Quantum-Enhanced Temporal Embeddings via a Hybrid Seq2Seq Architecture

This work investigates how shallow, NISQ-compatible quantum layers can improve temporal representation learning in real-world sequential data. We develop a QLSTM Seq2Seq autoencoder in which a depth-1 variational quantum circuit is embedded inside each recurrent gate, shaping the geometry of the learned latent manifold. Evaluated on fourteen rolling S and P 500 windows from 2022 to 2025, the quantum-enhanced encoder produces smoother trajectories, clearer regime transitions, and more stable, sector-coherent clusters than a classical LSTM baseline. These geometric properties support the use of a Radial Basis Function (RBF) kernel for downstream portfolio allocation, where both RBF-Graph and RBF-DivMom strategies consistently outperform their classical counterparts in risk-adjusted terms. Analysis across periods shows that compressed manifolds favor concentrated allocation, while dispersed manifolds favor diversification, demonstrating that latent geometry serves as a regime indicator. The results highlight a practical role for shallow hybrid quantum and classical layers in NISQ-era sequence modeling, offering a reproducible pathway for improving temporal embeddings in finance and other data-limited, noise-sensitive domains.

💡 Research Summary

The paper proposes a hybrid quantum‑classical sequence‑to‑sequence (Seq2Seq) autoencoder that embeds a depth‑1 variational quantum circuit (VQC) inside each gate of a Long Short‑Term Memory (LSTM) network, creating a Quantum‑Enhanced LSTM (QLSTM). The QLSTM encoder processes weekly return sequences of S&P 500 constituents (≈12‑13 observations per quarter) and compresses each sequence into a two‑dimensional latent vector. By inserting a shallow VQC (single‑layer ansatz with entangling CNOTs and Pauli‑Z expectation readout) into the input, forget, output, and candidate‑cell gates, the model introduces bounded non‑linearity that acts as an implicit regularizer, mitigating vanishing‑gradient issues common in classical LSTMs when data are scarce and noisy. The circuit depth is deliberately kept at one to stay within NISQ hardware constraints while still enriching the representational capacity of the recurrent unit.

After training on a rolling window of the previous twelve months, the encoder is frozen and used to generate out‑of‑sample embeddings for the subsequent quarter. Fourteen distinct QLSTM‑Seq2Seq models are trained sequentially from 2022Q2 to 2025Q2, ensuring that the learned latent space adapts to evolving market regimes. The two‑dimensional embeddings are then transformed into a Radial Basis Function (RBF) similarity kernel:
(K_{mn}= \exp\big(-|h_m-h_n|^2/(2\sigma^2)\big)), where (\sigma) is set to the median pairwise distance. This kernel captures similarity in the learned temporal dynamics rather than raw price movements, providing a principled basis for diversification.

Two downstream portfolio construction methods exploit the kernel:

1. RBF‑DivMom – a discrete selector that ranks stocks by short‑term momentum and penalizes selections that are too similar according to the RBF kernel. The penalty strength (\lambda) is tuned via a grid search; (\lambda=0.15) yields the best trade‑off between momentum capture and diversification.

2. RBF‑Graph – a continuous optimizer that treats the kernel matrix as a weighted adjacency matrix of a graph and maximizes a graph‑centrality objective (e.g., PageRank) to allocate capital continuously across all assets.

Both strategies are evaluated in a quarterly backtest framework, compounding the quarterly returns to obtain cumulative performance over the entire 2022‑2025 horizon. The results show that the quantum‑enhanced encoder produces smoother latent trajectories and more stable, sector‑coherent clusters than a classical LSTM baseline. Visualizations (Figures 1‑2) confirm that the geometry of the latent manifold remains consistent across regime shifts, indicating robust temporal generalization.

Performance metrics (Table I) reveal that RBF‑Graph achieves the highest risk‑adjusted returns: average Sharpe ratio ≈ 0.98, CAGR ≈ 1.07 % per quarter, and a final cumulative wealth of 2.40× the initial capital, outperforming the S&P 500 benchmark (1.45×) and the classical LSTM‑based RBF strategies. RBF‑DivMom, while delivering a slightly lower Sharpe (≈ 0.95) and CAGR (≈ 0.95 %), offers smoother equity curves and lower maximum drawdowns, making it attractive for more risk‑averse investors. The grid‑search over (\lambda) (Table III) confirms that modest similarity penalties (λ ≤ 0.30) keep the performance gap to RBF‑Graph minimal, whereas aggressive penalties erode returns.

Economic interpretation (Table II) links latent geometry to market regimes: compressed manifolds during bullish, growth‑driven periods enable RBF‑Graph to concentrate capital in high‑momentum clusters, while dispersed manifolds in volatile or sideways markets favor RBF‑DivMom’s diversification to mitigate drawdowns.

The authors acknowledge limitations: the need to retrain the model from scratch each quarter, the shallow depth of the VQC which may cap expressive power, and the focus on a single equity universe. Nonetheless, the study demonstrates that even depth‑1 quantum circuits can provide meaningful regularization and non‑linearity in sequence models, leading to tangible financial gains.

In conclusion, the work offers a reproducible pathway for leveraging NISQ‑compatible quantum layers in temporal representation learning. By coupling quantum‑enhanced embeddings with kernel‑based portfolio construction, the authors achieve superior risk‑adjusted performance and uncover a novel link between latent geometry and economic regime identification. Future research directions include deeper or more expressive quantum ansätze, multi‑modal data integration, and real‑time deployment on quantum‑ready hardware.