TFIM (blue): monotone growth from ( ) to ( ), quantifying increasing inaccessibility with system size. Heisenberg XXZ (orange): saturates at for all , representing maximal inaccessibility for any -anchored ansatz.
Figure 8: Effective Hilbert space expansion ( , ). (amplitude count above ) sampled every 5 steps. rises rapidly from at to by and remains saturated throughout the remaining 195 steps. This confirms Lemma 1 : smooth, monotone, bounded expansion with no discontinuous jump.
Figure 9: Von Neumann entropy vs. depth ( , 15 random samples per ). Architecture capacity measured at (A-H-EFT Phase II, blue) and (Static, green), vs. HEA (orange, uniform ). A-H-EFT Phase II and Static H-EFT-VA are both near , while HEA grows to at . The paradox is resolved by Fig. 10 : entanglement at initialization is not the operative metric.
Figure 10: Expressibility (mean purity ) vs. depth ( , 500 samples). Lower purity = higher expressibility. Static H-EFT-VA (green): purity , confined to pure product states. A-H-EFT Phase II (blue): purity , occupying the productive intermediate regime. HEA (orange): purity descends from to near the Haar limit ( , red dotted), maximally expressive but untrainable. The shaded region between Static and HEA represents the “trainability–expressibility tradeoff window,” and A-H-EFT is the first method to navigate it with a provable BP guarantee.
Figure 11: Noise robustness of adaptive training ( , ). Energy vs. step under depolarizing noise . A-H-EFT converges to for ( degradation) and for ( degradation), remaining trainable at all noise levels.
Figure 12: Finite-shot gradient estimator MSE ( , ). MSE vs. shot count (log-log). A-H-EFT (blue) and HEA (orange) both show shot-noise scaling. A-H-EFT MSE at , at . HEA’s lower MSE reflects larger gradient magnitudes, but those gradients carry no optimization signal in the BP regime.
Figure 13: Switch criterion sensitivity ( , , 10 seeds). Mean final energy vs. . Performance is flat at for . Only at does marginal degradation occur ( ), confirming robustness over three decades.
Figure 14: Growth constant sensitivity ( , , 10 seeds). Mean final energy vs. . Performance is identical at across all values. The safety clamp at (annotated) renders irrelevant to final outcome: any reaches the clamp within the 200-step budget.
Figure 15: Three-way statistical comparison ( seeds, Welch’s -test). Mean final energy s.d. (left axis) and -values (right axis, log scale). A-H-EFT vs. static H-EFT-VA (purple): from ( ) to ( ). A-H-EFT vs. HEA (red): from ( ) to ( ). Significance increases monotonically with , mechanistically consistent with the growing reference-state gap (Fig. 7 ).
Figure 16: Heisenberg XXZ gradient variance scaling. Phase I (left): clean power-law decay – (lower than TFIM due to larger ). Phase II (right): maintains throughout, confirming Theorem 1 on a second Hamiltonian family.
Figure 17: Heisenberg XXZ convergence ( ). A-H-EFT (solid): , , . Static H-EFT-VA (dashed): stagnates at positive energies , , —a qualitative failure when . A-H-EFT resolves this qualitative failure by Phase II expansion into the full Hilbert space.