Optimal training of finitely-sampled quantum reservoir computers for forecasting of chaotic dynamics

TL;DR

This work analyzes how finite sampling noise affects the chaotic time series prediction of the gate-based QRC and recurrence-free quantum reservoir computing (RF-QRC) models and demonstrates that the training and denoising of the noisy reservoir activation signals in RF-QRC are highly parallelizable on multiple quantum processing units (QPUs).

Abstract

In the current Noisy Intermediate Scale Quantum (NISQ) era, the presence of noise deteriorates the performance of quantum computing algorithms. Quantum Reservoir Computing (QRC) is a type of Quantum Machine Learning algorithm, which, however, can benefit from different types of tuned noise. In this paper, we analyse the effect that finite-sampling noise has on the chaotic time-series prediction capabilities of QRC and Recurrence-free Quantum Reservoir Computing (RF-QRC). First, we show that, even without a recurrent loop, RF-QRC contains temporal information about previous reservoir states using leaky integrated neurons. This makes RF-QRC different from Quantum Extreme Learning Machines (QELM). Second, we show that finite sampling noise degrades the prediction capabilities of both QRC and RF-QRC while affecting QRC more due to the propagation of noise. Third, we optimize the training of the finite-sampled quantum reservoir computing framework using two methods: (a) Singular Value Decomposition (SVD) applied to the data matrix containing noisy reservoir activation states; and (b) data-filtering techniques to remove the high-frequencies from the noisy reservoir activation states. We show that denoising reservoir activation states improve the signal-to-noise ratios with smaller training loss. Finally, we demonstrate that the training and denoising of the noisy reservoir activation signals in RF-QRC are highly parallelizable on multiple Quantum Processing Units (QPUs) as compared to the QRC architecture with recurrent connections. The analyses are numerically showcased on prototypical chaotic dynamical systems with relevance to turbulence. This work opens opportunities for using quantum reservoir computing with finite samples for time-series forecasting on near-term quantum hardware.

Figures (14)

• Figure 1: Schematic representation of a reservoir computer jaeger__2001. The input data $\pmb{u}_{in}$ is mapped to the reservoir matrix via $\pmb{W}_{in}$. The reservoir neuron connections governed by $\pmb{W}$ matrix allow the flow of information between neurons. The linear readout layer using the trained $\pmb{W}_{out}$ matrix is used to make output predictions $\pmb{u}_{p}$.
• Figure 2: Training phase in reservoir computing. Input time series $\pmb{u}_{in}(t)$ is mapped to a reservoir using $\pmb{W}_{in}$ matrix for CRC and suitable encoding schemes for QRC. Inside reservoir $r$, each neuron echoes with the input time series to generate a series of reservoir activation state signals. The reservoir activation state signals are concatenated in the reservoir state matrix $\pmb{(R)}$, which is then used for finding optimal output weight matrix $\pmb{W}_{out}$ in RC training using ridge regression.
• Figure 3: Prediction of the MFE time-series with (a) noise-free probability distribution (b) 0.5$\times 10^5$ shots (c) 2$\times 10^5$ shots (d) 4$\times 10^5$ shots.
• Figure 4: Averaged signal-to-noise ratios (SNR) of the reservoir states of QRC and RF-QRC with and without denoising filters. Underlying model is an MFE times series.
• Figure 5: Example of reservoir activation state time series with 50k shots at each timestep for RF-QRC of the MFE system. Top, visualization of the noisy reservoir activation signal and its noise component; bottom, visualization of the denoised reservoir activation signal and its noise component.