University of Tokyo Researchers Redefine Quantum Reservoir Computing with Non-Stationary Echo State Property

Researchers from the University of Tokyo have expanded the Echo State Property (ESP) concept in reservoir computing (RC) to include non-stationary systems. They introduced two new categories: non-stationary ESP for potentially non-stationary systems, and subspace-subset ESP for systems whose subsystems have ESP. The team numerically demonstrated the correspondence between non-stationary ESP in the quantum reservoir computer (QRC) framework with typical Hamiltonian dynamics and input encoding methods. The study offers a new understanding of the practical design of QRC and other possibly non-stationary RC systems.

What is the Echo State Property in Quantum Reservoir Computing?

The Echo State Property (ESP) is a fundamental concept in the reservoir computing (RC) framework. It ensures output-only training of reservoir networks by being agnostic to the initial states and far past inputs. However, the traditional definition of ESP does not describe possible non-stationary systems in which statistical properties evolve. To address this issue, researchers from the University of Tokyo have introduced two new categories of ESP: non-stationary ESP, designed for potentially non-stationary systems, and subspace-subset ESP, designed for systems whose subsystems have ESP.

The researchers numerically demonstrated the correspondence between non-stationary ESP in the quantum reservoir computer (QRC) framework with typical Hamiltonian dynamics and input encoding methods using nonlinear autoregressive moving-average (NARMA) tasks. They also confirmed the correspondence by computing linear-nonlinear memory capacities that quantify input-dependent components within reservoir states. This study presents a new understanding of the practical design of QRC and other possibly non-stationary RC systems in which non-stationary systems and subsystems are exploited.

How Does Physical Reservoir Computing Work?

Physical reservoir computing (PRC) utilizes nonlinear natural dynamics of physical substrate for temporal information processing. It has garnered much attention as a way to mitigate the massive computational resource needs of sophisticated machine learning methods such as deep learning. However, not all physical systems are effective as reservoir substrates due to potential initial-state sensitivity in their natural dynamics, such as in chaotic systems. One precondition for excluding such systems is the echo state property (ESP), which requires the initial state dependency to diminish over time.

Noisy intermediate-scale quantum (NISQ) describes non-fault-tolerant quantum computer environments. In the NISQ era, non-universal quantum computation schemes gained much attention because of their near-term feasibility on physical devices. Such computational procedures include, for instance, variational quantum computation (VQC) and quantum reservoir computing (QRC). VQC and QRC apply to one-shot and autoregressive quantum machine learning algorithms, which have also become a general prospective application of quantum computation.

What is Quantum Reservoir Computing?

Quantum Reservoir Computing (QRC) can be understood as a specific type of PRC that uses a quantum system as its physical reservoir. It has recently been shown to be capable of implementing temporal quantum tomography, predicting large-scale spatiotemporal chaos, and emulating functions requiring both classical and quantum inputs simultaneously with a single quantum reservoir. Other works on QRC include proposals of QRC in various physical apparatus, with some performing actual physical experiments and theoretical analyses.

Specifically, some works focus on the dissipative nature of the natural quantum system to find a relationship between the existence of dissipation and the trainability of QRC. Quantum systems have been attracting attention as promising substrates for PRC. However, quantum systems in general are not always stationary and in some cases, the traditional definition of ESP does not adequately ensure their capability for temporal information processing.

How Does Non-stationary ESP Work?

In this paper, the researchers define a two-way extension of traditional ESP. One direction is non-stationary ESP, which requires finite variance output signals relative to initial-state difference decay. Another direction is subset-subspace non-stationary ESP, which focuses on a situation in which a part of the system has a non-stationary ESP, leaving the entire system possibly initial-state dependent. The analysis includes a numerical study of non-stationary ESP and subset non-stationary ESP in a typical QRC setup with a specific type of Hamiltonian system.

The researchers have made the following contributions: They defined non-stationary and subspace-subset versions of ESP, which could be practical for QRC and other non-conventional systems. They showed a relationship between non-stationary ESP and the information processing capability of QRC using numerical experiments.

What are the Main Results of the Study?

First, the researchers present the definition of traditional ESP, which has been used extensively in the RC context. They argue that this ESP definition by state difference decay is general in that all known definitions of ESP are equivalent to this form.

Although ESP is supposed to work on stationary systems, a quantum system, for instance, is not always stationary even if the system state does not explicitly depend on time. A trivial example is the case in which a Pauli noise exists. The researchers depict an example where uniform depolarization of a quantum system occurs. In this paper, they define and analyze new conditions that secure such a possibly non-stationary system to function effectively as a practical reservoir.