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arXiv:2409.12079 · Physical Review Research

Engineering quantum reservoirs through Krylov complexity, expressivity, and observability

Saud Čindrak, Lina Jaurigue, Kathy Lüdge

Published Aug 22, 2025• 4 citations
arXivHTMLPDF

Abstract

This study employs Krylov-based information measures to understand task performance in quantum reservoir computing, a subfield of quantum machine learning. In our study, we show that fidelity and spread complexity can only explain the task performance for short time evolutions of the quantum systems. We then discuss two measures, Krylov expressivity and Krylov observability, and compare them to task performance and the information processing capacity. Our results show that Krylov observability exhibits almost identical behavior to information processing capacity, while being three orders of magnitudes faster to compute. In the case when the system is undersampled, Krylov observability best captures the behavior of the task performance.

DOI
10.1103/84f3-63mz

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This study employs Krylov-based information measures to understand task performance in quantum reservoir computing, a subfield of quantum machine learning. In our study, we show that fidelity and spread complexity can only explain the task performance for short time evolutions of the quantum systems. We then discuss two measures, Krylov expressivity and Krylov observability, and compare them to task performance and the information processing capacity. Our results show that Krylov observability exhibits almost identical behavior to information processing capacity, while being three orders of magnitudes faster to compute. In the case when the system is undersampled, Krylov observability best captures the behavior of the task performance.
We demonstrate that time-evolved operators can construct a Krylov space to compute operator complexity and introduce Krylov observability as a measure of effective phase space dimension in quantum systems. We test Krylov observability in the framework of quantum reservoir computing and show that it closely mirrors information processing capacity, a data-driven expressivity metric, while achieving computation times that are orders of magnitude faster. Our results validate operator complexity and give the interpretation that data in a quantum reservoir are mapped onto the Krylov space.
Quantum reservoir computing algorithms recently emerged as a standout approach in the development of successful methods for the noisy intermediate-scale quantum (NISQ) era because of its superb performance and compatibility with current quantum devices. By harnessing the properties and dynamics of a quantum system, quantum reservoir computing effectively uncovers hidden patterns in data. However, the design of the quantum reservoir is crucial to this end in order to ensure an optimal performance of the algorithm. In this work, we introduce a precise quantitative method with strong physical foundations based on the Krylov evolution to assess the wanted good performance in machine-learning tasks. Our results show that the Krylov approach to complexity strongly correlates with quantum reservoir performance, making it a powerful tool in the quest for optimally designed quantum reservoirs, which will pave the road to the implementation of successful quantum machine-learning methods.
Quantum reservoirs offer a hardware-friendly route to quantum machine learning, replacing trainable circuits with fixed random dynamics and a classical readout. Because the reservoir is not optimized, performance depends entirely on the choice of reservoir family, yet existing diagnostics demand resources that grow exponentially with system size. We introduce a scalable, hardware-agnostic framework built on two complementary quantities. The first is a task-independent order-statistics (ORS) expressivity score, which compares only the largest output probabilities of a reservoir ensemble against an analytical Haar baseline. It never reconstructs the full output distribution, is cost-independent of Hilbert-space dimension, and admits a closed-form depolarizing noise correction, making it directly usable on hardware. The second is the task-dependent effective rank ReffR_{\mathrm{eff}}Reff​ of the feature matrix, which measures how much input-dependent information reaches the readout. We validate the ORS score against established complexity diagnostics and confirm it remains informative under simulated noise and on IBM quantum hardware. Across synthetic and real quantum extreme learning machine and quantum reservoir computing benchmarks, ORS captures the intrinsic expressivity hierarchy of reservoir families while ReffR_{\mathrm{eff}}Reff​ determines when that expressivity becomes usable predictive information.
Quantum reservoir computing algorithms recently emerged as a standout approach in the development of successful methods for the NISQ era, because of its superb performance and compatibility with current quantum devices. By harnessing the properties and dynamics of a quantum system, quantum reservoir computing effectively uncovers hidden patterns in data. However, the design of the quantum reservoir is crucial to this end, in order to ensure an optimal performance of the algorithm. In this work, we introduce a precise quantitative method, with strong physical foundations based on the Krylov evolution, to assess the wanted good performance in machine learning tasks. Our results show that the Krylov approach to complexity strongly correlates with quantum reservoir performance, making it a powerful tool in the quest for optimally designed quantum reservoirs, which will pave the road to the implementation of successful quantum machine learning methods.
We compare Krylov's state complexity with an information-geometric (IG) measure of complexity for the quantum evolution of two-level systems. Focusing on qubit dynamics on the Bloch sphere, we analyze evolutions generated by stationary and nonstationary Hamiltonians, corresponding to geodesic and nongeodesic trajectories. We formulate Krylov complexity in geometric terms, both instantaneously and in a time-averaged sense, and contrast it with an IG complexity of quantum evolutions characterized in terms of efficiency and curvature. We show that the two measures reflect fundamentally different aspects of quantum dynamics: Krylov's state complexity quantifies the directional spread of the evolving state relative to the initial state, whereas the IG complexity captures the effective volume explored along the trajectory on the Bloch sphere. This geometric distinction explains their inequivalent behavior and highlights the complementary nature of state-based and information-geometric notions of complexity in quantum systems.
Abstract Quantum reservoir computing is a neuro‐inspired machine learning approach harnessing the rich dynamics of quantum systems to solve temporal tasks. It has gathered attention for its suitability for NISQ devices, for easy and fast trainability, and for potential quantum advantage. Although several types of systems have been proposed as quantum reservoirs, differences arising from particle statistics have not been established yet. In this work, the ability of bosons, fermions, and qubits are assessed and compared to store information from past inputs by measuring linear and nonlinear memory capacity. While, in general, the performance of the system improves with the Hilbert space size, it is shown that also the information spreading capability is a key factor. For the simplest reservoir Hamiltonian choice, and for each boson limited to at most one excitation, fermions provide the best reservoir due to their intrinsic nonlocal properties. On the other hand, a tailored input injection strategy allows the exploitation of the abundance of degrees of freedom of the Hilbert space for bosonic quantum reservoir computing and enhances the computational power compared to both qubits and fermions.