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Cost-Function Locality and Trainability Diagnostics for Quantum Neural Networks: An Empirical and Artifact-Level Study of Layerwise and Local-Cost Training

Fahad Abdullah, Asma Zubair, Farhan Riaz

Published Oct 19, 2026• 0 citations

Abstract

We report an empirical study of how quantum neural networks (QNNs) train under global and local costfunctions, with layerwise training included as a third point of comparison. The main experiment is a set of 180 training runson binary MNIST (digits 3 and 6), using an 8-qubit hardware-efficient ansatz written in TensorFlow Quantum. A second,separate study measures gradient landscapes at a fixed input across 48 circuit configurations with 4 to 10 qubits. Wecompare three procedures: end-to-end training with a global product-Z readout (the baseline), a layerwise curriculum thatreuses trained parameters between stages, and training on a local averaged-Z cost. Test accuracy barely moves betweenmethods. The global baseline still reaches 87.2% at 8 layers, so we see no accuracy collapse in this regime, andmatched-seed tests at 8 layers find no significant difference (Friedman p = 0.654; every pairwise paired test isnon-significant, as are the tests at 4 and 6 layers). Gradient statistics tell the methods apart much more clearly. At depths 4,6 and 8, the local cost ends training with 1.5 to 2.4 times lower gradient variance than the global baseline (paired p <= 9.4x 10⁻³ after Holm correction; Cohen's d_z up to 2.3). It also costs far more to simulate at depth: 6.57 h per run at 8 layers,against 1.66 h for the baseline. In the landscape study, the local observable's gradient variance is 1.4 to 10 times lower thanthe global observable's over the whole configuration grid, and 4 to 10 times lower at depth 8. Layerwise training follows adifferent path, with gradient variance falling during optimization, yet it gives no

DOI: 10.5281/zenodo.23477928

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We report an empirical study of how quantum neural networks (QNNs) train under global and local costfunctions, with layerwise training included as a third point of comparison. The main experiment is a set of 180 training runson binary MNIST (digits 3 and 6), using an 8-qubit hardware-efficient ansatz written in TensorFlow Quantum. A second,separate study measures gradient landscapes at a fixed input across 48 circuit configurations with 4 to 10 qubits. Wecompare three procedures: end-to-end training with a global product-Z readout (the baseline), a layerwise curriculum thatreuses trained parameters between stages, and training on a local averaged-Z cost. Test accuracy barely moves betweenmethods. The global baseline still reaches 87.2% at 8 layers, so we see no accuracy collapse in this regime, andmatched-seed tests at 8 layers find no significant difference (Friedman p = 0.654; every pairwise paired test isnon-significant, as are the tests at 4 and 6 layers). Gradient statistics tell the methods apart much more clearly. At depths 4,6 and 8, the local cost ends training with 1.5 to 2.4 times lower gradient variance than the global baseline (paired p <= 9.4x 10⁻³ after Holm correction; Cohen's d_z up to 2.3). It also costs far more to simulate at depth: 6.57 h per run at 8 layers,against 1.66 h for the baseline. In the landscape study, the local observable's gradient variance is 1.4 to 10 times lower thanthe global observable's over the whole configuration grid, and 4 to 10 times lower at depth 8. Layerwise training follows adifferent path, with gradient variance falling during optimization, yet it gives no

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Abstract Quantum neural networks (QNNs) are often challenged with the problem of flat cost function landscapes during training, known as barren plateaus (BP). A solution to potentially overcome the problem of the BP has recently been proposed by Cerezo et al In this solution, it is shown that, for an arbitrary deep quantum layer(s) in QNNs, a global cost function (all qubits measured in an n-qubit system) will always experience BP, whereas a local cost function (single qubit measured in an n-qubit system) can help to alleviate the problem of BP to a certain depth ( )). In this paper, we empirically analyze the locality and globality of the cost function in hybrid quantum neural networks. We consider two application scenarios namely, binary and multi-class classification, and show that for multiclass classification, the local cost function setting does not follow the claims of Cerezo et al; that is, the local cost function does not result in an extended quantum layer’s depth. We also show that for multiclass classification, the overall performance in terms of accuracy for the global cost function setting is significantly higher than the local cost function setting. On the other hand, for binary classification, our results show that the local cost function setting follows the claims of Cerezo et al, and results in an extended depth of quantum layers. However, the global cost function setting still performs slightly better than the local cost function.

Training quantum neural networks (QNNs) on near-term hardware remains hampered by two compounding difficulties: the exponential vanishing of gradient variance known as the barren plateau, and the

\mathcal{O}(L \cdot 2^n)

time and memory cost of differentiating through an

n

-qubit,

L

-layer circuit. We propose RLQ-Grad, a reinforcement-learning-based optimizer in which a classical policy

π_φ

(a spectrally-normalized PPO agent) learns to propose parameter updates directly, conditioned on the QNN's current parameters, loss, accuracy, and previous update. Because the surrogate gradient is emitted by a classical network rather than obtained by differentiating through the unitary

U(θ)

, its variance is not constrained by the barren plateau concentration bound, and its cost scales with the number of trainable parameters rather than the Hilbert-space dimension. We prove these properties formally and verify them on a hardware-efficient ansatz across four supervised benchmarks with up to

n=20

qubits. RLQ-Grad preserves a near-flat gradient-variance curve where backpropagation, parameter-shift, and adjoint differentiation decay by 1 to 2 orders of magnitude. Accounting for the full training pipeline (PPO rollouts, actor-critic updates, and optimizer states), RLQ-Grad needs under 2 MB of memory and runs

2490\times

,

7876\times

, and

673\times

faster per iteration than these three methods at

n=20

. It improves top-1 accuracy by up to

+10\%

over gradient-based baselines on circuits of up to 12 qubits, and matches dedicated barren plateau mitigation methods on CIFAR-10 at 14 to 20 qubits, where evolutionary and gradient-free optimizers collapse to chance.

Abstract With the increased focus on quantum circuit learning for near-term applications on quantum devices, in conjunction with unique challenges presented by cost function landscapes of parametrized quantum circuits, strategies for effective training are becoming increasingly important. In order to ameliorate some of these challenges, we investigate a layerwise learning strategy for parametrized quantum circuits. The circuit depth is incrementally grown during optimization, and only subsets of parameters are updated in each training step. We show that when considering sampling noise, this strategy can help avoid the problem of barren plateaus of the error surface due to the low depth of circuits, low number of parameters trained in one step, and larger magnitude of gradients compared to training the full circuit. These properties make our algorithm preferable for execution on noisy intermediate-scale quantum devices. We demonstrate our approach on an image-classification task on handwritten digits, and show that layerwise learning attains an 8% lower generalization error on average in comparison to standard learning schemes for training quantum circuits of the same size. Additionally, the percentage of runs that reach lower test errors is up to 40% larger compared to training the full circuit, which is susceptible to creeping onto a plateau during training.

Variational quantum algorithms (VQAs) optimize the parameters

\boldsymbol{\theta}

of a quantum neural network

V(\boldsymbol{\theta})

to minimize a cost function

C

. While VQAs may enable practical applications of noisy quantum computers, they are nevertheless heuristic methods with unproven scaling. Here, we rigorously prove two results, assuming

V(\boldsymbol{\theta})

is a hardware-efficient ansatz composed of blocks forming local 2-designs. Our first result states that defining

C

in terms of global observables leads to an exponentially vanishing gradient (i.e., a barren plateau) even when

V(\boldsymbol{\theta})

is shallow. This implies that several VQAs in the literature must revise their proposed cost functions. On the other hand, our second result states that defining

C

with local observables leads to at worst a polynomially vanishing gradient, so long as the depth of

V(\boldsymbol{\theta})

is

\mathcal{O}(\log n)

. Taken together, our results establish a connection between locality and trainability. Finally, we illustrate these ideas with large-scale simulations, up to 100 qubits, of a particular VQA known as quantum autoencoders.