Predicting Chaotic Systems with Quantum Echo-state Networks

TL;DR

This paper tackles time-series prediction of chaotic systems by introducing a Quantum Echo-State Network (QESN), a gate-based quantum reservoir that processes input through a context-window embedding into memory and readout registers, using data reuploading and sparse entanglement to generate fading memory. The main contribution is a NISQ-friendly architecture that yields reservoir features from random sparse weights, with outputs sampled from the quantum reservoir and mapped to targets via classical elastic-net regression. Key findings show that full probability-distribution readouts improve predictive RMSE over simple qubit expectations, and that noiseless QESN simulations can outperform a classical ESN when reservoir size matches the readout qubits, though real hardware noise degrades performance. The work suggests quantum reservoirs can scale to more complex dynamics and PDE predictions, with future work on entanglement schemes and data embeddings to enhance hardware robustness.

Abstract

Recent advancements in artificial neural networks have enabled impressive tasks on classical computers, but they demand significant computational resources. While quantum computing offers potential beyond classical systems, the advantages of quantum neural networks (QNNs) remain largely unexplored. In this work, we present and examine a quantum circuit (QC) that implements and aims to improve upon the classical echo-state network (ESN), a type of reservoir-based recurrent neural networks (RNNs), using quantum computers. Typically, ESNs consist of an extremely large reservoir that learns high-dimensional embeddings, enabling prediction of complex system trajectories. Quantum echo-state networks (QESNs) aim to reduce this need for prohibitively large reservoirs by leveraging the unique capabilities of quantum computers, potentially allowing for more efficient and higher performing time-series prediction algorithms. The proposed QESN can be implemented on any digital quantum computer implementing a universal gate set, and does not require any sort of stopping or re-initialization of the circuit, allowing continuous evolution of the quantum state over long time horizons. We conducted simulated QC experiments on the chaotic Lorenz system, both with noisy and noiseless models, to demonstrate the circuit's performance and its potential for execution on noisy intermediate-scale quantum (NISQ) computers.

Figures (4)

• Figure 1: Proposed QESN framework. Data is input as a sliding context window over time through a fully connected classical layer, and then passed through a dynamic quantum reservoir that is repeatedly sampled to generate a probability distribution over the $2^\frac{n_q}{2}$ Hilbert space at each time step. With the collected probability distributions, a classical least-squares optimization process fits the reservoir states to a target signal.
• Figure 2: Test prediction from X to Y component in the Aer simulator with 16 qubits.
• Figure 3: Test prediction from X to Z component in the Aer simulator with 16 qubits.
• Figure 4: Test set loss with classical reservoir, basic linear regression, and QESN architecture.