Physics-anchored residual quantum neural networks for cloud-cover parameterization

Zhuo Wang; Ce Yu; Bin Yang

quant-phOct 19, 2026

Physics-anchored residual quantum neural networks for cloud-cover parameterization

Authors: Zhuo Wang, Ce Yu, Bin Yang

Organizations: Tianjin University

Abstract

Compact quantum neural networks have been explored for cloud-cover parameterization, but their predictive value depends on how physical knowledge is incorporated and which classical models provide the reference. We investigate an offline residual formulation that retains a fitted \XR\ cloud-cover estimate and learns a correction to its logit in transformed target space. Using released ICON/DYAMOND-derived data, we compare ten model variants in six- and eight-input settings, with three matched replicates and 10,000 training samples per replicate. All quantum circuits use local Z readout and noiseless exact expectation values. Under a common 30-epoch protocol, residual prediction reduced physical-space test mean squared error relative to direct QNN prediction by 54.3% and 38.8% for six-input XYZ and ZZXY circuits, and by 49.2% and 38.9% for their eight-input versions. Residual QNNs also outperformed the selected compact residual multilayer perceptron in every paired replicate. A larger, longer-trained Grundner-Q1 adaptation nevertheless achieved lower error: residual Q1 reached mean MSEs of 0.00778 and 0.00791, compared with 0.01124 and 0.01082 for residual XYZ. The results support residual prediction for compact simulated QNNs within the tested budget and establish a clear boundary set by stronger classical references. They do not establish computational quantum advantage or online climate-model skill.

Figures & tables

Figure 1: Physics-anchored residual QNN. The fixed branch supplies

\operatorname{logit}a(x)

. The quantum branch supplies local expectations

m_{i}=\langle Z_{i}\rangle

to one affine head. Its zero initialization recovers the clipped anchor. The branches combine before the sigmoid, and the inverse target transform returns physical cloud cover. Training loss is evaluated before that inverse transform; the dashed arrow updates only trainable parameters. Overlapping blocks and representative wires abbreviate the six- or eight-qubit circuit, not its actual layer count. The figure’s

T

and

T_{x}

denote the target and input transforms.

Model family	Six inputs	Eight inputs	Training protocol
Xu–Randall	0	0	Fixed calibration
Linear residual	7	9	Adam, 30 epochs
Compact MLP, direct or residual	119	203	Adam, 30 epochs
XYZ QNN, direct or residual	109	201	Adam, 30 epochs
ZZXY QNN, direct or residual	114	200	Adam, 30 epochs
Q1, direct or residual	8,961	9,089	Adam, at most 300 epochs

Table 1: Trainable parameters and training protocols. Every residual also uses two fixed Xu–Randall constants; Q1 adds 128 non-trainable normalization states. Compact models are near-sized across families and exactly size-matched within each direct/residual pair.

Model	Six inputs		Eight inputs
	MSE $\downarrow$	$R^{2}$ $\uparrow$	MSE $\downarrow$	$R^{2}$ $\uparrow$
Xu–Randall	$0.04804\pm 0.00067$	$0.504\pm 0.008$	$0.04804\pm 0.00067$	$0.504\pm 0.008$
Compact MLP, direct	$0.04277\pm 0.00599$	$0.558\pm 0.062$	$0.02753\pm 0.00813$	$0.715\pm 0.085$
XYZ QNN, direct	$0.02547\pm 0.00530$	$0.737\pm 0.054$	$0.02165\pm 0.00374$	$0.776\pm 0.038$
ZZXY QNN, direct	$0.02002\pm 0.00329$	$0.793\pm 0.034$	$0.01815\pm 0.00168$	$0.812\pm 0.017$
XR + linear	$0.01663\pm 0.00008$	$0.828\pm 0.000$	$0.01667\pm 0.00014$	$0.828\pm 0.001$

Table 2: Physical-space performance, mean

\pm

sample SD over three matched replicates of 20,000 test cells each. All QNNs use local Z readout. Compact models share 30 epochs; Q1 uses its separate longer protocol.

Figure 2: Test-set performance across input configurations. Each row groups six-input (open) and eight-input (filled) bars, whose lengths show mean physical-space MSE. Small points represent three matched replicates; whiskers show

\pm

one sample SD (

n=3

, 20,000 test cells per replicate). QNNs are blue, Q1 adaptations ochre, and remaining controls gray. Every residual model is anchored to Xu–Randall; all QNNs use local Z readout. Q1 uses a larger architecture and a separate, longer training protocol. Axis values are expressed in units of

10^{-2}

; lower values indicate lower error. Source data: runs.csv and summary.csv .

Model	Six inputs		Eight inputs
	Hellinger $\downarrow$	Wasserstein $\downarrow$	Hellinger $\downarrow$	Wasserstein $\downarrow$
Xu–Randall	$0.0972\pm 0.0011$	$0.0189\pm 0.0012$	$0.0972\pm 0.0011$	$0.0189\pm 0.0012$
Compact MLP, direct	$0.2320\pm 0.0655$	$0.0716\pm 0.0141$	$0.2130\pm 0.0902$	$0.0339\pm 0.0022$
XYZ QNN, direct	$0.0800\pm 0.0084$	$0.0247\pm 0.0018$	$0.0713\pm 0.0074$	$0.0220\pm 0.0019$
ZZXY QNN, direct	$0.0908\pm 0.0065$	$0.0279\pm 0.0015$	$0.0774\pm 0.0096$	$0.0249\pm 0.0014$
XR + linear	$0.0865\pm 0.0023$	$0.0246\pm 0.0008$	$0.0845\pm 0.0026$	$0.0242\pm 0.0009$

Table 3: Distributional distances in physical cloud-fraction space. Values are means

\pm

sample SD over three matched replicates, with 20,000 test cells per replicate. Both Hellinger and Wasserstein distances are lower-is-better (

\downarrow

), unlike

R^{2}

. Hellinger uses 100 equal bins on

[0,1]

; Wasserstein is the empirical one-dimensional distance. Models and training protocols match Table 2 . All QNNs use local Z readout.

Figure 3: Paired effects of residual prediction and input configuration. a–d , direct-to-residual QNN comparisons for six-input XYZ, six-input ZZXY, eight-input XYZ, and eight-input ZZXY, respectively. Each thin line connects one matched replicate; small endpoints show its physical-space MSE. Large open circles (six inputs) and filled squares (eight inputs) show means, with whiskers indicating

\pm

one sample SD. e , percentage MSE reductions for eight inputs relative to six, computed within replicate before averaging. Bars and right-hand labels show mean reductions; small points show individual pairs, and whiskers indicate

\pm

one sample SD. Positive values favor eight inputs; colors identify QNNs (blue), Q1 (ochre), and other controls (gray). All panels use three matched replicates with 20,000 test cells each. QNN inputs, qubit count, and circuit depth change together in panel e.

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An architectural capacity ceiling, not a barren plateau: why a fixed-encoding variational quantum circuit cannot fit the Lorenz-63 attractor

Tushar Pandey

Texas A&M University, College Station, TX 77843, USA

Quantum Circuit Expressibility Variational Quantum Circuits Physics-Informed ML Quantum Circuit Trainability Chaotic Time-Series Prediction

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Variational quantum circuits train poorly on chaotic forecasting, usually blamed on barren plateaus (exponentially vanishing gradients). Using an exactly simulable four-qubit variational quantum physics-informed circuit fit to Lorenz-63, we show the barren-plateau explanation fails: the failure is an architectural capacity ceiling fixed by the circuit time-encoding, not its trainable depth. Four measurements support this. (i) A McClean-comparable gradient-variance estimator sits at the local-cost Haar/2-design scale 2^(-2n)=3.9e-3 at n=4; on structurally live parameters it decays about ninefold with depth then saturates there, large enough to train, not an exponential collapse. (ii) At a common budget of 200 optimiser iterations (600, in three stages, for layer-wise), gradient descent, layer-wise, and SPSA reach the same order of magnitude of loss, so no optimiser unlocks a better basin. (iii) The output-Jacobian rank saturates at 33 from five layers on, so depth buys no new output directions. (iv) A Fourier analysis explains why: the qubit-1 phase encoding acts on the initial |0> and is inert, so the maximum accessible frequency is 2.5/t_max=0.83 Hz, identical at every depth and about 4.4x below the narrowest Lorenz component bandwidth. The corrected band has dimension 1+2x5=11 per observable, and 3x11=33 equals the measured rank ceiling exactly, unifying the two diagnostics. A trained depth sweep agrees: mean loss improves with depth then flattens once the rank saturates. We correct our earlier preprint diagnosis, which compared unnormalised gradient norms to the McClean threshold, and place the advantage of fixed reservoirs and classical echo-state networks in architecture, not quantum mechanics.

Abstract

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