From Trainability Diagnostics to Optimization Claims: Boundaries and Controls in Variational Quantum Optimization

Pilsung Kang

quant-phSep 18, 2026

From Trainability Diagnostics to Optimization Claims: Boundaries and Controls in Variational Quantum Optimization

Authors: Pilsung Kang

Abstract

Barren plateau diagnostics characterize whether gradient signal remains available for training, but surviving signal need not translate into successful optimization. We study this trainability--optimization gap at the level of optimizer steps. Treating coefficient-weighted Hamiltonian-term gradients as task-like components, we introduce step-level diagnostics and derive an exact bridge between signed termwise organization, directional activity, and first-order descent. Resolving this bridge into standard first-order geometry shows that the apparent organization--activity factors are not independent optimization axes and that, at fixed state and update norm, the raw gradient maximizes first-order descent of the summed objective. We compare vanilla gradient descent, a deterministic Hamiltonian-term PCGrad variant, and probe-gated LSO-PCGrad on transverse-field Ising model instances with hardware-efficient and Hamiltonian variational ansatzes, together with matched controls for update norm and probe budget. Blind projection can improve an organization diagnostic while worsening final energy and first-order predictability. After conditioning on standard first-order geometry, residual term-space composition shows no reproducible material incremental association with realized descent, while optimizer-relative update norm shows positive material associations in some settings without cross-regime reproducibility. Matched controls provide no resolved final-energy benefit attributable to the projected direction, and the improvement of LSO-PCGrad is more consistent with probe-based search and step-norm adaptation than with Hamiltonian-term projection itself. These results show that gradient-structure diagnostics can characterize trainability and update geometry without serving as standalone evidence of optimization benefit, which requires controls matched on update norm and search budget.

Figures & tables

Panel of Figure 1 — Figure 1: Overview of the analysis. The exact step-level bridge is rewritten in standard first-order geometry, which exposes a fixed-norm first-order boundary and, on the nondegenerate domain, a residual term-space coordinate $C$ . The lower branches summarize the association and matched-control tests and their main conclusions. Notation is defined in Sections 3 and 4 .

Update rule	Update direction	Projection control
Vanilla	$\mathbf{u}_{\mathrm{van}}=\mathbf{g}$	None (equiv. $\lambda=0$ )
PCGrad	$\mathbf{u}_{\mathrm{pc}}$	Full projection (uncapped)
LSO-PCGrad	$\mathbf{u}_{\mathrm{van}}+\lambda\,\tilde{\boldsymbol{\delta}}$	Probe-gated capped interpolation, $\lambda\in[0,1]$
Matched control rules (Section 3.3 )
Norm-matched PCGrad	$\\|\mathbf{u}_{\mathrm{van}}\\|\,\mathbf{u}_{\mathrm{pc}}/\\|\mathbf{u}_{\mathrm{pc}}\\|$	Full projection; norm matched, Eq. ( 16 )
Grid-based LSO	$\mathbf{u}_{\mathrm{van}}+\lambda\,\tilde{\boldsymbol{\delta}}$	Probe-selected on the fixed grid $\lambda\in\{0,0.25,0.5,0.75,1\}$

Table 1: Primary update rules (upper block) and matched control rules (lower block) compared in this work. All methods use the same Hamiltonian-term gradient decomposition

\mathbf{g}=\sum_{\alpha}\mathbf{g}_{\alpha}

, but differ in how the update direction is constructed. Here

\mathbf{u}_{\mathrm{van}}=\mathbf{g}

denotes the vanilla direction,

\mathbf{u}_{\mathrm{pc}}

the projected aggregate direction in Eq. ( 13 ),

\tilde{\boldsymbol{\delta}}

the capped displacement in Eq. ( 14 ), and

s(\lambda)

the step-length factor of Eq. ( 18 ).

Role	Prior parameter-level diagnostics	This work (step level)
Termwise contribution	$a_{\alpha,k}$	$d_{\alpha}(\mathbf{u})=\mathbf{g}_{\alpha}^{\top}\mathbf{u}$
Survival ratio	$R_{k}$ (nonnegative)	$S_{\mathrm{step}}$ (signed)
Effective term count	$N_{\mathrm{eff},k}$	$N_{\mathrm{eff}}^{\mathrm{step}}$
Organization diagnostic	$B_{\mathrm{eff},k}$ (nonnegative)	$U_{\mathrm{step}}$ (signed)
Directional activity	$Q_{k}$	$Q_{\mathrm{step}}$
Bridge identity	$(\partial_{k}\langle H\rangle)^{2}=B_{\mathrm{eff},k}^{2}Q_{k}$	$\mathbf{g}^{\top}\mathbf{u}=U_{\mathrm{step}}\sqrt{Q_{\mathrm{step}}}$

Table 2: Correspondence between prior parameter-level diagnostics and the step-level diagnostics introduced in this work.

Ansatz	Method	Mean $\Delta B_{\mathrm{eff}}$	Mean $\Delta E_{\mathrm{final}}$	Worse conditions	Winner cases
HEA	PCGrad	$+0.451$	$-0.200$	$7/12$	$80/360$
HEA	LSO-PCGrad	$-0.029$	$-0.626$	$0/12$	$242/360$
HVA	PCGrad	$+0.697$	$+1.377$	$12/12$	$0/360$
HVA	LSO-PCGrad	$-0.002$	$-0.037$	$2/12$	$263/360$

Table 3: Summary of optimization outcomes relative to vanilla gradient descent, based on final-step values (see Section 10.3 for the aggregation protocol).

\Delta E_{\mathrm{final}}=E^{\mathrm{final}}_{\mathrm{method}}-E^{\mathrm{final}}_{\mathrm{vanilla}}

, so positive values indicate worse final energy;

\Delta B_{\mathrm{eff}}

is defined analogously. Winner cases count how often each method achieves the lowest final energy among the three primary update rules across 360 seed–condition cases (30 seeds over 12

(n,d)

settings).

Figure 2: Seed-averaged optimization trajectories for the three primary update rules at the largest tested condition,

(n,d)=(10,8)

: energy (left) and parameter-averaged

B_{\mathrm{eff}}

(right) for HEA (top) and HVA (bottom). Lines are means over 30 seeds and bands indicate one standard deviation across seeds. In HVA, PCGrad sustains an elevated

B_{\mathrm{eff}}

while its energy plateaus above the vanilla and LSO-PCGrad trajectories. The late-step oscillation in the former trajectory reflects a two-step alternation present in individual seed trajectories rather than an averaging artifact. In HEA, this condition is among the minority in which PCGrad improves final energy (cf. Table 3 ). Corresponding figures for the other conditions can be generated from the released data and plotting code.

		$\log\sqrt{Q_{\mathrm{step}}}=\log R+\log A$			$\log R=\log G+\log\nu$
Ansatz	Rule	$s_{R}$	$s_{A}$	$s_{\mathrm{cov}}$	$s_{G}$	$s_{\nu}$	$s^{\prime}_{\mathrm{cov}}$
HEA	vanilla	0.355	0.179	0.464	1.000	0.000	0.000
	pcgrad	0.173	0.491	0.378	1.309	0.650	-0.895
	lso_pcgrad	0.439	0.146	0.422	0.639	0.082	0.261
	pcgrad_nm	0.239	0.453	0.269	1.000	0.000	0.000
	lso_grid	0.456	0.147	0.407	0.639	0.083	0.259

Table 4: Log-variance decomposition of the step-level directional activity. The outer decomposition resolves

\log\sqrt{Q_{\mathrm{step}}}=\log R+\log A

, whereas the nested decomposition resolves

\log R=\log G+\log\nu

Ansatz	Rule	Est.	Median	Max. abs.	Material	Est.	Median	Max. abs.	Material
		$\rho_{C}$				$\rho_{\nu}$
HEA	lso_grid	11/12	$-0.027$	$0.181$	0/12	11/12	$0.357$	$0.637$	8/12 $(+)$
	vanilla_ls_nm	12/12	$-0.009$	$0.272$	1/12 $(-)$	12/12	$0.194$	$0.686$	6/12 $(+)$
HVA	lso_grid	12/12	$-0.027$	$0.096$	0/12	12/12	$0.173$	$0.290$	4/12 $(+)$
	vanilla_ls_nm	11/12	$0.014$	$0.062$	0/12	11/12	$0.083$	$0.140$	0/12

Table 5: Prespecified association summary for the optimizer-controlled rules eligible for the full-coordinate adjudication. Cell-level values

\bar{\rho}

are seed-aggregated partial Spearman associations. A cell is classified as material when

|\bar{\rho}|\geq 0.20

and its seed-bootstrap confidence interval excludes zero.

Ansatz	Rule	$\Delta E_{\mathrm{final}}$	Worse cond.	$\mathrm{corr}$ at $n=4$ / $6$ / $8$ / $10$
HEA	PCGrad	$-0.200$	$7/12$	$0.871$ / $0.861$ / $0.655$ / $0.791$
HEA	Norm-matched PCGrad	$+0.390$	$11/12$	$0.970$ / $0.990$ / $0.959$ / $0.972$
HVA	PCGrad	$+1.377$	$12/12$	$0.854$ / $0.805$ / $0.812$ / $0.798$
HVA	Norm-matched PCGrad	$+1.480$	$12/12$	$0.955$ / $0.923$ / $0.936$ / $0.919$

Table 6: Q1 adjudication. The first two data columns report the condition-balanced mean final-energy difference relative to vanilla gradient descent (positive values indicate worse final energy) and the number of

(n,d)

conditions ending above vanilla. The last column reports first-order predictability

\mathrm{corr}(\Delta E_{\mathrm{pred}},\Delta E_{\mathrm{act}})

by system size, pooled over depths, seeds, and transitions within each

n

Ansatz	Mean (grid $-$ vls)	95% CI	Grade	vls seed wins
HEA	$+0.267$	$[+0.1251,\,+0.4337]$	practical separation	$267/360$
HVA	$+0.034$	$[+0.0055,\,+0.0669]$	direction consistent, size unresolved	$308/360$

Table 7: Q2 adjudication. Condition-balanced mean paired final-energy difference between lso_grid and vanilla_ls_nm (positive values favor the line search on the vanilla direction), the two-level hierarchical-bootstrap 95% confidence interval, the resulting CI–margin grade at

m=0.02

, and the number of paired seeds (out of

360

) in which vanilla_ls_nm reaches the lower final energy. Column headings abbreviate lso_grid as grid and vanilla_ls_nm as vls.

Variational Quantum Algorithms

Variational quantum algorithms use parameterized quantum circuits and classical optimization to approximate solutions to quantum and optimization problems.

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Apr 20, 2026

Trainability Beyond Linearity in Variational Quantum Objectives

Barren-plateau results have established exponential gradient suppression as a widely cited obstacle to the scalability of variational quantum algorithms. When and whether these results extend to a given objective has been addressed through loss-specific arguments, but a general structural characterization has remained open. We show that the objective itself admits a fixed-observable representation if and only if the loss is affine in the measured statistics, thereby identifying the exact boundary of the standard concentration-based proof template. Existing transfer results for non-affine losses achieve this reduction under additional assumptions; our characterization implies that such a reduction is not structurally available for a class of non-affine objectives, placing them outside the automatic reach of the existing proof template. Beyond the affine regime, a chain-rule decomposition reveals three governing factors -- model responsivity, loss-side signal, and transmittance -- and induces a loss-class dichotomy: bounded-gradient losses inherit suppression, while amplification-capable losses can in principle counteract it. In the exponentially wide setting, both classes fail, but for different structural reasons. When the interface is instead designed at polynomial width -- exposing coarse-grained statistics rather than individual bitstring probabilities -- the exponential-dimensional obstruction is relaxed and the dichotomy plays a genuine role. In a numerical demonstration on a charge-conserving quantum system, the amplification-capable objective produces resolved gradients several orders of magnitude larger than affine and inheriting baselines at comparable shot budgets. Over the tested interval, its scaling trend is statistically distinguished from the exponential trend of both alternatives. The boundary is affine; what lies beyond it is a representation-design problem.

Gordon Ma, Xiufan Li

Centre for Quantum Technologies, National University of Singapore, Singapore

Barren Plateaus Variational Quantum Circuits Variational Quantum Algorithms Quantum Circuit Trainability

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Jun 8, 2026

Adaptive directional gradients for parameterised quantum circuits

Training parameterised quantum circuits (PQCs) on quantum hardware is bottlenecked by the measurement cost of gradient estimation, which under the parameter-shift rule scales linearly in the number of trainable parameters and dominates the total shot budget of training at scale. In this work, we propose a framework of forward gradient estimators for PQCs, based on the forward mode of automatic differentiation, that yields an unbiased estimator of the gradient by averaging a freely tunable number of random directional derivatives and recovers SPSA, random coordinate descent, and the parameter-shift rule as limiting cases, with no ancilla qubits or controlled-gate overhead. We prove that stochastic quantum forward gradient descent converges under standard assumptions, with an explicit second-moment expansion that interpolates between the single-direction extreme of SPSA and the full-gradient extreme of parameter-shift. Within this framework we derive QUIVER (Quantum Iterative V-adaptive Estimator Rule), an adaptive optimiser for parameterised circuits whose update rule follows from a closed-form minimum measurement-cost allocation. We show numerically that forward gradients train Hamming-weight-preserving orthogonal quantum neural networks with up to 60 qubits and 1770 parameters on the ECG5000 and MNIST datasets orders of magnitude more efficiently than the parameter-shift rule. We also demonstrate that our proposed QUIVER optimiser can outperform iCANS and gCANS measurement-frugal optimisers on optimisation problems using the quantum approximate optimisation algorithm and quantum simulation with the variational quantum eigensolver.

Brian Coyle, Snehal Raj, Virag Umathe +2

School of Informatics, University of Edinburgh, Edinburgh, UK · LIP6, CNRS, Sorbonne Université, Paris, France · QC Ware, Palo Alto, USA and Paris, France +1

Variational Quantum Circuits Adaptive Gradient Methods Variational Quantum Algorithms Quantum Neural Networks Quantum Circuit Optimization

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Mar 20, 2025

Reducing QPU Calls in Variational Quantum Algorithms via Classical Offloading of Gradient Computation

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 Hardware Efficient and dynamical LIe algebra supported Ansatz (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\%

relative to Full-PSR training of the same ansatz. For classification, the largest improvement in mean peak test accuracy relative to Full-PSR is

2.8

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.

Rahul Bhowmick, Harsh Wadhwa, Avinash Singh +3

Quantum Lab, Fujitsu Research of India · Quantum Laboratory, Fujitsu Research, Fujitsu Limited Japan

Variational Quantum Algorithms Quantum Circuit Trainability Hybrid Quantum-Classical ML

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