Variational Quantum Dimension Reduction for Recurrent Quantum Models

TL;DR

A variational quantum dimension reduction framework that identifies and removes irrelevant memory degrees of freedom while preserving the recurrent dynamics of the target model, enabling variational circuit optimization and quantum process compression for near-term quantum devices.

Abstract

Recurrent quantum models (RQMs) realize sequential quantum processes through repeated application of a unitary operation on a memory system coupled with a series of output registers. However, such models often rely on unnecessarily large memory spaces, introducing redundancy and limiting scalability. Here, we introduce a \textit{variational quantum dimension reduction} framework that identifies and removes irrelevant memory degrees of freedom while preserving the recurrent dynamics of the target model. Our approach employs two parameterized quantum circuits: a decoupling unitary $V(θ_1)$ that isolates the essential memory subspace; and a compressed recurrent unitary $\tilde{U}(θ_2)$ that reconstructs the dynamics in the reduced space. The optimization is guided by a unified cost function combining decoupling fidelity and dynamical accuracy, evaluated using the \textit{Quantum Fidelity Divergence Rate} (QFDR), a metric that quantifies long-term fidelity per time step. Applied to a cyclic random walk model, our framework achieves up to three orders of magnitude smaller QFDR compared to variational matrix product state truncation, while requiring only trajectory samples rather than explicit state reconstructions. This establishes a scalable, data-driven paradigm for learning minimal recurrent quantum architectures, enabling variational circuit optimization and quantum process compression for near-term quantum devices.

Figures (6)

• Figure 1: Recurrent quantum model: A sequential quantum circuit where a hidden memory state is iteratively updated by repeated unitary blocks $U$ coupling to input qubits initialised in $|0\rangle$ to produce the target data stream (Q-Sample).
• Figure 2: The objective of this work is to determine an accurate, dimension-reduced approximation of a target RQM governed by a black-box unitary $U$ (left), via a variational quantum circuit realization $\tilde{U}$ (right).
• Figure 3: Schematic illustration of the circuit constructions used to evaluate the variational cost function. The figure shows, in abstract form, how the original update $U$, the decoupling unitary $V(\boldsymbol{\theta_1})$, and the reduced update $\widetilde{U}(\boldsymbol{\theta_2})$ are combined to generate the quantities entering the cost.
• Figure 4: Quantum fidelity divergence rate $R_F$ versus original memory size $n$ for the cyclic walk model. Solid lines denote our approach, dashed lines the baseline; colors indicate reduced dimensions ($\tilde{n}=2$ in blue, $\tilde{n}=1$ in orange). Our method achieves consistently lower divergence, especially under stronger compression. Error bars show one standard deviation.
• Figure 5: Variational ansatz used for both $V(\boldsymbol{\theta_1})$ and $\widetilde{U}(\boldsymbol{\theta_2})$. Each layer consists of single-qubit $U3$ rotations followed by alternating two-qubit entangling blocks across adjacent qubits.