Learning Nonlinear Input-Output Maps with Dissipative Quantum Systems

TL;DR

We address learning nonlinear input-output maps with fading memory using dissipative quantum systems. The authors develop a rigorous learning framework, prove universality for a provably universal quantum reservoir class, and validate performance through numerical experiments showing competitive results with classical schemes using far more tunable parameters. The work highlights the potential advantage of the exponentially growing Hilbert space in quantum systems for time-series emulation and points to practical pathways for implementing such models on NISQ devices. Overall, the paper provides both a theoretical foundation and empirical evidence that dissipative quantum systems can approximate any fading-memory I/O map and may surpass classical approaches when scaling harnesses the quantum state space.

Abstract

In this paper, we develop a theory of learning nonlinear input-output maps with fading memory by dissipative quantum systems, as a quantum counterpart of the theory of approximating such maps using classical dynamical systems. The theory identifies the properties required for a class of dissipative quantum systems to be {\em universal}, in that any input-output map with fading memory can be approximated arbitrarily closely by an element of this class. We then introduce an example class of dissipative quantum systems that is provably universal. Numerical experiments illustrate that with a small number of qubits, this class can achieve comparable performance to classical learning schemes with a large number of tunable parameters. Further numerical analysis suggests that the exponentially increasing Hilbert space presents a potential resource for dissipative quantum systems to surpass classical learning schemes for input-output maps.

Figures (12)

• Figure 1: Typical SA outputs during the evaluation phase, for the (a) LRPO, (b) Missile (c) NARMA15 and (d) NARMA20 tasks. The leftmost, middle and rightmost panels show the outputs for timesteps 1501-1530, 2001-2030 and 2471-2500, respectively
• Figure 2: Average SA NMSE for the (a) LRPO, (b) Missile, (c) NARMA15 and (d) NARMA20 tasks, the error bars represent the standard error. For comparison, horizontal dashed lines labeled with "E$m$" indicate the average performance of ESNs with $m$ computational nodes, and horizontal dot-dashed lines labeled with "V$o,p$" indicates the performance of Volterra series with kernel order $o$ and memory $p$. Overlapping dashed and dot-dashed lines are represented as dashed lines
• Figure 3: Average SA NMSE for the LRPO, Missile, NARMA15 and NARMA20 tasks under decoherence. For comparison, the average SA NMSE without the effect of noise is also plotted. In all plots, the error bars represent the standard error
• Figure 4: Average sum of complex modulus of off-diagonal elements in the system density operator for timesteps 1501-1550, under the (a) dephasing noise, (b) decaying noise, (c) GAD with $\lambda=0.4$ and (d) GAD with $\lambda=0.6$. Row $n-1$ in the figure corresponds to the average sum for $n$-qubit SA
• Figure 5: Average NMSE for different input encodings, for approximating the (a) LRPO, (b) Missile (c) NARMA15 and (d) NARMA20 tasks. Error bars represent the standard error

Theorems & Definitions (20)

• Definition 1: Convergence
• Definition 2: Mixing
• Theorem 3: Convergence property
• proof
• Lemma 4
• proof
• Definition 5: Weighted norm
• Definition 6: Fading memory
• Lemma 7: Compactness
• Theorem 8: Stone-Weierstrass