Exploring quantum mechanical advantage for reservoir
computingNiclas Götting1 , Frederik Lohof1 , Christopher Gies1
1. Institute for Theoretical Physics, University of Bremen, Bremen, Germany

Summary. Quantum reservoir computing is an emerging field in machine learning with quantum
systems. Here, we establish a link between quantum properties of a quantum reservoir, namelyentanglement and its occupied phase space dimension, and its classical linear short-term memory
capacity. We find that a high degree of entanglement in the reservoir is a prerequisite for a more
complex reservoir dynamics that is key to unlocking the exponential phase space and higher short-term
memory capacity. We quantify these relations and discuss the effect of dephasing on the performanceof physical quantum reservoirs.

The machine learning paradigm of reservoir computing has provento be well-suited for various applications such as autonomous learningand time-series prediction [1, 2]. While classically implemented reservoir
computers [3] have been researched extensively, their quantum mechanical counterparts [4] only now are coming into the focus of the researchcommunity. The promises they bring to the table are, among others:• exponential phase space dimension scaling with system size,• entanglement as a resource with no classical analogue.
The study of quantum reservoir computers (QRC) is now more im- Figure 1: A possible physicalportant than ever, as with the advent of sophisticated semiconduc- implementation of a quantum
tor fabrication techniques for quantum photonic systems like coupled- reservoir computer: coupledcavity arrays (see Fig. 1) QRCs are at the brink of realization. cavity arrays. Image taken fromIn preparation to this, we investigate the fundamental properties [5].
of QRCs on the basis of the transverse-field Ising model (TFIM).
Using the linear short-term memory (STM) capacity CSTM as an established, classical benchmark, wemeasure the QRC’s performance in relation to several of its key figures, specifically the TFIM couplingstrength J0 , the mean logarithmic negativity ĒN , and the covariance dimension Dc – the latter of whichallow to probe the intrinsic quantum properties of the QRC [6].
The logarithmic negativity EN ∈ [0, 1] is given byĒNI0.07IIEN (ρ) = log2 ρΓA 1 ,(1)where ρΓA is the partial transpose of ρ with respect to subsystem A and ∥·∥1 denotes the trace norm. As an entanglement
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monotone, it provides a method for quantifying entanglement
between any two bipartitions of the quantum system. Here,
0.01691215
we calculate the average logarithmic negativity over all biparJ0Dc
2
0.5
0.4
0.3
0.2
titions and the whole reservoir dynamics to obtain a single
number ĒN as a measure for the entanglement strength during
4IVIII
execution of the short-term memory task.
6The time evolution of the QRC state can be understoodas a path in its 4N -dimensional phase space, where N is the8number of qubits in the system. To investigate, whether
CSTM
the QRC actually takes advantage of the full phase space or
only evolves on a lower-dimensional manifold, we employ aFigure 2: All relations between coupling measure called the covariance dimension. For this, random
strength J0 , mean logarithmic negativity clusters from the QRC’s discretely sampled state evolution
ĒN , covariance dimension Dc and linear are chosen and their spatial extent is approximated by ashort-term memory capacity CSTM .
principal component analysis (PCA). By counting the numberof principal components which are above a certain threshold,the covariance dimension for each cluster is then determined. The overall covariance dimension Dc isobtained by averaging over all individual covariance dimensions of the clusters.
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As can be seen for the example of a 3-qubit TFIM in Fig. 2,
all four aforementioned properties are directly linked to eachother. Upon increasing the qubit coupling strength, we finda stronger mean entanglement of the QRC (see quadrant I)
and a higher dimensionality of the phase space submanifold(see quadrant II). We explain this by the fact that the input
injection method applied here for the execution of the STM
task is given by projective measurement, which by definition
destroys the entanglement of the input qubit with the rest ofthe system. As we keep the input injection frequency constant,the increased speed of the reservoir dynamics resulting fromlarger coupling strengths allows the quantum system to build Figure 3: Influence of dephasing on the 3up more entanglement and explore a higher dimensional phase qubit QRC for different coupling strengths.
space before the input injection collapses the state again.
Interestingly, the stronger expression of these quantum properties also results in a better performanceof the QRC in the classical linear short-term memory task as shown in quadrants III and IV of Fig. 2,respectively.
In real-world scenarios, the interaction of the QRC with the environment plays an important role andshould therefore be investigated systematically. For this matter, we introduce a channel to the system,
which simulates a loss of quantum entanglement due to interaction with the environment. This so-calleddephasing channel can be tuned by the dephasing strength γ. While we see a decrease in performance ofstrongly coupled TFIM-QRCs when increasing the dephasing, the opposite effect appears in the weaklycoupled regime for a small amount of dephasing, as can be seen in Fig. 3. This poses the possibility ofusing the ubiquitous decoherence effects in noisy intermediate-scale quantum devices as a resource forquantum reservoir computing, which are otherwise undesired e.g. for gate based quantum computing.
In summary, we make a first step to assess the influence of quantum effects on the performance ofQRCs in classical tasks. Future research will concern the question of how to translate between descriptionsof coherent quantum systems on the one hand and nonlinear network dynamics on the other. We aim toidentify effects of tunable parameters also for photonic semiconductor systems such as the coupled-cavity
array.

References[1] K. C. Chatzidimitriou und P. A. Mitkas, ”Adaptive reservoir computing through evolution andlearning”, Neurocomputing, 2013.
[2] D. J. Gauthier, E. Bollt, A. Griffith, und W. A. S. Barbosa, ”Next generation reservoir computing”,
Nat. Comm., 2021.
[3] H. Jaeger und H. Haas, ”Harnessing Nonlinearity: Predicting Chaotic Systems and Saving Energy in
Wireless Communication”, Science, 2004.
[4] K. Fujii und K. Nakajima, ”Harnessing Disordered-Ensemble Quantum Dynamics for Machine
Learning”, Physical Review Applied, 2017.
[5] T. Heuser, J. Große, S. Holzinger, M. M. Sommer and S. Reitzenstein, ”Development of Highly
Homogenous Quantum Dot Micropillar Arrays for Optical Reservoir Computing,” IEEE Journal of
Selected Topics in Quantum Electronics, 2020.
[6] N. Götting, F. Lohof and C. Gies, ”Exploring quantum mechanical advantage for reservoir computing,”
arXiv:2302.03595, 2023.