Opportunities in Quantum Reservoir Computing and Extreme Learning Machines
Pere Mujal, Rodrigo Martínez-Peña, Johannes Nokkala, Jorge García-Beni, Gian Luca Giorgi, Miguel C. Soriano, Roberta Zambrini
OOpportunities in Quantum Reservoir Computing and ExtremeLearning Machines
Pere Mujal, Rodrigo Mart´ınez-Pe˜na, Johannes Nokkala, Jorge Garc´ıa-Beni, Gian Luca Giorgi,Miguel C. Soriano, Roberta Zambrini*
Dr. P. Mujal, R. Mart´ınez-Pe˜na, Dr. J. Nokkala, J. Garc´ıa-Beni, Dr. G. L. Giorgi, Dr. M. C. Soriano, Prof. R.ZambriniIFISC, Instituto de F´ısica Interdisciplinar y Sistemas Complejos (UIB-CSIC)UIB Campus, E-07122 Palma de Mallorca, SpainEmail Address: roberta@ifisc.uib-csic.esDr. J. NokkalaTurku Centre for Quantum Physics, Department for Physics and Astronomy,University of Turku, FI-20014, Turun Yliopisto, Finland
Keywords:
Quantum machine learning, unconventional computing, information processing, reservoircomputing, extreme learning machines, NISQ, neural networks.
Quantum reservoir computing (QRC) and quantum ex-treme learning machines (QELM) are two emerging ap-proaches that have demonstrated their potential bothin classical and quantum machine learning tasks. Theyexploit the quantumness of physical systems combinedwith an easy training strategy, achieving an excellentperformance. The increasing interest in these uncon-ventional computing approaches is fueled by the avail-ability of diverse quantum platforms suitable for imple-mentation and the theoretical progresses in the studyof complex quantum systems. In this perspective arti-cle, we overview recent proposals and first experimentsdisplaying a broad range of possibilities when quantuminputs, quantum physical substrates and quantum tasksare considered. The main focus is the performance ofthese approaches, on the advantages with respect toclassical counterparts and opportunities.
In recent years, we are witnessing an explosion of uncon-ventional computing methods and systems [1, 2]. One ofthe driving motivations for these efforts on unconventionalcomputing is to go beyond von Neumann architectures,physically co-locating processing and memory operations[3, 4]. For the unconventional computing revolution to oc-cur, computational models and computing substrates areto be considered as a whole. Neuromorphic, and more gen-erally neuro-inspired, computing is one of such fields wherethe computational paradigm goes hand in hand with thedesign of the physical substrate, aiming at approaching thecomputational power of the human brain [5].Machine learning, and in particular the field of artifi-cial neural networks (NN), can similarly benefit from theprogress in neuro-inspired computing devices [6–8]. Thepotential to build systems that are orders of magnitudemore energy efficient than traditional ones is a major keymotivation [9]. As computing approaches get closer to con-siderations on their physical substrates, the analog prop- erties of physical systems come into focus [10]. Thanks torecent advances, artificial NN are envisioned to be run evenon top of analog quantum computing devices [11–13], withthe possibility to exploit the advantages of superpositionin quantum computing and the parallelism in neural com-puting. The implications of combining machine learningand quantum physics indeed represents a major avenue forresearch in the coming years [14–17].In this perspective article, we concentrate on the poten-tial of quantum devices for reservoir computing (RC) andextreme learning machines (ELM). RC and ELM are ma-chine learning paradigms that exploit the natural dynam-ics of input-driven randomly connected NN for informationprocessing [18, 19]. The main advantage of the RC andELM concepts in the context of artificial NN is their mini-mal requirements for learning (usually referred as trainingin the machine learning literature) [20, 21]. Figure 1 il-lustrates the three layers typical of RC and ELM, namelyan input layer, a hidden layer or substrate, and an outputlayer. In RC and ELM, one only needs to adjust the weightsof the output connections via e.g. linear regression (see Fig-ure 1), while the rest of the connections can be initializedwith random weights and are not optimized. Despite thesimplicity in the training, these methods have been success-ful in numerous practical applications [22, 23]. AlthoughRC and ELM share many features, a main difference re-sides in the fact that RC exploits the natural dynamics ofthe substrate as an internal memory of past input informa-tion while ELM does not. Both RC and ELM are amenableto dedicated hardware implementations in e.g. digital elec-tronics [24, 25] or nonlinear analog systems [26, 27]. Thequantum counterparts of RC and ELM will be referred inthe following as QRC and QELM, respectively. As commonin the literature on these topics, we consider as quantumRC and ELM if they are based on quantum substrates andthis will be the main focus of this article. We note thatthis is a more restrictive definition with respect to what isusually considered as quantum in machine learning (whereeither data or computing device can be non-classical).1 a r X i v : . [ qu a n t - ph ] F e b NPUT
SUBSTRATE
OUTPUT T R A I N I N G Figure 1: Schematic representation of the basic ingredientsfor RC and ELM. The information from the input is fedinto the substrate, which acts as a hidden layer or reser-voir. The response of the substrate, through a selectionof observables, is then used to produce the desired outputafter optimization of the output connections by training.Quantum systems exhibit a large number of degrees offreedom that can be exploited for QRC and QELM. Sincethe pioneering proposal of networks of quantum spins asreservoir substrates [28], there have been a variety of worksexploring the possibilities that quantum mechanics can of-fer to this research area. In particular, the spin-based im-plementation is the most analyzed platform for QRC at themoment [29–36], with the continuous-variable systems be-coming another promising option [37,38]. In the context ofQELM, both fermionic and bosonic setups have been pro-posed for instance in entanglement detection [39] or lightfield phase estimation [40]. Inspired by these neuromorphicapproaches other quantum tasks have been reported such asquantum states preparation [41, 42] or reconstruction [43].Proof-of-principle experiments are also ongoing [34, 44]. Inparticular, QELM has been reported in a NMR experi-ment [44], while QRC has been realized on the quantumcomputation platform of IBM [32–34].Exploring extended quantum systems as substrates formachine learning in QRC and QELM represents a timelyand potentially disruptive opportunity. First of all, givena substrate of N nodes, the exponential size of the Hilbertspace allows in principle for a substantially enlarged outputwhen compared to classical ones [28]. The extent of thisadvantage when accessing a large number of output degreesof freedom has been quantified for instance in QRC withspin [35] as well as Gaussian networks [37]. While deco-herence is a strong limiting factor in gate-based quantumcomputing, in principle QRC and QELM are well suited forNISQ [45]. Furthermore quantum substrates can interactwith quantum inputs and are naturally suited for quan-tum tasks, providing new avenues for edge computing inquantum systems with increasing complexity. Beyond theopportunities in the context of quantum technologies forcomputation with a quantum advantage, QRC and QELMoffer a new and original perspective to characterize quan-tum physical systems in terms of their operation as sub-strates [46].In the context of quantum machine learning, four ap-proaches are often identified referring to the different com-binations of classical (C) or quantum (Q) data as wellas to the classical (C) or quantum (Q) computing de-vices [47]. This classification in either CC, CQ, QC or QQ (data/device) is also useful to frame the recent advancesin QRC and QELM. Here we will further label the clas-sical or quantum nature of the computational task. Thisinput/substrate/task classification will guide the discussionof existing approaches in this perspective article.The rest of the article is organized as follows. In Sec-tion 2, we give a brief description of the mathematical for-malism for the full classical case and we comment on how toextend it to the quantum case. In Section 2.2 and 2.3, weconcentrate on the different input forms and their treat-ment in order to do a given computational task, respec-tively. In Section 2.4, we analyze the substrates that havebeen proposed and their requirements. The performanceof QRC and QELM in some exemplifying works is under-lined in Section 2.5 and 2.6. In Section 3, we point outthe current challenges in both experimental and theoreti-cal contexts. We end the article by highlighting potentialopportunities for this research field.
In this section, we provide an overall picture of the formal-ism behind RC and ELM before moving on to their quan-tum counterparts in the following sections. We will usethis general framework to assess previous works in QRCand QELM and provide an umbrella for the classificationof future works.All possible situations regarding the classical (C) orquantum (Q) character of the input, the substrate and thetask, respectively are summarized in Table 1. TraditionalRC and ELM fall into the first column of the table, whereasQRC and QELM correspond to the second one. For in-stance, the first proposal of QRC [28] belongs to the CQCclass: a C input is fed into a Q substrate to do a C task . Wewill elaborate on the content of Table 1 along the followingsections. The overview provided by the classification in thetable reflects the richness of possibilities and therefore ofopportunities in this research field.
Classical reservoir computing :In general, RC deals with (physical or artificial) dy-namical systems that can map a given input to their statespace [18, 57]. Here we briefly review RC definition in thebroadly studied CCC scenario. If the states are vectors of N real numbers, x k , and inputs sequential real scalars (forsimplicity), { s k } , then the most general form of such a mapwould be [49] x k = f ( s k , x k − ) , (1)where f is a fixed function determined by, e.g., the sys-tem itself and how the input is injected. Notice that thestate of the substrate at a given time, x k , depends also onpast inputs. This is seen recurrently iterating Eq. 1, e.g.for three steps x k = f ( s k , f ( s k − , f ( s k − , x k − ))) . As anillustration, the input might be a temporal signal such as2able 1: All possible combinations of input, substrate andtask being classical (C) or quantum (Q). Each category islabeled with three letters by specifying the character of theinput, substrate and task, in this order. The sub-index k labels each time step or instance. The sub-index i is asso-ciated to the internal degrees of freedom of the substrate.For the classical input, { s k } is a series of real numbers. Inthe quantum case, ρ ink is a density matrix representing aninput state. The state of the substrate at a given instant isdefined by x k in the classical regime and by the density ma-trix ρ R in the quantum one. For the training process with aclassical substrate, a selection of the substrate variables areused, x outk . With a quantum substrate, the readout for thetraining is obtained after a set of measurements, { O outi } .We distinguish between classical tasks, T C , and quantumtasks, T Q . Classical Substratex k Quantum Substrate ρ S ClassicalInput CCC { s k } , x outk → T C [19, 21, 48–54] CQC { s k } , { O outi } → T C [28–38, 40, 45] s k CCQ { s k } , x outk → T Q CQQ { s k } , { O outi } → T Q [41, 42] QuantumInput QCC { ρ ink } , x outk → T C QQC { ρ ink } , { O outi } → T C ρ ink QCQ { ρ ink } , x outk → T Q [55] QQQ { ρ ink } , { O outi } → T Q [39, 43, 56] speech, and the corresponding task might be to interpretoral conversations into text in real time. As is typical forRC, in this example each desired output (words) dependson multiple past inputs (recent sounds), i.e. the input his-tory, s i ≤ k , so the state of the system is used as memory.Additionally, the map f should be contracting such that x k to a good approximation depends only on recent inputhistory [50–52]; this ensures both convergence and indepen-dence of distant past such as the initial state of the system.Finally, the map should also guarantee separability [49],which implies that the reservoir computer separates anypair of different inputs into different outputs.RC builds on this simple approach to solve machinelearning tasks: the bulk of the processing is offloaded toa fixed complex system and desired input-output maps areachieved by only adjusting how its state is post-processed—this is how the output of such systems is trained. Theoutput sequence, { y k } , is obtained from some of the statevariables of the substrate, x outk , y k = h ( x outk ) , (2)where x outk is a vector of L ≤ N states, i.e. all state vari-ables may not be accessible for the output, and the function h contains free parameters that are optimized during thetraining process. The supervised learning is done by mini-mizing a cost function between the true target output, ¯ y k , and the predicted one, y k . It is worth mentioning that thegeneralization of this framework to a vectorial input, { s k } ,or output, { y k } , is straightforward.The overall objective of RC is to provide a high-dimensional mapping of the input s k to x k , and ultimatelyapproximate the desired output y k . To this end, the func-tions f and h provide memory and usually define a non-linear transformation from input to output. Memory andnon-linearity resources play a key and distinct role in dif-ferent tasks. A useful quantifier of the memory of a givensystem, used also in works considering quantum reservoirsas will be seen, is called information processing capacity(IPC) [58]. The IPC facilitates the analysis of what kindof linear and nonlinear memory functions a given systemcan approximate. Classical extreme learning machines :The main difference between ELM and RC is that inan ELM the state of the substrate, x l , is uniquely definedby only its corresponding input, s l . Therefore, the mapbetween the two is given by [19, 53, 54] x l = f ( s l ) , (3)where l denotes different instances of the input. In ELM,temporal dependencies between different input instancesare not captured and are irrelevant, so it specializes in adifferent kind of tasks. The set of inputs, { s l } , is not givenin any particular order. For instance, in a classificationproblem, in which we want to know if there is a car in apicture or not, the presence of a car in the previous pictureis not related to the current one. It should be stressed thata RC can be used as an ELM by, e.g., simply re-initializingit between inputs, and an ELM can be used as a RC pro-vided that f is contracting as described above and the stateis allowed to retain memory of past inputs. Indeed, situ-ations can be envisioned where the same physical systemcan be used for both [27].In the following sections, we analyze and discuss the ex-tension of the two approaches to the quantum domain, i.e.QRC and QELM, where the substrate is a quantum system.We start discussing how to encode the input information,which needs to trigger the substrate properly. We thendiscuss recent proposals of quantum tasks as well as somerequirements of a physical system to be used as a substrate.Once the main elements of QRC and QELM have been in-troduced, two relative examples of tasks –namely timer andclassification– will be shown in Section 2.5. These tasks arecommon in the classical framework and we will show themfor a quantum reservoir of spins and of harmonic oscillators,respectively. Once we have a computational problem to be tackled withthe use of a classical or quantum substrate as a processingunit, the first issue to be addressed is how to introduce theinput into the system and different strategies have been de-vised. One possibility, as we explain in more detail below,3s by directly accessing to all or a part of the substrate. Al-ternatively, an external system can serve as an ancilla andbe used to encode the input, which is then coupled to thesubstrate. In both cases we can distinguish between clas-sical inputs and quantum ones, as consistently classified inTable 1, indicating them through a scalar or a quantumstate.When the input is classical, the information is usuallygiven as a series of real numbers, { s k } , with k = 0 , , ... ,as explained in Section 2.1. For QRC, each instance of theinput is fed into the substrate at consecutive time steps,∆ t , so that time is given by t = k ∆ t , whereas for a QELMtask, the time does not play a relevant role and k representseach instance.In the first proposal of QRC [28], the designed algorithmis suited for processing classical temporal information, e.g.given a sequential input { s k } the goal is to reproduce aclassical target output { y k } . In order to input the classicalsequence into the quantum substrate (of interacting qubits)the input is first encoded in the quantum state throughone (or more) of the qubits. The state of this selectedqubit is recurrently prepared in a different superpositionof the basis states | (cid:105) and | (cid:105) . The value of the classi-cal input, s k ∈ [0 ,
1] for a normalized continuous variable,or s k ∈ { , } for a binary one, fixes the components, i.e. | ψ k (cid:105) = √ − s k | (cid:105) + √ s k | (cid:105) . Subsequent works concern-ing time-series predictions have employed a similar inputencoding procedure [30, 35] in pure states. An alternativeway to encode the classical input is used in [29, 31–34, 45],where the input qubit is consecutively initialized in the gen-eral mixed state ρ k = (1 − s k ) | (cid:105) (cid:104) | + s k | (cid:105) (cid:104) | . The stateis pure when s k = 0 or s k = 1, and is completely mixed for s k = 1 /
2. In continuous variable systems, the input canbe encoded, for instance, in coherent states amplitude, insqueezing strength or phase, in thermal excitation. Theseencodings have been analyzed in the case of single-modeGaussian states in harmonic networks in Ref. [37], where itwas shown that the encoding choice can display a signifi-cant effect on the QRC performance (see Section 2.6).Another less explored possibility to fed a classical inputinto a quantum substrate is by exciting the substrate withan external field, such as a light beam modeled by a time-dependent parameter in the Hamiltonian that describes thereservoir. As an example, the input of the anharmonic os-cillator processing system in Refs. [38, 40] is the phase of amonochromatic light beam driving of the dynamics. Eachphase instance is learned through measurements at differenttimes, while the system is reset between inputs, represent-ing an example of QELM with time multiplexing.Some approaches inspired by QRC and QELM have alsoaddressed problems where the input is quantum, being anancilla mode in a quantum state. The dynamical evolutionof the density matrix of the coupled quantum substrate andthe quantum input modes enables different tasks [39, 43].For instance, quantum input states can be classified whenencoded into an ancilla interacting with a fermionic net-work [39]. Further, the input quantum state, either infinite dimension or in continuous variable, can be recon- structed [43]. Recently, a quantum input has been alsoconsidered in classical RC. For instance, in continuouslymonitored superconducting qubits [55, 59], the informationabout the quantum state is carried by a microwave signalresulting from the measurement process.
In this section, we will describe recently proposed compu-tational tasks that are suitable for QRC and QELM. In thefirst place, it is worth introducing a criterion to establishwhether a given task is classical or quantum. We considera task to be quantum when it exploits the quantum infor-mation content of input and/or output states. Examplesare the identification of quantum correlations or quantumstate preparation. This definition is adopted in Table 1.We notice that, classical, T C , and quantum, T Q , tasks canbe performed with both classical and quantum substrates.Starting with classical tasks, there are several recentstudies about the performance of quantum reservoirs assubstrates for (classical) time-series processing [28–38, 40,45,46], so they are in the CQC class. These include bench-mark tasks commonly considered in classical RC such as thetimer task (see Section 2.5.2), realizing nonlinear functionsof past inputs such as the nonlinear autoregressive movingaverage (NARMA) [60], and chaotic time series predictionbased on, e.g. the Mackey-Glass system [61]. As for QELM,there are at present no attempts to apply quantum reser-voirs to classical tasks such as, for instance, classificationof images [53]. Further examples of classification taskshave been reported in other NN settings [62] as well as inrecently reported quantum classifiers [63, 64]. This CQCframework corresponds to the extension of the full classicalapproach [19, 21, 48–54] by replacing the classical substratewith a quantum one. The main goal is the realization ofsimilar classical tasks with an improved performance (Sec-tion 2.6).There are also recent proposals of quantum tasks thatcan be considered as QELM or inspired by it. In Ref. [39]the quantum task consists in the detection of entanglementand the computation of entanglement-related quantities,which are typically difficult to extract from experimentalsetups. The setup consists of a quantum input ancillacoupled to a quantum network. Similarly, also quantumstate tomography based on a quantum reservoir has beenreported, another example of a QQQ protocol in a frame-work similar to QELM. Conventional quantum tomographyschemes are challenging but, with the quantum reservoirapproach in [43], the unknown density matrix of the inputquantum state can be reconstructed after a single measure-ment on local observables of the reservoir nodes without theneed of any correlation detection. The characterization ofquantum state is indeed an important quantum task, alsoexplored in other platforms, like for instance the quantumneuron proposed in Ref. [65].Beyond detection schemes, another example of a T Q is the preparation of desired quantum states [41, 42] inancilla systems interacting with different quantum sub-4trates. In [41], for instance anti-bunched and cat stateshave been reported while Ref. [42] addresses maximally en-tangled states, NOON, W, cluster, and discorded states.The main difference with QRC and QELM frameworks isthat in spite of the presence of an input signal in the sys-tem, this is just driving it in a suitable operation regimebut it is not an information input: the classical input doesnot encode which is the target quantum state, being thisindependently imposed through the proper choice of a costfunction. As no quantum resource is needed in the input,in Table 1, these examples are listed in the CQQ category.As a note, a QQC protocol could be devised as the in-verse process to the previous one. For instance, given ageneric (quantum) state of an ancilla system as input, thetask could be to find a classical control parameter in thesubstrate Hamiltonian preserving that quantum state.QRC has also been considered in the context of quan-tum state measurement, as a strategy for processing in-formation coming from continuously monitored supercon-ducting qubits coupled to a Kerr network acting as a reser-voir [55]. The substrate of Josephson parametric oscilla-tors is operating into a classical regime and then classifiedas QCQ in Table 1. We also mention a recent experimentwith a single qubit coupled to a superconducting cavitywhose output is fed into a recurrent NN (implemented insoftware) [59]. The NN was able to predict measurementoutcome probabilities as well as to estimate the initial stateof the qubit and parameters such as Rabi frequency, de-phasing rate and measurement rate.Finally, a quantum substrate can also provide a generalframework for quantum computing [56], achieving a univer-sal set of quantum gates and also non-unitary operations,of interest to simulate open quantum systems. Althoughthis is a prominent quantum task (QQQ) inspired by ex-treme learning techniques, the model of Ref. [56] escapesthe basic three-layer scheme of Figure 1, as also the con-nections between input layer and substrate need to passthrough the training process. In QRC and QELM, quantum dynamical systems amenableto input driving and output extraction are exploited tosolve the tasks at hand. The exponential growth of Hilbertspace dimension obtained by increasing the number ofquantum elements endows them with a large state spacethat plays a determinant role in the performance [28]. Suchquantum substrates require several ingredients to solvecomputational tasks.The first one is the need of a rich spatial and/or tempo-ral dynamical behavior, which is frequently achieved intro-ducing disorder in the interactions between different systemcomponents [26]. In fact, symmetries in the quantum sub-strate can cause the appearance of local conserved quanti-ties that would reduce the number of exploitable degrees offreedom, which are responsible for a richer dynamics [46].The second one has to do with how the state of the (a) (b)(d)(c)Nuclear magnetic resonancein molecules Trapped ionsSuperconducting qubitsFermions or bosonsin latticesQuantum circuits(e) (f) Photonic platforms
Figure 2: (a)-(f) Various examples of platforms to beused as quantum substrates for information processing.Two suitable candidates to implement spin-network mod-els [28–32, 35, 45] are (a) nuclear magnetic resonance inmolecules [44], and (b) trapped ions [66]. (c) State-of-the-art ultracold atomic setups in optical lattices with bosonicand fermionic species are well-suited for Bose-Hubbardand Fermi-Hubbard discrete models [39, 41–43, 56], respec-tively. Additionally, other physical implementations arepossible, e.g. in arrays of quantum dots in semiconductordevices, and in (d) coupled superconducting qubits [63].(e) Quantum circuits [33,67] have been used as a substratein the IBM platform [34,36]. (f) Continuous-variable mod-els [37,40,67] could be engineered in photonic experimentalsetups as well [68–70].substrate depends on the input. As in the classical case,QRC requires contractiveness. Contraction of the dynam-ics, i.e., the convergence of two different initial states af-ter repeated input injections, guarantees that the systempresents a fading memory and allows the proper learningof temporal functions [52]. In this manner, initial condi-tions of the quantum system are erased and the state ofthe system only depends on the input history. At variance,in QELM it is usually assumed that the substrate is ini-tialized into some fixed state before an input is processed.In this case, the state depends on the most recent inputonly. Finally, the state of the system should be nonlinearin input to achieve nontrivial information processing witha simple—and consequently easy to train—readout [18].With these requirements in mind, different systems andmodels have been proposed that could be implemented inseveral platforms. Figure 2 illustrates a selection of thephysical platforms that are considered for QRC and QELM.Experimental implementations of such quantum substrates5or QRC and QELM can be modeled as few-body systems.The first proposal for QRC was based on quantum spin net-works [28]. The size of the Hilbert space of these spin-basedsystems grows as 2 N , where N is the number of spins, butthe total number of degrees of freedom amounts to 4 N − z direction (cid:104) σ zj (cid:105) . Such performance partly comesfrom the exploitation of a technique called temporal mul-tiplexing, where a number of V evenly spaced snapshots ofthe dynamics are taken between two input injections, thusincreasing the number of computational nodes by a fac-tor of V . The spin-based approach has since been furtherrefined. In [29], spatial multiplexing is introduced, wheremultiple networks receive the same input and output isgathered from all of them in a single output layer. In [31],spatial multiplexing is combined with classical connectionsbetween the networks to enable multidimensional inputs.Nuclear magnetic resonance in molecules, as depicted inFigure 2(a), and trapped ions, as depicted in Figure 2(b),are potential platforms for spin-based QRC and QELM sys-tems.Most works on spin-based systems for QRC mainly ex-plore their applicability and computational performancefrom a numerical perspective, but efforts in the analyti-cal direction have also been made. One of the main the-oretical problems in the field of machine learning is theuniversal approximation property: the possibility to findexamples of a class of systems that can approximate ele-ments of a given class of functions with arbitrary precision.In Ref. [32], focusing on a setup of interacting spins similarto the proposal of [28], sufficient conditions for this kindof universality were reported. Later, a different variationof the QRC spin system in terms of quantum circuits, asdepicted in Figure 2(e), was envisioned in [33], being theprecedent of the IBM experimental implementation of [34].In this last work, a family of QRC circuit models has beenidentified where the universality property holds. The ef-forts to analyze and optimize the performance of the orig-inal spin model are still ongoing [30, 35, 46].All the previously mentioned spin-based references fallinto the category of CQC in our classification of Table 1,focusing on solving classical temporal tasks in the QRCframework. Going beyond classical tasks, discrete-variablemodels of fermions, as depicted in Figure 2(c), have beenintroduced to solve non-temporal quantum tasks [39,43,56].The authors of [39] proposed a 2D Fermi-Hubbard modelwith nearest-neighbor couplings as a substrate, where eachlattice site is pumped by incoherent excitations, while theinput is written in two bosonic modes that are coupledto the reservoir via the dissipative “cascaded formalism”.The whole system is modelled by a Lindblad master equa-tion and the output is built as a function of the fermionicoccupation numbers. This formalism allows the authorsto perform the task of recognizing entangled states and it is potentially useful for any estimation of nonlinear func-tions of the input state. The same substrate is employedin [43], harnessing the time multiplexing technique to in-crease the number of output nodes for quantum state to-mography and can be applied to reconstruct both fermionicand bosonic states. Finally, the Fermi-Hubbard model isalso employed in [56], where a quantum reservoir model isproposed such that a universal set of quantum gates can berealized. We point out variational quantum circuits con-sisting of coupled superconducting transmons, which aredepicted in Figure 2(d), as another promising candidatefor QELM, since fully trained variational quantum circuitssolving non-temporal tasks have already been implementedon such platforms [63]. Elementary building blocks for aquantum counterpart of a network of spiking neurons in-tended to process quantum input were proposed in [65]however, based on simple spin Hamiltonians that imple-ment the transformations under certain conditions involv-ing in particular the interaction time.We now move to discuss the substrates that are usu-ally modelled as continuous variables. Proposals to real-ize either basic building blocks [71] or full scale quantumNN [67] using variational (i.e. parameterized) quantumcircuits have recently been reported. In these proposals,linear transformations and displacements of the input areachieved by Gaussian gates. Nonlinearity is introducedwith non-Gaussian gates in [67]. Measurement-inducednonlinearity is proposed as an alternative to experimen-tally difficult non-Gaussian gates in [71], however no ad-vantage over classical perceptrons was observed. Althoughintended to be trained by adjusting the circuit parame-ters, these proposals could be adapted to realize continu-ous variable QELM or QRC by starting from a randomlyinitialized but fixed feedforward or recurrent architecture,respectively, and training only the readout layer. A firstproposal in this direction is the use of a Gaussian bosonsampler as a QELM [13], where classical input could beencoded in, e.g., squeezing and quantum input directly in-jected and the output would be a function of the expectedvalues of detected photons.Additionally, the dynamics of a generic continuous vari-able system may be harnessed directly thanks to the flex-ibility offered by both ELM and RC frameworks. A QRCwith Gaussian states of random linear networks to performclassical tasks, i.e. CQC in Table 1, was proposed in [37].In this approach, nonlinearity originates from the input andreadout layers whereas the network state provides memory.The input layer consists of a generally nonlinear mapping ofthe classical input sequence into quantum states, which arethen injected into the network with periodic state resetsof one of the network oscillators. The observables of therest of the oscillators are combined with a trained functionto match a desired output. Already with these minimalresources, an improvement over purely classical resourceswas found (see Section 2.6 for more details), and the reser-voir family was found to have the universal approximationproperty. This approach is suited for realization in pho-tonic devices, promising candidate platforms to be used as6ubstrates for information processing. The different fre-quency or temporal modes found in photonic systems, asdepicted in Figure 2(f), could serve are reservoir for QRCand QELM. For instance, it was recently proposed the gen-eration of re-configurable complex networks in Ref. [69]based for instance on frequency combs [70]. As a furtherexample, lattices of coplanar waveguide resonators have al-ready been produced for photons in the microwave regime,as in [68]. This kind of systems are experimentally flexibleplatforms where different geometric configurations are pos-sible between the network connections, such as hyperboliclattices. The discussion in [68] also points to the possi-bility of including interactions between photons by meansof nonlinear materials, and of coupling the resonators tosuperconducting qubits, ideal for QRC and QELM.A different scheme, based on a single Kerr nonlinearoscillator, for CQC was adopted in [40]. Although only asingle observable is considered, the scheme allows for itscontinuous monitoring; the output signal was discretizedto facilitate training with the usual methods, leading to asituation not unlike the temporal multiplexing mentionedearlier for the spin based approach. First considered forQELM [40], more recently, also temporal tasks have beenreported in this scheme [38]. Substrates that can be mod-elled using continuous variables have also been used totackle quantum tasks. A random bosonic lattice of nonlin-ear nodes excited by a randomly distributed classical fieldhas been proposed for CQQ [41]. Following the principlesof ELMs and RCs, only the readout layer, i.e. the basischange implemented by the linear optics, is trained. Thearchitecture is feed forward, making the scheme an exampleof a QELM. In Section 2.3, we described different computational taskswith a focus on their classical or quantum nature withoutmaking an explicit distinction between tasks that belongto QRC from those that are proper to QELM. Based onthe differences underlined in Section 2.1, we can find ex-amples of both approaches. In particular, several worksreports on temporal tasks employing the memory of thereservoir, a defining feature of QRC [28–38, 45, 46, 55]. Incontrast, when the tasks that are performed do not requirememory from past inputs to generate the output, we aredealing with a QELM [38–44, 56]. Below, for the sake ofconcreteness, we present an example of squeezing classi-fication using QELM and an example of a temporal taskrealized with QRC.
Dynamics of quantum systems that do not retain memoryof past inputs can be used to solve non-temporal tasks suchas classification, where the inputs are categorized followingsome given rule. In Figure 3, we show results of using theharmonic oscillator network of Ref. [37] in such a fashion S u cc e ss r a t e Classification of squeezed states
Figure 3: Squeezed vacuum states with either a fixed (yel-low, left bars) or a random (blue, right bars) phase areclassified according to the magnitude of squeezing using aQELM. 100 random realizations of both the states to beclassified and of the oscillator network acting as QELM areconsidered for each case. The columns and error bars indi-cate the success rate and standard deviation, respectively.to classify squeezed states.Specifically, the input is a squeezed vacuum state withsqueezing magnitude r ≤
2, which can take a finite numberof different values, and phase ϕ , which is either constantor selected uniformly at random from the interval [0 , π/ r ,and the oscillator network is trained to classify the statesaccordingly. The network consists of N = 4 oscillators, ini-tially in its ground state. The input state is injected intothe network by setting the state of one of the oscillatorsto it. The network is then allowed to evolve for a fixedtime before the output is extracted. The output, i.e. theclass assigned to the input state, is a trained function of 6of the network observables. As can be seen, if the phaseis constant the network succeeds in the task in all cases.With random phase the success rate remains high for asmall number of classes. This simple classification exampledisplays the versatility of QRC platforms also serving forQELM: indeed the difference with the original model stud-ied in Ref. [37] is merely that here we reset the state of thenetwork to the ground state between inputs and with thepurpose of a classification task. Further details are givenin Appendix A.1. The natural dynamics of quantum systems can be employedto solve temporal tasks. Here, we show the performance ofthe quantum spin model of Ref. [28] in one of the bench-mark tasks used to characterize the memory capacity ofreservoir computers: the timer task.The timer task is a relatively simple task that consistsin obtaining a response from the system after a countdownfinishes. The indication to start the countdown is given bythe input sequence { s k } (dashed-grey line) going from zero7 a)(b) Figure 4: Timer task trajectories for a quantum spin net-work of N = 10 spins. In panel (a) the countdown time is τ = 5, whereas in panel (b) we used τ = 20. The inputsequence { s k } (dashed line), the target { ¯ y k } (solid line),and outputs { y k } (symbols) are shown in both situations.We used three different instances of the output layer withdifferent number of O observables in each instance, namely O = 10, squares, O = 30, triangles, and O = 75 circles (seeAppendix A.2 for more details).to one, and the countdown finishes when the target se-quence { ¯ y k } shows a spike (black-solid line). Figure 4(a)and (b) account for different countdown times (5 and 20time steps, respectively), testing the limits of the memorycapabilities of our system. Three instances of the outputlayer were employed for a network of N = 10 spins, where O denotes the number of observables. These output lay-ers were constructed with different combinations of localobservables, see Appendix A.2 for further details. In bothsituations, O = 75 outperforms the smaller output layers,observing a drastic decrease of capabilities for O = 10 and O = 30 when we extend the countdown time to τ = 20.These results are a simple but convincing evidence of thepositive influence on the system performance of using alarger number of observables. The possibility to use a largenumber of observables relies on the large number of ex-ploitable degrees of freedom that quantum systems contain. A key motivation in considering quantum substrates in ma-chine learning is the possible advantage these can represent.In this section, we summarize some recent results on theperformance of quantum substrates both in QELM and inQRC. Where available, we report on the performance ofa quantum substrate and its classical counterpart on theexecution of the same task. A potential advantage of theextension of RC and ELM to the quantum domain is thepossibility of dealing with an exponentially large number ofdegrees of freedom even for small networks of coupled units.The way these degrees of freedom can be computationallyexploited will not only depend on the size of the Hilbertspace of the reservoir, but also on the way information isencoded in, processed by and extracted from the system. The study of the computational power in relation withthe system size is indeed the target of several works onQRC. In the pioneering work of Ref. [28], a spin networkwas employed to solve problems such as the short-termmemory task and the parity check task. These tasks serveto estimate the capacity of the system to store the in-put sequence and to perform a nonlinear mapping, respec-tively. By tuning the input injection rate and the couplingstrength, spin networks of a few qubits (
N <
10) exhibitsimilar computational capabilities to echo state networks(ESN) of 100 −
500 nodes. The performance of a 500-nodeESN can be matched by a spin network of only N = 7qubits, for instance measuring at several times along theresponse of each qubit. The possibility to extract addi-tional information from the dynamics of each spin at differ-ent times emphasizes the role played by the hidden degreesof freedom of the system. In a similar spirit, the poten-tial of exploiting the large amount of computational nodesthat are available in a quantum substrate was quantifiedin [35] through the IPC, a tool well suited to assess theperformance in RC (see Section 2.1). Taking as observ-ables quantum correlations as wells as spin projections,and considering time multiplexing techniques, it was in-deed demonstrated that the capacity of spin networks canbe extended without indications of saturation. Hence thesystem computational capability increases as the outputsize is increased, thus successfully exploiting the potentialof the quantum Hilbert space. However, this is not alwaysthe case. For instance, in Ref. [38], it is shown that for asingle-qudit reservoir the performance quickly saturates asthe number of considered levels grows.The relevance of the state space size and encoding is alsomanifested in [37], where a network of quantum harmonicoscillators is compared with an ESN with the same numberof computational nodes, i.e., the same number of oscillatorsas the number of nodes in the ESN. The quantum systempossesses a larger state space even when restricted to Gaus-sian states. Furthermore, the way in which the input isencoded in the quantum network plays a determinant rolein exploiting all degrees of freedom. Computing the IPCfor systems with N = 8 computational nodes (both for theESN and the quantum network), it was shown that encod-ing in the mean amplitude increases the total capacity onlyby a factor of 2 respect to the ESN. In contrast, encodingin the fluctuations of the input oscillator allows to accessto N observables. In particular, squeezed vacuum encod-ing increases the nonlinear memory of the system, whileclassical thermal fluctuations only provide linear memory.As we have seen, in the case of classical tasks and out-puts the performance achievable with quantum substratesis quantified using the same tools that have been devel-oped for classical RC and ELM, and direct comparisonswith classical substrates can be done. On the contrary,for quantum tasks we have seen a broader spectrum of ap-proaches and anticipated some of their specific advantagesin Section 2.4. In the case of the quantum tasks mentionedabove (entanglement detection, quantum state tomogra-phy, quantum state preparation) no RC or ELM protocols8ave been devised so far that are based on classical sub-strates (to the best of our knowledge, Ref. [55] is the onlyexample of a classical RC approach to a quantum problem).In order to assess the usefulness and the advantage of theprotocols proposed in Refs. [39, 41, 42], a comparison canbe made with previous results on the same tasks that havebeen obtained using more standard (fully-trained) classicalNN architectures. As an example, the task of classifying,through a NN, whether a (multipartite) state is entangledor not was addressed in Ref. [72]. There, the classificationis limited to pure states and the extension to mixed onesrepresents a nontrivial issue. In contrast, the simple modelof [39], based on a quantum substrate, works irrespectiveof the mixedness of the state under study. As for quan-tum state preparation, the QELM of [41] does not requireany additional resource apart from the reservoir itself, incontrast with other NN proposals, which for instance relyon number-resolved measurements [73], or conditional mea-surements [74].The impact of noise has also been addressed in most ofthe platforms for QRC and QELM. In qubit based reser-voirs, both effects such as dephasing [28, 32] and additiveexternal noise in observables [28, 31, 43] have been consid-ered. Ref. [32] compared effects of dephasing to (gener-alized) amplitude damping and found them to be similar,whereas Ref. [43] compared systematic and random noise,finding that the reservoir learns to compensate for the for-mer and tolerates the latter. Importantly, noise does notnecessarily rule out universal QRC [34] and it has beensuggested [36] that a small amount of noise can preventoverfitting in NISQ systems, similarly to the classical neu-ral networks case. Sometimes computation speed can besacrificed for better noise tolerance by simply increasingthe number of repetitions for the protocol: this approachwas found to make nontrivial information processing possi-ble even when the signal-to-noise ratio was low in an NMRexperiment [44]. Additive external noise has also been con-sidered in continuous variables systems [40, 41, 67, 75] andit has been found that performance degrades gracefully. Inparticular, impact of external noise in observables can bereduced if the memory of the reservoir can be tuned to beclose to the requirements of the task at hand [75]. Further-more, simulations suggest that a quantum neural networkimplemented in an optical platform can learn to compen-sate for photon loss [67]. Many opportunities arise when RC and ELM are extendedinto the quantum regime but still there are several openchallenges. One of the main drawbacks for experimen-tal implementations of QRC and QELM comes from theoutput extraction. In most cases, the readout data takenfrom the quantum substrates are typically expectation val-ues of local node observables or their correlations. Thisimplies that, due to the stochastic nature of quantum mea- surements, several copies of the system or multiple mea-surements are generally needed. Furthermore, in tempo-ral tasks, the system output needs to be measured in asequence of times. As measurements introduce a (evenstrong) back-action in the reservoir dynamics, their effectneeds to be accounted for in on-line time series process-ing. In order to avoid it, the protocol would have to berepeated many times until each timestep that one wishesto consider. Finally, the interaction with external degreesof freedom of the quantum reservoir induces decoherenceand dissipation in it. These issues are important whendealing with quantum substrates and become particularlychallenging for on-line time series processing as in QRC.A way of dealing with the measurement problem con-sists in taking a large number of quantum substrates andperforming ensemble computing, as considered in the firstexperimental implementation of QELM on NMR platforms[44], where the observable averages are taken over a largenumber of identical molecules in a solid. Back-action effectsare negligible in those cases, as weak measurements areperformed over the substrate [76]. Although promising, noimplementations of temporal data processing (QRC) havebeen reported on NMR platforms so far.NISQ schemes for QRC are also promising and have alsobeen experimentally implemented on current IBM quantumcomputers [34, 36]. The output of quantum computationon the IBM platform is so far obtained through projectivemeasurements, so the measurement back-action problemdiscussed above represents a serious issue when comparedwith on-line processing in classical RC. For instance, theexperimental implementation reported in Ref. [34] requiredfor each time a different ensemble of realizations, a ratherlaborious procedure; the system needs be restarted fromtime 0 to be measured at time k ∆ t so the substrate retainsmemory from the past.Most theoretical proposals stand on the assumptionthat it is possible to average over several identical realiza-tions. The number of repetitions required to obtain a re-liable estimation of expectation values will depend on thesystem as well as on the observable considered. For in-stance, the estimation of variances in continuous variablesof quantum vacuum states will require, in general, more ef-fort than the estimation of a mean quadrature of an intensemode with small quantum fluctuations [40]. Depending onthe specific features of each platform, different strategies foroutput extraction and measurement can be devised and thisrepresents an almost completely unexplored field. Alterna-tive implementations in which statistics can be performedover a single realization would be a significant advance inthe field. In the case of QRC, experimental platform pro-posals allowing on-line data processing are also desirable.These frameworks could be based on sequential measure-ments of the substrate, implemented in a way in which thereservoir does not completely lose all information aboutpast inputs after each detection. This would most likelyyield back-action effects on the dynamics of the substrate,which should also be taken into account.A field so far unexplored in QRC is represented by the9rocessing of temporal sequences of inputs that can bringinformation, for instance, about the stochastic source thatgenerates them. For this class of problems, the measure-ment back-action problem is in general less worrying thanin the time series processing protocols described so far. In-deed, if information were encoded in the entire time series,the final state of the substrate would be determined by thewhole input injection process, so that an output could bebuilt at the final computing stage after a single measure-ment. This class of problems is not necessarily limited toprocessing of classical inputs, as it can also include quan-tum time series processing. For instance, one may wantto determine whether a quantum channel is Markovian ornot, or whether noise in a multipartite quantum registeris correlated or uncorrelated. This family of cases wouldrepresent QRC tasks within the QQQ cell of Table 1.Broadly speaking, a general theory for quantum tasks, T Q , is yet to be formulated. In this regard, a new conceptanalog to the universal approximation property should bedefined. For instance, in the case of quantum state tomog-raphy [43], a universality property would ensure that anyarbitrary quantum state belonging to a given Hilbert spacecan be reconstructed. Furthermore, there is still no theo-retical magnitude that can estimate the performance of agiven quantum substrate by computing how well it approxi-mates any arbitrary solution, similar to the IPC in classicalRC that provides an overall overview of the linear and non-linear memory properties of the system [58]. A step towardsthis direction has been recently reported in [13] where aninformation-theoretic quantifier of the memory of a quan-tum or classical learning machine processing quantum orclassical inputs is introduced. Although its use for suchtasks is discussed, it has yet to be applied in this manner.Finally, in situations where the output is a quantumstate, training cannot be done with linear regression be-cause the readout layer might be, e.g., an interactionHamiltonian [56], which also makes such class of protocolshighly unorthodox with respect to the standard definitionof reservoir computing. There might also be additionalrestrictions for the trained parameters to be admissible.Although in such situations general purpose optimizationmethods, such as the Nelder-Mead algorithm, have beensuccessfully used to train the system [56], it could be won-dered if theoretical work on the optimal training methodcould further improve results in these cases. This perspective article on QRC and QELM covers thestate-of-the-art and provides an outlook of future researchavenues in this emerging field. We have introduced a clas-sification of the current literature in terms of the classicalor quantum nature of the input, substrate and task, re-spectively, that provides a general overview and points outunexplored directions. Recent works have shown that sev-eral classical and quantum tasks can be performed success-fully. A relevant feature of QRC and QELM is the combi-nation of the benefits of simple training requirements, com- pared to other machine learning methods, with the poten-tial of improved performance of quantum substrates. Sev-eral platforms are good candidates to be used as quantumsubstrates and facilitate the practical implementation incurrently available NISQ. An additional envisioned advan-tage of the QRC framework applied to NISQ devices is tobypass the strict requirements of quantum error correctionor error mitigation techniques, since time-invariant readouterrors can be learned during training and are compensatedfor by the output layer [34]. Further opportunities remainto be explored including photonic platforms, particularlysuccessful for classical RC [57].We note that quantum substrates allow one to tackleproblems that are incompatible with their classical counter-parts. Conventional classical machine-learning algorithmsare nowadays being using for tasks related to assist quan-tum experimental realization. Tasks include e.g. quan-tum state preparation, quantum state tomography, calcu-lation of entanglement-related quantities from accessibleobservables, and continuous monitoring of systems. QRCand QELM proposals widen the opportunities as can bemore naturally adapted to interact with quantum inputsor embedded in experimental platforms. According to thepresent state of the art, while QRC has been mainly em-ployed in classical temporal tasks, QELM has found itsrealm in quantum information tasks. However, the useof QRC is expected to display a disruptive potential inquantum temporal tasks as, for instance, in quantum se-cure communication, quantum channel tomography, non-Markovianity detection, or quantum error correction. Clas-sical or quantum RC substrates embedded in quantum sys-tems can indeed allow for low-latency processing of quan-tum signals at the computational edge with high fidelity[55], operating at the timescales of the measured quantumsystem and providing a speed-up.To conclude, the field of QRC and QELM is still mov-ing its first steps, so it is premature to make quantitativecomparisons with their much more advanced classical coun-terparts in terms of efficiency or performance, especiallybecause of the absence of fully working experimental im-plementations. However, the perspective we have presentedhere gives us a glimpse of the potential and the versatilityof these frameworks together with the new avenues thatdeserve to be explored.
Acknowledgements
We acknowledge the Spanish State Research Agency,through the Severo Ochoa and Mar´ıa de Maeztu Pro-gram for Centers and Units of Excellence in R&D (MDM-2017-0711) and through the QUARESC project (PID2019-109094GB-C21 and -C22/ AEI / 10.13039/501100011033);We also acknowledge funding by CAIB through theQUAREC project (PRD2018/47). The work of MCS hasbeen supported by MICINN/AEI/FEDER and the Univer-sity of the Balearic Islands through a “Ramon y Cajal” Fel-lowship (RYC-2015-18140). GLG is funded by the SpanishMinisterio de Educaci´on y Formaci´on Profesional / Minis-10erio de Universidades and co-funded by the University ofthe Balearic Islands through the Beatriz Galindo program(BG20/00085).
Conflict of Interest
The authors declare no conflict of interest.
A Appendix
A.1 Details on the QELM classifier
The classification task of Fig. 3 is defined by ρ in = | r, ϕ (cid:105) (cid:104) r, ϕ | , ¯ y = r, (4)where the input state, ρ in , is a single-mode Gaussian statein squeezed vacuum, determined by squeezing magnitude r ∈ [0 ,
2] and phase ϕ ∈ [0 , π/ r can take only a fi-nite number of equally spaced different values which are theclasses. For three classes the possible values are r ∈ { , , } and so on. The phase ϕ is either uniformly random or takesa constant value ϕ = 0. The target output ¯ y is the class,and should coincide with r .The starting point of the used QELM is the oscilla-tor network presented in Section IIA of [37], where it wasused for QRC. Here it is converted into a QELM by reset-ting its state to the ground state between inputs, removingany memory of input history. The network is completelyconnected and consists of N = 4 oscillators. The bare fre-quencies of all oscillators is set to ω = 0 .
25, whereas theinteraction strengths in the network are chosen uniformlyat random from g ∈ [0 , . ρ in . The network is allowed to evolve for a time ∆ t ,after which the diagonal elements of the covariance matrixof the rest of the oscillators are used to form the output.The value of ∆ t is chosen by considering a large set ofpossible values and choosing the one that minimizes thespectral radius specified in Lemma 1 of [37]. The networkis first trained as in [37] (except that there is no prepara-tion phase since the network state is reset between inputs)using random input states. Typically the outputs are realnumbers close to r . The predicted class is taken to be thenearest class to the network output; the training is finishedby shifting the bias term such that this leads to the correctclassification as often as possible. The trained weights arethen tested by classifying a new set of random inputs.The success rate in the test phase is shown in Fig. 3.The network is first trained with 500 inputs and then usedto classify 200 fresh inputs. Both inputs with a constantphase ϕ = 0 and with a random phase are considered andthe number of classes, i.e. possible values for r , is var-ied. 100 random realizations of both the inputs and thenetworks were considered for each of the 8 different cases. A.2 Details on the QRC timer
The input/output map of the timer task of Figure 4 can berepresented by s k = (cid:40) k ≥ c, k < c, ¯ y k = (cid:40) k = c + τ, , (5)where the values of the inputs { s k } indicate when thecountdown starts ( k = c = 500). Once the countdownis initialized, the target is to obtain an isolated response ofthe output sequence with value ¯ y c + τ = 1.The system we considered here is the quantum spinmodel of [28]. We used a network of N = 10 spins, withan homogeneous external magnetic field h = 10, couplingstrength between qubits given by a random uniform distri-bution J ij ∈ [ − / , /
2] and input injection rate ∆ t = 10,everything in normalized units. Three different instancesof the output layer were employed in Figures 4(a) and (b),where O denotes the number of observables. First, bluesquares correspond to the case of using only the spin pro-jections in the z -axis (cid:104) σ zi (cid:105) (1 ≤ i ≤ N ). Second, yellowtriangles correspond to all the local spin projections (cid:104) σ xi (cid:105) , (cid:104) σ yi (cid:105) and (cid:104) σ zi (cid:105) . Third, the red dots were obtained with thetwo-spin correlations along the z -axis (cid:104) σ zi σ zj (cid:105) (1 ≤ i, j ≤ N , i (cid:54) = j ) in addition to all the local spin projections.Figures 4(a) and (b) represent the results for the timertasks of τ = 5 and τ = 20, respectively. We fed an inputsequence of 800 time steps to the system, where the first400 were a used as warming up to remove the initial con-dition. The output layers were trained only with the last400 time steps, and we took 10 different realizations of thecouplings of the network to obtain averages of the outputtrajectories { y k } . References [1] A. Adamatzky, L. Bull, B. De Lacy Costello, S. Step-ney, C. Teuscher,
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