Data-driven Quantum Dynamical Embedding Method for Long-term Prediction on Near-term Quantum Computers

TL;DR

This work introduces data-driven quantum dynamical embedding (QDE), a fixed-depth, trainable-quantum-circuit framework designed to forecast long-term time series on near-term quantum devices by embedding non-Markovian data into an extended Markovian state space. It achieves long-horizon predictions with circuit depth that scales as $\mathcal{O}(\Gamma)$ rather than with sequence length $L$, enabling practical use on NISQ hardware. The authors demonstrate cosine-wave, composite, and NARMA2 predictions, show denoising capabilities, and present a superconducting-qubit proof-of-concept with LECL-based noise mitigation, along with dynamical analyses and universality results. While highlighting advantages in depth efficiency and noise resilience, they also address limitations such as barren plateaus and memory-loss compared to full quantum reservoirs, outlining paths for future improvements and applications in quantum-enhanced time-series analysis.

Abstract

The increasing focus on long-term time series prediction across various fields has been significantly strengthened by advancements in quantum computation. In this paper, we introduce a data-driven method designed for time series prediction with quantum dynamical embedding (QDE). This approach enables a trainable embedding of the data space into an extended state space, allowing for the recursive retrieval of time series information. Based on its independence of time series length, this method achieves depth-efficient quantum circuits that are crucial for near-term quantum computers. Numerical simulations demonstrate the model's capability to predict not only wave signals but also more complex signals such as NARMA. Prediction accuracy improves with model scaling, and notably, the model achieves better accuracy on wave signal tasks with fewer parameters compared to QRC. Additionally, the model shows promising potential for denoising classical noise in wave signals, and when combined with error mitigation techniques for typical quantum noise, it enables reliable long-term prediction of wave signals. We implement this model, restricted to 2 qubits, on the Origin ``Wukong" superconducting quantum processor as a simple proof-of-concept on NISQ devices. Furthermore, we provide theoretical analysis of the QDE's dynamical properties for the 2-qubit case and discuss its potential universality. Overall, this study represents our first step towards leveraging near-term quantum devices for time series forecasting, offering insights into integrating data-driven learning with quantum dynamical embeddings.

Figures (20)

• Figure 1: Circuits for QDE and Learning Protocols. (a) Illustration of a generic QDE architecture. The qubits for encoding the classical state are divided into two sets: the memory register (top, with $n_m$ qubits) for inputting memory $\boldsymbol{m}_t$, and the data register (bottom, with $n_x$ qubits) for inputting data $\boldsymbol{x}_t$ at time $t$. The circuit then undergoes evolution through a unitary operator $U(\boldsymbol{\theta})$ under a specified ansatz. Finally, selected observable ensembles are measured to extract classical information. With this architecture and the initial point $(\boldsymbol{m}_0, \boldsymbol{x}_0)$, the state pairs $(\boldsymbol{m}_1,\boldsymbol{x}_1),(\boldsymbol{m}_2,\boldsymbol{x}_2),(\boldsymbol{m}_3,\boldsymbol{x}_3),\dots$ are obtained in an autoregressive way. (b) The protocol of QRC and QRNNs. The data is injected into the quantum system and measured to retrieve classical information. The memory register maintains the quantum state form and will lose coherence in the long run. (c) An enhanced QDE architecture with $M$ linear-composite channels. In this design, the QDE is available to explore the dynamics with higher degrees of freedom by either scaling the register qubit number or the composite channel number. The final signal $\boldsymbol{x}_t$ is retrieved through the superposition of sub-signals $\boldsymbol{x}_t^1, \dots, \boldsymbol{x}_t^M$ from different channels. (d) Training and predicting protocols. In the training stage, initial data $\boldsymbol{x}_0$ is fed into the QDE block, and the output data $\hat{\boldsymbol{x}}_{i}$ is used to compute the cost function, with $i= {1,\cdots,L}$ and time updated over $L$ steps. After the training, the system undergoes an additional $T$ steps of evolution to predict future values. Note that while the memory evolves simultaneously with the data in both stages, it does not contribute to the cost function calculation.
• Figure 2: Demonstrations of QDE for Time Series Prediction: (a) $(1,1)$-QDE application on cosine-wave prediction with $x(t) = 0.5\cos(\omega t)$, where $\omega = \pi/25$ and the value of time axis is rescaled by $\Delta=0.04$. The training and predicting lengths are set to $L=T=100$ in this and the following tests, unless otherwise specified. (b) Composite periodic signal predicting, $x(t) = 0.2\cos(\omega t) + 0.3\sin(2\omega t)$, where $\omega = \pi/25$. (c) Aperiodic time series predicting, $x(t) = 0.2\cos(\omega t) + 0.3\sin(\sqrt{5}\omega t)$, with the same architecture and settings as (b). (d), (e), and (f) correspond to the noisy versions of the scenarios described in (a), (b), and (c), respectively, trained with noise uniformly distributed within $[-0.1, 0.1]$. For these versions, the number of training points is increased to enhance the capture of signal characteristics buried in noise. The prediction part of each subplot is juxtaposed with theoretical clean signals for comparison. The red dashed line represents the deviation $\delta = \hat{x}(t)-x_{\text{ref}}(t)$, where $\hat{x}(t)$ is the output of the QDE model and $x_{\text{ref}}(t)$ is the reference, the noisy signal in training stage and theoretical signal in predicting stage. (g) Prediction of NARMA2 data using the $(2, 1)$-QDE model. The training length is set to 500 steps, with a fixed prediction length. For clarity, only the last 200 training points are shown. The signal is generated following the formula described in the main text.
• Figure 3: Performance Scaling of the QDE Model for Cosine-wave Signal Prediction Under Two Scenarios. (a)-(c) The first row demonstrates the effects of increasing the total number of channels in a $(1,1)$-QDE configuration. (d)-(e) The second row illustrates performance changes when scaling up the number of internal qubits within a single QDE block (i.e., increasing $n_x$ and $n_m$). Each violin plot represents the distribution of MSE outcomes across 50 trials, highlighting median trends and variability associated with different circuit depths and qubit counts.
• Figure 4: Comparison of QDE with QRC on Cosine-Wave Signal Prediction. (a) Error map for different values of the Hamiltonian hyper-parameters of QRC, with the transverse field $h$ and the disorder bound $W$ varying logarithmically in the range $[10^{-2}, 10^2]$. Results are averaged over 100 realizations. (b) and (c) With $h/J_s$ ($W/J_s$) fixed at the minimum error from map (a), the other hyper-parameter $W/J_s$ ($h/J_s$) is varied. The average results are shown with the minimum to maximum error range (grey shadows) and standard deviation (red shadows). The reference line for the QDE model represents the average performance over different circuit parameter initializations using a single $(1,1)$-QDE channel. Since the QDE used here is based on a HEA architecture without hyper-parameters, it is naturally independent of the x-axis values.
• Figure 5: Performance of the QDE model on NARMA2. The four subfigures are arranged from left to right by increasing qubit count. Each violin plot represents the distribution over 50 trials, consistent with previous tests.

Theorems & Definitions (11)

• Definition 1: Training a data-driven model
• Lemma 1
• Lemma 2
• Theorem 1: Single mode cosine-wave approximation
• Theorem 2: Composite cosine-wave approximation
• Theorem 3: Approximating any continuous function
• proof
• proof
• proof
• proof