A Quantum Variational Approach to Prototypical Recurrent Unit

Mahyar Sadeghi Garjan; Tommaso Cesari; Michel Barbeau

cs.LGSep 3, 2026

A Quantum Variational Approach to Prototypical Recurrent Unit

Authors: Mahyar Sadeghi Garjan, Tommaso Cesari, Michel Barbeau

Organizations: School of Electrical Engineering and Computer Science, University of Ottawa, Canada · School of Computer Science, Carleton University, Ottawa, Canada

Abstract

We introduce a lightweight Quantum Prototypical Recurrent Unit (QPRU) that requires significantly fewer parameters than both classical recurrent architectures, such as Long Short- Term Memory (LSTM) and Gated Recurrent Unit (GRU), and quantum variants, including Quantum LSTM (QLSTM) and Quantum GRU (QGRU). Despite its compact design, the QPRU achieves competitive forecasting performance, matching state-of-the-art baselines while offering important structural and practical advantages, including enhanced scalability and a reduced number of trainable parameters.

Figures & tables

Figure 1: Architecture of the proposed QPRU.

Figure 2: VQC exploited in QPRU.

Model	Quantum Params	Total Params
QPRU	$2ql$	$q(2l+d_{\text{in}}+2d_{\text{hid}}+1)+d_{\text{hid}}$
QGRU	$3ql$	$q(3l+2d_{\text{hid}}+d_{\text{in}}+1)+d_{\text{hid}}$
QLSTM	$4ql$	$q(4l+2d_{\text{hid}}+d_{\text{in}}+1)+d_{\text{hid}}$

Table 1: Parameter complexity of QLSTM, QGRU, and the proposed QPRU model.

Model	Classical	Quantum	Total
	Params	Params	Params
QPRU (PRU)	$53\,(80)$	$16\,(0)$	$69\,(80)$
QGRU (GRU)	$53\,(120)$	$24\,(0)$	$77\,(120)$
QLSTM (LSTM)	$53\,(60)$	$32\,(0)$	$85\,(160)$

Table 2: Parameter comparison of classical and quantum recurrent units for hidden state dimension

5

, input feature dimension

1

4

qubits, and

2

ansatz layers. Classical counterparts are shown in parentheses.

Figure 3: Sinusoidal Function Forecasting using QPRU . The red solid line represents the absolute error, and the dashed red line represents the training/validation split.

Model	$\boldsymbol{\ell_{1}}(\times 10^{-2}$ )	$\boldsymbol{\ell_{2}}(\times 10^{-4}$ )	$\boldsymbol{\ell_{\infty}}(\times 10^{-2})$
QPRU	$1.2949\pm 0.4543$	$2.63\pm 1.83$	$2.8277\pm 0.9683$
QGRU	$1.5572\pm 0.6885$	$3.66\pm 2.59$	$3.1338\pm 1.3277$
QLSTM	$1.5772\pm 0.6093$	$4.01\pm 2.74$	$3.7471\pm 1.3887$
PRU	$1.4474\pm 0.5951$	$3.52\pm 3.01$	$3.2643\pm 1.2826$
GRU	$1.7772\pm 0.6611$	$5.10\pm 3.48$	$3.6358\pm 1.2057$
LSTM	$1.5491\pm 0.4759$	$3.36\pm 1.75$	$3.0791\pm 0.8634$

Table 3: Experiment I results. Norm values are averaged over 10 independent validation runs for each model. Values are expressed in

(mean\pm STD)\times 10^{-n}

format.

Figure 4: Forecasting of AAPL closing prices. Predictions on training and validation sets are shown, with the dashed line marking the split. Validation MSE and MAE (computed on denormalized data) are

4.4370\pm 7.920

and

1.6265\pm 1.338

, respectively, indicating consistent performance on unseen data.

Model	$\boldsymbol{\ell_{1}}$	$\boldsymbol{\ell_{2}}$	$\boldsymbol{\ell_{\infty}}$
QPRU	$0.0461\pm 0.0151$	$0.0032\pm 0.0017$	$0.1559\pm 0.0203$
QGRU	$0.1086\pm 0.1017$	$0.0462\pm 0.1082$	$0.3472\pm 0.3590$
QLSTM	$0.1198\pm 0.0906$	$0.0552\pm 0.0837$	$0.5473\pm 0.5261$
PRU	$0.0667\pm 0.0194$	$0.0064\pm 0.0031$	$0.1768\pm 0.0292$
GRU	$0.0606\pm 0.0222$	$0.0052\pm 0.0034$	$0.1573\pm 0.0283$
LSTM	$0.0553\pm 0.0047$	$0.0065\pm 0.0295$	$0.1816\pm 0.1271$

Table 4: Experiment II results. Norm values are averaged over 10 independent validation runs for each model and are computed on data normalized to the range

[-1,1]

. Values of the table are of the form mean

\pm

STD.

Figure 5: Confidence intervals for stock forecasting for different models. The intervals are computed using

r=2\sqrt{\ln(2/\delta)/20}

, with

\delta=0.0005

, representing the half-width of the interval at a high confidence level. The intervals for all methods overlap, indicating that none of the pairwise differences is statistically significant. Within this experimental setup, our QPRU model performs on par with state-of-the-art baselines while retaining the structural and practical advantages discussed earlier.

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Recent advances in quantum computing and machine learning have motivated the development of quantum models for sequential data processing. In this paper, we propose a Recursive Quantum Long Short-Term Memory model, or Recursive QLSTM, which extends QLSTM through metacore-based recursive constructions. We numerically test the model under different input sequence lengths, metacore designs, and recursive rules, and identify the best-performing architecture among these variants. For this selected model, we further provide theoretical arguments explaining why its recursive structure improves temporal information propagation and enhances learning performance. Our results suggest that Recursive QLSTM offers a flexible and effective framework for quantum recurrent learning over input time series of various lengths.

Abstract

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