QLIF-CAST: Quantum-Classical Recurrent Forecasting

• QLIF-CAST is a hybrid quantum–classical recurrent forecaster that replaces classical LIF neurons with single-qubit quantum LIF elements for continuous time-series regression.
• It embeds a shallow quantum circuit using two Rx gates and T1-style damping within a seven-layer recurrent architecture, achieving lower forecasting error and faster training than classical baselines.
• Empirical results on weather, air quality, and wind-speed datasets demonstrate its ability to capture rich temporal structure and sensor uncertainty effectively.

Searching arXiv for the exact QLIF-CAST paper and closely related context papers. QLIF-CAST is a hybrid quantum–classical recurrent forecasting architecture that adapts the Quantum Leaky Integrate-and-Fire (QLIF) neuron from classification to continuous-valued time-series regression, with a specific emphasis on short-term multivariate weather prediction. In this formulation, neuron excitation states are encoded as single-qubit quantum superpositions updated by $R_x$ rotation gates and a $T_1$ relaxation leak, and the resulting excitation probabilities are embedded within a seven-layer recurrent network for next-step forecasting (Marchisio et al., 18 May 2026). The model was evaluated both against a parameter-matched classical LIF baseline on multivariate weather data and against state-of-the-art quantum LSTM and quantum neural network baselines on air-quality and wind-speed benchmarks, where it achieved lower error than the matched classical LIF model and substantially shorter training times than deeper variational quantum alternatives (Marchisio et al., 18 May 2026).

1. Definition and conceptual role

QLIF-CAST is defined as a hybrid quantum–classical recurrent forecaster in which the conventional leaky-integrate-and-fire neuronal update is replaced by a single-qubit quantum LIF element. The central design choice is to represent excitation probability through a qubit state whose dynamics are driven by shallow quantum evolution rather than by a purely classical membrane-potential recursion (Marchisio et al., 18 May 2026).

The motivating task is short-term multivariate time-series forecasting of environmental variables, including temperature, humidity, wind speed, and pressure, as well as single-target regression tasks in air quality and wind speed (Marchisio et al., 18 May 2026). The paper positions the model between two existing regimes: classical LSTM-based architectures, which offer strong predictive accuracy at higher computational cost, and classical LIF-based spiking models, which are computationally efficient but have more limited state dynamics. QLIF-CAST is presented as an attempt to retain shallow, efficient neuron-level computation while introducing richer temporal state evolution through superposition, interference, and probabilistic measurement (Marchisio et al., 18 May 2026).

A plausible implication is that QLIF-CAST is best understood not as a deep variational quantum model, but as a shallow quantum-neuronal replacement inside an otherwise standard recurrent regression stack. This interpretation is consistent with the fact that the architecture contains no trainable quantum angles and relies on classical backpropagation through surrounding layers rather than parameter-shift-based quantum optimization (Marchisio et al., 18 May 2026).

2. Quantum neuron formulation

The quantum component is a single-qubit representation of excitation. The qubit state is written as

$$|\psi\rangle = \alpha_0 |0\rangle + \beta_1 |1\rangle,$$

with excitation probability

$p_1 = |\beta_1|^2.$

In QLIF-CAST, the excitation probability $\alpha$ is represented through a rotation angle $\phi$ such that

$$\alpha = \sin^2(\phi/2), \qquad \phi = 2 \arcsin(\sqrt{\alpha}).$$

The per-timestep circuit is

$$|0\rangle \rightarrow R_x(\phi_t) \rightarrow R_x(\theta_{\text{input},t}) \rightarrow \text{Measure},$$

yielding

$$\alpha_{t+1} = \sin^2\!\left(\frac{\phi_t + \theta_{\text{input},t}}{2}\right).$$

A spike is emitted if $\alpha_{t+1} \ge 0.75$, after which the state is reset (Marchisio et al., 18 May 2026).

The gate dynamics use the standard Pauli-$T_1$0 rotation

$T_1$1

This is a depth-2, single-qubit construction, and each QLIF neuron executes two $T_1$2 gates per timestep (Marchisio et al., 18 May 2026).

Leak is implemented through a $T_1$3-style amplitude damping mechanism. The Kraus operators are

$T_1$4

with

$T_1$5

In the operational model, leak acts in probability space as $T_1$6, and the corresponding decay angle is

$T_1$7

The effective input rotation is then

$T_1$8

so that a previous spike causes the input-driven angle $T_1$9 to dominate, while otherwise the evolution is governed by decay (Marchisio et al., 18 May 2026).

The recurrent update for neuron $$|\psi\rangle = \alpha_0 |0\rangle + \beta_1 |1\rangle,$$0 is therefore specified by

$$|\psi\rangle = \alpha_0 |0\rangle + \beta_1 |1\rangle,$$1

$$|\psi\rangle = \alpha_0 |0\rangle + \beta_1 |1\rangle,$$2

$$|\psi\rangle = \alpha_0 |0\rangle + \beta_1 |1\rangle,$$3

$$|\psi\rangle = \alpha_0 |0\rangle + \beta_1 |1\rangle,$$4

If $$|\psi\rangle = \alpha_0 |0\rangle + \beta_1 |1\rangle,$$5, then $$|\psi\rangle = \alpha_0 |0\rangle + \beta_1 |1\rangle,$$6 and $$|\psi\rangle = \alpha_0 |0\rangle + \beta_1 |1\rangle,$$7; otherwise $$|\psi\rangle = \alpha_0 |0\rangle + \beta_1 |1\rangle,$$8 (Marchisio et al., 18 May 2026).

For comparison, the classical reference neuron follows the discrete LIF update

$$|\psi\rangle = \alpha_0 |0\rangle + \beta_1 |1\rangle,$$9

with $p_1 = |\beta_1|^2.$0, and emits a spike when $p_1 = |\beta_1|^2.$1 crosses threshold (Marchisio et al., 18 May 2026).

3. Network architecture and learning pipeline

QLIF-CAST is embedded in a seven-layer recurrent network whose total parameter count is approximately $p_1 = |\beta_1|^2.$2, and this total is identical for the quantum and classical baselines (Marchisio et al., 18 May 2026). The layer stack is fixed except for the neuronal update in the second layer.

Layer	Function	Parameters
L1	TimeDistributed Dense(48, ReLU) + Dropout(0.1)	240
L2	48-neuron QLIF layer or classical LIF layer	2,400
L3	BatchNormalization + Dropout(0.2)	96
L4	LSTM(24)	7,008
L5	Dense(32, ReLU) + Dropout(0.2)	800
L6	Dense(16, ReLU)	528
L7	Linear Dense($p_1 = |\beta_1|^2.$3) output	$p_1 = |\beta_1|^2.$4

Each input window has length $p_1 = |\beta_1|^2.$5. For weather and wind forecasting, this corresponds to 12 hours; for air quality, 12 days (Marchisio et al., 18 May 2026). Each window is standardized to zero mean and unit variance. At each timestep $p_1 = |\beta_1|^2.$6, the input $p_1 = |\beta_1|^2.$7 is first projected through

$p_1 = |\beta_1|^2.$8

after which the QLIF layer computes 48 excitation trajectories in parallel, producing a sequence of shape $p_1 = |\beta_1|^2.$9. This sequence is normalized, passed to an LSTM with hidden size 24, and then processed by two dense regression heads to produce the next-step forecast $\alpha$0 (Marchisio et al., 18 May 2026).

Training uses mean squared error,

$\alpha$1

with mean absolute error, RMSE, and $\alpha$2 reported as evaluation metrics on inverse-standardized predictions (Marchisio et al., 18 May 2026). Optimization uses Adam with initial learning rate $\alpha$3, exponential decay, batch size 64, and early stopping with patience 5. L2 regularization is set to $\alpha$4, and dropout ranges from 0.1 to 0.2 (Marchisio et al., 18 May 2026).

Because the quantum angles are not trainable, the model does not use parameter-shift rules. Instead, backpropagation flows through classical parameters, while the spike threshold is differentiated through a surrogate gradient based on

$\alpha$5

This design is one of the main reasons the architecture trains substantially faster than variational quantum sequence models (Marchisio et al., 18 May 2026).

4. Datasets, implementation, and hardware execution

The primary controlled experiment uses the D1 Weather History dataset from Kaggle, containing 96,453 hourly samples from April 2006 to September 2016. Inputs and targets are Temperature ($\alpha$6C), Relative Humidity (fraction), Wind Speed (km/h), and Pressure (mb). The forecasting target is the next hour’s 4-variable vector, and the chronological split uses 10,000 training samples and 2,000 test samples (Marchisio et al., 18 May 2026).

Two additional datasets are used for cross-domain comparison. D2 is the Bangkok Air Quality dataset from WAQI, containing approximately 3,290 daily records from July 2016 to January 2026, with inputs PM10, O$\alpha$7, and NO$\alpha$8, and target next-day PM2.5. D3 is an Open-Meteo wind-speed dataset with 35,064 hourly samples from January 2020 to December 2023 at 100 m altitude, with next-hour wind speed as the target (Marchisio et al., 18 May 2026).

Simulation is performed with PennyLane on the lightning.qubit state-vector backend, and the surrounding network is implemented in TensorFlow/Keras (Marchisio et al., 18 May 2026). The reported training platform is an Intel Core i9-13900H CPU with 16 GB RAM. Crucially, the per-timestep quantum workload is shallow: for 48 neurons, there are 96 $\alpha$9 gates and 48 measurements per timestep, with no entanglement and no multi-qubit gates (Marchisio et al., 18 May 2026).

The implementation uses vectorized execution by flattening angles across neurons, timesteps, and batch elements into a single parallel PennyLane call, producing approximately $\phi$0 speedup relative to naïve sequential circuit execution (Marchisio et al., 18 May 2026). This is a central implementation detail: the practical viability of the model depends less on raw qubit count than on the fact that the quantum subroutine is embarrassingly parallel and extremely shallow.

Hardware verification was performed on IBM Marrakesh, a 156-qubit QPU, by running the core single-qubit sequence

$\phi$1

with 1,000 shots per test case (Marchisio et al., 18 May 2026). The average deviation between QPU-measured $\phi$2 and ideal simulation was $\phi$3, with representative comparisons $\phi$4 vs. $\phi$5, $\phi$6 vs. $\phi$7, and $\phi$8 vs. $\phi$9 (Marchisio et al., 18 May 2026). This supports the claim that the neuron-level circuit is hardware-compatible in the NISQ setting.

5. Empirical performance

In the parameter-matched comparison on the multivariate weather task, QLIF-CAST achieved MSE $$\alpha = \sin^2(\phi/2), \qquad \phi = 2 \arcsin(\sqrt{\alpha}).$$0, MAE $$\alpha = \sin^2(\phi/2), \qquad \phi = 2 \arcsin(\sqrt{\alpha}).$$1, RMSE $$\alpha = \sin^2(\phi/2), \qquad \phi = 2 \arcsin(\sqrt{\alpha}).$$2, and $$\alpha = \sin^2(\phi/2), \qquad \phi = 2 \arcsin(\sqrt{\alpha}).$$3, whereas the classical LIF baseline achieved MSE $$\alpha = \sin^2(\phi/2), \qquad \phi = 2 \arcsin(\sqrt{\alpha}).$$4, MAE $$\alpha = \sin^2(\phi/2), \qquad \phi = 2 \arcsin(\sqrt{\alpha}).$$5, RMSE $$\alpha = \sin^2(\phi/2), \qquad \phi = 2 \arcsin(\sqrt{\alpha}).$$6, and $$\alpha = \sin^2(\phi/2), \qquad \phi = 2 \arcsin(\sqrt{\alpha}).$$7 (Marchisio et al., 18 May 2026). These correspond to a $$\alpha = \sin^2(\phi/2), \qquad \phi = 2 \arcsin(\sqrt{\alpha}).$$8 reduction in MSE, a $$\alpha = \sin^2(\phi/2), \qquad \phi = 2 \arcsin(\sqrt{\alpha}).$$9 reduction in MAE, and an $$|0\rangle \rightarrow R_x(\phi_t) \rightarrow R_x(\theta_{\text{input},t}) \rightarrow \text{Measure},$$0 reduction in RMSE for the quantum model under identical parameter count and training schedule (Marchisio et al., 18 May 2026).

The per-variable MAE breakdown is more mixed. QLIF-CAST improves Temperature from $$|0\rangle \rightarrow R_x(\phi_t) \rightarrow R_x(\theta_{\text{input},t}) \rightarrow \text{Measure},$$1 to $$|0\rangle \rightarrow R_x(\phi_t) \rightarrow R_x(\theta_{\text{input},t}) \rightarrow \text{Measure},$$2 and Pressure from $$|0\rangle \rightarrow R_x(\phi_t) \rightarrow R_x(\theta_{\text{input},t}) \rightarrow \text{Measure},$$3 to $$|0\rangle \rightarrow R_x(\phi_t) \rightarrow R_x(\theta_{\text{input},t}) \rightarrow \text{Measure},$$4, but performs worse on Humidity ($$|0\rangle \rightarrow R_x(\phi_t) \rightarrow R_x(\theta_{\text{input},t}) \rightarrow \text{Measure},$$5 vs. $$|0\rangle \rightarrow R_x(\phi_t) \rightarrow R_x(\theta_{\text{input},t}) \rightarrow \text{Measure},$$6) and Wind Speed ($$|0\rangle \rightarrow R_x(\phi_t) \rightarrow R_x(\theta_{\text{input},t}) \rightarrow \text{Measure},$$7 vs. $$|0\rangle \rightarrow R_x(\phi_t) \rightarrow R_x(\theta_{\text{input},t}) \rightarrow \text{Measure},$$8) (Marchisio et al., 18 May 2026). The paper interprets this as indicating that gains are concentrated on variables with stronger temporal structure, especially temperature and pressure (Marchisio et al., 18 May 2026).

Cross-domain comparisons place QLIF-CAST in a different region of the speed–accuracy trade-off from deeper quantum sequence models. On the Bangkok PM2.5 task, QLIF-CAST obtained MAE $$|0\rangle \rightarrow R_x(\phi_t) \rightarrow R_x(\theta_{\text{input},t}) \rightarrow \text{Measure},$$9 and RMSE $$\alpha_{t+1} = \sin^2\!\left(\frac{\phi_t + \theta_{\text{input},t}}{2}\right).$$0, compared with published QLSTM values of MAE $$\alpha_{t+1} = \sin^2\!\left(\frac{\phi_t + \theta_{\text{input},t}}{2}\right).$$1 and RMSE $$\alpha_{t+1} = \sin^2\!\left(\frac{\phi_t + \theta_{\text{input},t}}{2}\right).$$2. However, QLIF-CAST converged in 26 epochs versus 100 epochs for QLSTM (Marchisio et al., 18 May 2026). On the wind-speed benchmark, QLIF-CAST achieved RMSE $$\alpha_{t+1} = \sin^2\!\left(\frac{\phi_t + \theta_{\text{input},t}}{2}\right).$$3, MAE $$\alpha_{t+1} = \sin^2\!\left(\frac{\phi_t + \theta_{\text{input},t}}{2}\right).$$4, and $$\alpha_{t+1} = \sin^2\!\left(\frac{\phi_t + \theta_{\text{input},t}}{2}\right).$$5, while the published LSTM-QNN achieved RMSE $$\alpha_{t+1} = \sin^2\!\left(\frac{\phi_t + \theta_{\text{input},t}}{2}\right).$$6, MAE $$\alpha_{t+1} = \sin^2\!\left(\frac{\phi_t + \theta_{\text{input},t}}{2}\right).$$7, but required approximately $$\alpha_{t+1} = \sin^2\!\left(\frac{\phi_t + \theta_{\text{input},t}}{2}\right).$$8 minutes of training, compared with $$\alpha_{t+1} = \sin^2\!\left(\frac{\phi_t + \theta_{\text{input},t}}{2}\right).$$9 minutes for QLIF-CAST (Marchisio et al., 18 May 2026). The reported training-time reduction is $\alpha_{t+1} \ge 0.75$0, or approximately $\alpha_{t+1} \ge 0.75$1 (Marchisio et al., 18 May 2026).

The paper therefore does not claim uniform dominance in absolute forecasting accuracy. Instead, it presents QLIF-CAST as occupying a distinct Pareto position in which shallow quantum neuronal dynamics improve over a matched classical LIF baseline, while training much faster than parameterized quantum recurrent baselines (Marchisio et al., 18 May 2026).

6. Interpretation, limitations, and research significance

The proposed explanation for QLIF-CAST’s behavior has three components. First, sequential $\alpha_{t+1} \ge 0.75$2 and $\alpha_{t+1} \ge 0.75$3 operations introduce constructive and destructive interference in amplitude space, which provides richer temporal modulation than scalar exponential decay (Marchisio et al., 18 May 2026). Second, the amplitude-damping leak acts nonlinearly in probability space, so the effective decay differs qualitatively from the classical affine update $\alpha_{t+1} \ge 0.75$4 (Marchisio et al., 18 May 2026). Third, the excitation probability $\alpha_{t+1} \ge 0.75$5 is interpreted as a probabilistic state variable that can naturally encode sensor uncertainty and may regularize optimization (Marchisio et al., 18 May 2026).

Several limitations are explicit. End-to-end QPU training was not demonstrated; hardware use was limited to circuit verification (Marchisio et al., 18 May 2026). The cross-domain comparisons with QLSTM and LSTM-QNN rely on published numbers rather than retraining under identical conditions (Marchisio et al., 18 May 2026). The architecture also remains shallow and single-qubit at the neuron level, so scaling to larger neuron counts or longer sequences may encounter hardware-throughput limits, and the effect of stronger decoherence remains an open question (Marchisio et al., 18 May 2026).

The paper suggests several future directions: multi-step forecasting horizons, ensembles of QLIF forecasters, hybrid QLIF–variational spiking-LSTM models, and multi-qubit neuron designs (Marchisio et al., 18 May 2026). This suggests a broader research program in which quantum spiking elements are treated as modular replacements for classical neuronal dynamics rather than as end-to-end quantumized recurrent blocks.

Within the immediate literature supplied here, QLIF-CAST is distinct from work that studies QLIF as an information-flow diagnostic in many-body quantum chaos (Yi, 16 Mar 2026). In that latter context, QLIF denotes Quantum Liang Information Flow, whereas in QLIF-CAST it denotes Quantum Leaky Integrate-and-Fire (Yi, 16 Mar 2026, Marchisio et al., 18 May 2026). The acronym overlap is purely nominal; the two uses refer to unrelated theoretical objects.

In summary, QLIF-CAST establishes that a shallow single-qubit spiking neuron can be integrated into a classical recurrent forecasting stack for regression, can outperform a parameter-matched classical LIF baseline on multivariate weather prediction, can train far faster than deeper variational quantum sequence models, and can be executed on present-day hardware with only $\alpha_{t+1} \ge 0.75$6 average deviation from simulation (Marchisio et al., 18 May 2026). This suggests a design space in quantum machine learning where practical value may emerge less from deep trainable quantum circuits than from carefully engineered shallow quantum dynamical primitives embedded within classical sequence architectures.