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Quantum Wiener architecture for quantum reservoir computing

Alessio Benavoli, Felix Binder

TL;DR

The paper addresses temporal processing with quantum reservoir computing by introducing a quantum Wiener architecture (qWiener) that pairs quantum linear dynamics with weak measurements and a classical nonlinear readout. It proves fading-memory and universal approximation despite quantum constraints and provides a kernel-theoretic interpretation showing deep kernels emerge from the reservoir, thereby linking quantum RC to kernel methods. The analysis of a harmonic-oscillator instantiation demonstrates passivity and practical implementability, while experiments show qWiener outperforms prior quantum and classical RC models on standard benchmarks, with expressivity growing with reservoir size. By casting the readout within a Gaussian-process framework, the approach affords principled uncertainty quantification and suggests a path toward deeper quantum RC architectures and Bayesian analyses.

Abstract

This work focuses on quantum reservoir computing and, in particular, on quantum Wiener architectures (qWiener), consisting of quantum linear dynamic networks with weak continuous measurements and classical nonlinear static readouts. We provide the first rigorous proof that qWiener systems retain the fading-memory property and universality of classical Wiener architectures, despite quantum constraints on linear dynamics and measurement back-action. Furthermore, we develop a kernel-theoretic interpretation showing that qWiener reservoirs naturally induce deep kernels, providing a principled framework for analysing their expressiveness. We further characterise the simplest qWiener instantiation, consisting of concatenated quantum harmonic oscillators, and show the difference with respect to the classical case. Finally, we empirically evaluate the architecture on standard reservoir computing benchmarks, demonstrating systematic performance gains over prior classical and quantum reservoir computing models.

Quantum Wiener architecture for quantum reservoir computing

TL;DR

The paper addresses temporal processing with quantum reservoir computing by introducing a quantum Wiener architecture (qWiener) that pairs quantum linear dynamics with weak measurements and a classical nonlinear readout. It proves fading-memory and universal approximation despite quantum constraints and provides a kernel-theoretic interpretation showing deep kernels emerge from the reservoir, thereby linking quantum RC to kernel methods. The analysis of a harmonic-oscillator instantiation demonstrates passivity and practical implementability, while experiments show qWiener outperforms prior quantum and classical RC models on standard benchmarks, with expressivity growing with reservoir size. By casting the readout within a Gaussian-process framework, the approach affords principled uncertainty quantification and suggests a path toward deeper quantum RC architectures and Bayesian analyses.

Abstract

This work focuses on quantum reservoir computing and, in particular, on quantum Wiener architectures (qWiener), consisting of quantum linear dynamic networks with weak continuous measurements and classical nonlinear static readouts. We provide the first rigorous proof that qWiener systems retain the fading-memory property and universality of classical Wiener architectures, despite quantum constraints on linear dynamics and measurement back-action. Furthermore, we develop a kernel-theoretic interpretation showing that qWiener reservoirs naturally induce deep kernels, providing a principled framework for analysing their expressiveness. We further characterise the simplest qWiener instantiation, consisting of concatenated quantum harmonic oscillators, and show the difference with respect to the classical case. Finally, we empirically evaluate the architecture on standard reservoir computing benchmarks, demonstrating systematic performance gains over prior classical and quantum reservoir computing models.
Paper Structure (30 sections, 14 theorems, 99 equations, 4 figures, 3 tables)

This paper contains 30 sections, 14 theorems, 99 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Background and Novel Preliminaries
  3. Classical Wiener architecture and connection to kernel methods
  4. Quantum systems
  5. Linear quantum dynamical systems
  6. Quantum Kalman filter
  7. Main results
  8. One-dimensional quantum harmonic oscillator
  9. Limit Kernel
  10. Universality of quantum Wiener architectures
  11. Connection to previous work
  12. Numerical experiments
  13. Exploiting universality
  14. Implementation
  15. Conclusions
  16. ...and 15 more sections

Key Result

Lemma 1

For $i=1,\dots,n_c$, assume that the parameters $\alpha_{i},\omega_{i}$ of the impulse response eq:h are jointly sampled from a PDF $\phi_{\boldsymbol{\theta}}(\alpha,\omega)$ (depending on some hyperparameters ${\boldsymbol{\theta}}$), and $\beta_{i},\gamma_{i} \sim N(0,1/\sqrt{n_{c}})$ independent for all time instants $t,s$. Then, in the limit as $n_c \rightarrow \infty$, $E[h(t)h(s)]$ tends to

Figures (4)

  • Figure 1: Wiener architecture: connection of $d$ parallel linear SISO dynamical systems ($h_i$) with a memoryless nonlinear function $f$. $u(t),o(t)$ are the input and output signal respectively.
  • Figure 2: Quantum Wiener architecture obtained by concatenating $d$ quantum harmonic oscillators.
  • Figure 3: Prediction on a test set for the NARMA10 benchmark: true-signal in gray; HqW($24$) in orange; LqW($24$) in green; PadeqW($14$) in red; ESN($200$) in blue.
  • Figure 4: Prediction on a test set for the NARMA10 benchmark for HqW($24$): the mean is the orange line and the orange-shaded region is the 95% credible interval.

Theorems & Definitions (26)

  • Definition 1: Gaussian Process
  • Lemma 1: zorzi2018harmonic
  • Proposition 1: zorzi2018harmonic
  • Remark 1
  • Proposition 2
  • Remark 2
  • Proposition 3
  • Remark 3
  • Remark 4
  • Remark 5
  • ...and 16 more