Information Processing Capacity of Spintronic Oscillator

Physical reservoir computing is a framework that enables energy‐efficient information processing by using physical systems. Nonlinear dynamics in physical systems provide a computational capability that is unique to reservoirs. It is, however, difficult to find an appropriate task for a reservoir because of the complexity of nonlinear information processing. The information processing capacity has recently been used to clarify systematically the tasks that are solved by reservoirs; it quantifies the memory capacity of reservoirs in accordance with the order of nonlinearity. Herein, an experimental evaluation of the information processing capacity of a spintronic oscillator consisting of nanostructured ferromagnets is reported. The spintronic reservoir state is electrically manipulated by adding a delayed‐feedback circuit. The total capacity reaches a maximum of 5.6 at the edge of the echo state property. A trade‐off between the linear and nonlinear components of the capacity is also found. The result can be used to better understand the nonlinear information processing in reservoirs and to find good matches between reservoirs and tasks. As an example, a function‐approximation task is performed and it is found that it can be efficiently solved when the reservoir state is appropriately tuned so that its information processing capacity matches that of the task.


Introduction
Physical reservoir computing is an emerging form of computing that is capable of energy-efficient and real-time information processing.It is based on the framework of recurrent neural networks, [1][2][3][4] where a nonlinear dynamical system, called a reservoir layer, such as a photonic or other quantum system replaces an intermediate layer.  Reser46][47][48][49][50][51] In particular, it has been shown that a spintronic oscillator can perform a human-voice recognition task with high accuracy, 99.6%. [35]In addition, a unique and practical feature of these devices is that they are nanometer scale.However, spintronic reservoir computing is a recent development, and its range of application to real-world tasks is unclear.
One of the most critical issues in reservoir computing is how to find problem that can be solved by a reservoir with low error.Even if a reservoir can handle some tasks efficiently, it is not guaranteed to be useful in other tasks.To clarify which tasks can be handled by a particular reservoir, a number of numerical simulations [7,11,14,15,[20][21][22]31,[52][53][54] and a few experiments [14,55,56] have calculated the information processing capacity (IPC). [57] The C is a task-independent computational capability, and classifies the memory capacity of reservoirs in accordance with the order of nonlinearity.[57] Therefore, the IPC consists of linear and nonlinear componentwise capacities. The linea (nonlinear) capacity quantifies the number of the linearly (nonlinearly) transformed data from the inputs a reservoir can store.Roughly speaking, a large total IPC means the reservoir has a high computational capability.0] More importantly, the IPC can be used to identify the components of the memory capacity that are necessary for completing a task.When the order of the dominant capacity in a reservoir matches the (non)linear order of the target data, the reservoir might be able to perform the task efficiently even if its total IPC is small.For example, in a numerical simulation, a spintronic reservoir was able to perform a sensing task with high accuracy even though its total IPC was relatively small (about 2 at maximum).[31] However, numerical simulations have recently been revealed to have limitations, [58] wherein the calculations sometimes fail to predict the magnetization dynamics quantitatively.For example, it may be that the calculations cannot accurately quantify material parameters or that structural inhomogeneities among the samples lead to sampledependent behavior.Therefore, an experimental IPC evaluation is necessary to directly prove the applicability of a reservoir to a task.
In this work, we performed an experimental evaluation of the IPC of a spintronic oscillator.The IPC was quantified as the reconstruction rate of target data from the output voltage from the oscillator.Here, we selected 4711 linear and nonlinear independent target data from a series of random voltage-pulse inputs to the oscillator.As a result, the linear and nonlinear components of the IPC were evaluated.The total IPC was maximized to be 5.6 by changing the gain of a delayed-feedback circuit attached to the oscillator.A trade-off between the linear and nonlinear components was also found.The results indicate that the reservoir's IPC can be tuned by the feedback circuit.We then performed a function-approximation task and found that the approximation error was minimized when the feedback gain was tuned so that the IPCs of the oscillator and the task matched.

Input Signals to Spintronic Oscillator
The spintronic oscillator (physical reservoir) used in this work consists of ferromagnetic and nonmagnetic materials on a nanometer scale.Figure 1a schematically shows the main structure of the oscillator.It has two ferromagnetic layers, called free (top) and reference (bottom) metallic layers, separated by a thin insulating layer called a tunnel barrier (middle).The magnetization in the free layer oscillates when a direct voltage is applied to the system.On the other hand, the magnetization direction of the reference layer is fixed.The magnetization oscillation in the free layer can be electrically detected through the tunnel magnetoresistance effect, [59,60] where the resistance of the system is low (high) when the magnetizations in the free and reference layers are aligned (anti)parallel.The spintronic oscillator used in the present work had an oscillation frequency of 383 MHz, full width at half maximum of 1.3 MHz, and integrated power of 0.30 μW when a direct voltage of V 0 ¼ þ275 mV was applied (see also the Experimental Section for a detailed description of the materials and dimensions of the oscillator).
In addition to the direct voltage, we apply a magnetic field that is generated in the delayed-feedback circuit shown in Figure 1a.Here, the output voltage from the oscillator is amplified and injected into a metallic line placed above the oscillator to generate radiofrequency magnetic field with a delay time of 29 ns.The feedback input-signal changes the magnetization dynamics from ordered to chaotic, as has recently been predicted theoretically [61,62] and shown experimentally. [63]Note that the nonlinearity in the dynamics determines the computational capability of the reservoir. [23]We will show in Section 3 that the feedback gain can be used to tune the components of the IPC.
A sequence of time-dependent inputs is also injected; it is used to define the target data of the IPC evaluation.In the experiment, it was an independent and identically distributed random pulse sequence with uniform distribution À1 ≤ u k ≤ þ1 that was used as a modulating voltage.The bias voltage V 0 ¼ 275 mV was modulated as V in ðtÞ ¼ V 0 þ uðtÞΔV, where ΔV corresponds to the modulation amplitude (125 mV), and uðtÞ [À1 ≤ uðtÞ ≤ 1] is a pulse u k in the time range of ðk À 1ÞΔt ≤ t < kΔt with a pulse of Δt ¼ 50 ns.The number of random pulses was 5280.Summarizing the above, the spintronic oscillator has three inputs, i.e., a direct voltage exciting the auto-oscillation of the magnetization, a feedback magnetic field for manipulating the dynamical state, and a random pulse voltage as the input for computing.While the magnetization dynamics in the presence of the direct voltage and feedback were previously studied, [61][62][63] the effect of the random input on the dynamics has not been investigated yet.Therefore, we first studied the dynamical state of the spintronic oscillator when the feedback signal and random input data were simultaneously injected.

Power Spectral Density
Figure 1b shows examples of the random input signal and the output voltage from the spintronic oscillator at a feedback gain of g fb ¼ À40 dB.The changes in the output voltage follow those of the input signal.Figure 1c shows the power spectral density of the output voltage.The linewidth (26 MHz) of the output voltage is relatively wide compared with that in the absence of the random inputs (1.3 MHz mentioned above).This is because the random inputs modulate the instantaneous frequency.Figure 1d summarizes the power spectral density versus the gain.In the small feedback region (g fb < 0 dB), a single broad peak appears around 390 MHz.The peak splits into two around g fb ¼ 0 dB, and then into multiple peaks above a feedback gain of 20 dB.[63] In Section 4, we will show that the efficiency of performing a function-approximation task changes at these gains.Moreover, in Section 5, we will show that the change in the dynamical state relates to the breakdown of the echo state property.
One might be interested in distinguishing the dynamical states in terms of, for example, periodic, periodic doubled, and chaos.It is, however, difficult because the dynamics are driven by a series of pulse-shaped random inputs, and therefore, inputs are discontinuous when they are switched.A mathematical definition of, for example, chaos in such a switched dynamical system is still growing.In addition, the randomness of the inputs makes it difficult to define, for example, periodicity in outputs.Therefore, we did not label the dynamical states in this work, in contrast with our previous work where the dynamics were distinguished as chaotic or nonchaotic through an evaluation of noise limit. [63]

Dynamical Trajectory in an Embedding Space
The changes in the dynamical state can also be found by examining dynamical trajectories in an embedding space. [63]Figure 2a shows the autocorrelation function, vðt 0 Þvðt 0 þ tÞ h i , of the output voltage vðtÞ from the spintronic oscillator for various feedback gains, where t 0 is the initial time for the evaluation.The correlation remains finite for a long time for feedback gains above 0 dB.Here, let us estimate the minimum time τ at which the autocorrelation function becomes zero, i.e., vðt 0 Þvðt 0 þ τÞ h i ¼ 0. The 2D plot of vðtÞ and vðt þ τÞ in Figure 2b corresponds to the dynamical trajectory produced in the embedding space.The trajectory is concentrated around a circle when the feedback gain is relatively small.This implies that the magnetization dynamics are approximately periodic.The slightly broadened trajectory is due to thermal activation and amplitude modulation by the random inputs.The maximum amplitude remains the same for g fb ≲ 20 dB, although the trajectory becomes broad with increasing feedback gain.The maximum amplitude of the trajectory increases for g fb ≳ 20 dB.These results indicate again that the dynamical state changes at g fb ¼ 0 dB and 20 dB.

Information Processing Capacity
Here, we describe the evaluation of the IPC of the spintronic oscillator.

Target Data of IPC
The IPC quantifies the reconstruction rate of the target data from the output voltage of the spintronic oscillator.The target data were orthogonal polynomials that covered all possible linear and nonlinear combinations of the inputs where P n is an orthogonal polynomial of degree n, prepared using Gram-Schmidt orthogonalization based on the given input stream (see also the Experimental Section), and m, d, and D are the index of the polynomial, delay, and maximum delay.Here, P n ðu kÀd Þ with n ¼ 1ð≥ 2Þ is a linear (nonlinear) function of the random sequence u k , while P 0 ¼ 1.The degree n m,d ≥ 0 is a series of d for y m IPC , and the degree of nonlinearity N m for y m IPC can be expressed as N m ¼ P D d¼0 n m,d .Note that the target data are distinguished by the degree n and delay d.Therefore, we will use a label ffn, dgg to characterize the target data.For example, P 1 ðu kÀ1 Þ and P 2 ðu kÀ1 Þ are labeled ff1, 1gg and ff2, 1gg, respectively.
The red lines in Figure 3a,b are examples of linear (N m ¼ 1) and nonlinear (N m ¼ 2) target data, which are characterized by ffn, dgg ¼ ff1, 1gg and ffn, dgg ¼ ff2, 1gg, respectively.The random inputs u k are also shown in Figure 3c.The linear (N m ¼ 1) target data in Figure 3a are u k itself in Figure 3c with the delay d, whereas the target data with N m > 1 in Figure 3b are a nonlinear function of u k .

System Output
Next, we tried to reproduce the target data of IPC from the oscillator's output.The oscillator's output was the amplitude of the output voltage estimated by a Hilbert transformation. [36,37,64]ote that thermal activation in a magnetic material adds noise to the amplitude.Therefore, we measured the output signal from the oscillator 60 times for the same input signal.The output amplitude AðtÞ for the kth input, averaged over 60 trials, was decomposed into I ¼ 200 virtual nodes as follow: Using A k,i , we define the system output as where w i was determined so that the error, The blue lines in Figure 3a,b show examples of the system outputs.The feedback gain in this case was 0 dB.As shown in Figure 3a, the linear target data are closely approximated by the system output, whereas the approximation of the nonlinear target data in Figure 3b is poorly approximated.

Component-Wise Capacity and Total IPC
The approximation of each target data by the system output can be quantified by the component-wise capacity Here, min w i MSE and ⋅ h i refer to the minimization of the mean-square error against w i and the average, respectively.The range of C m is from 0 to 1, and C m becomes 1 when the system output completely reproduces the target output.For example, the component-wise capacities estimated from Figure 3a,b are 0.88 and 0.32, respectively.
The component-wise capacity is also distinguished by the label ffn, dgg.For example, the capacity for y 0 IPC,k ¼ P 2 ðu k Þ is labeled as ffn, dgg ¼ ff2, 0gg, while the capacity for y 2 IPC,k ¼ P 1 ðu k ÞP 1 ðu kÀ1 Þ is labeled as ff1, 0g, f1, 1gg.The componentwise capacity for N m ¼ 1 is the linear component of the IPC, C m with N m ≥ 2 is classified as the nonlinear component.The total IPC is defined as where M ¼ 4711 is the total number of combinations of orthogonal polynomials tested in the experiment (see the Experimental Section).The total IPC of a physical reservoir is bounded by the number of linearly independent variables used as the output, which is a feature called the completeness property. [53,57]

IPC of Spintronic Oscillator
Figure 4a plots IPC as a function of the feedback gain.The component-wise capacities are distinguished by their color.The total IPC is approximately constant, C tot ¼ 3.5, when g fb < 0 dB, and is maximized, C tot ≃ 5.6, near g fb ¼ 20 dB.Above g fb ¼ 20 dB, the total IPC drastically decreases.Note that the feedback gains (0 and 20 dB) corresponding to the local maxima of C tot are those at which the dynamical state of the magnetization changes, as discussed in Section 2. This correspondence between the computational capability and the dynamical state is similar to the one found in the previous work [65][66][67] where the computational capability is enhanced near the edge of chaos.In Section 5, we will revisit this point and show that the total IPC is maximized at the edge of the echo state property.Figure 4b-f summarizes the linear (N m ¼ 1) and nonlinear (N m = 2, 3, 4, and 5) components of the IPC versus g fb , where these components are the sum of the component-wise capacities with the same N m .For the feedback of g fb < 0 dB, the linear (first-order, N m ¼ 1) and second-order (N m ¼ 2) capacities are dominant and third-order (N m ¼ 3) capacity has a finite value.Higher order (N m ≥ 4) nonlinear components appear as the feedback gain increases.This can be interpreted to be a result of highly nonlinear dynamics caused by a large feedback effect.We can also see a trade-off between the IPCs; the linear and second-order components of the IPC decrease for a feedback gain range around 0 < g fb < 20 dB, while the higher order IPCs increase.This might reflect the fact that the total IPC is bounded as a result of the completeness property mentioned above.
We notice that the total IPC is much lower than the node number, I ¼ 200.It implies that some nodes are not linearly independent each other and thus, the effective dimension is smaller than the node number (see also Supporting Information for rank analyses).A way to enhance the total IPC might be to use spatially distributed nodes, in addition to (or instead of ) virtual nodes defined by time-multiplexing method here.For example, large memory capacities were found in a reservoir consisting of coupled spintronic oscillators. [33,38]A reservoir with spatial nodes detecting spin-wave propagation was found to have a large memory capacity. [49]While these results were obtained by performing numerical simulations, it is currently difficult to fabricate spatially distributed and coupled nodes in spintronics devices experimentally.We will keep this issue as a future work.Another possible reason of the low total IPC is a lack of the echo state property.While the definition and analysis of the echo state property will be mentioned in Section 5, here we briefly mention that the echo state property relates to the computational reproducibility of reservoir computing.It is often broken by, for example, noise and chaotic dynamics, which might be non-negligible in spintronics devices.Remind noise is suppressed in the present study as much as possible by applying an averaging technique to the output amplitude, [36] as mentioned above.It is, however, difficult to make noise completely zero.On the other hand, chaotic dynamics might be unavoidable in the present study, [63] although it is difficult to specify chaotic region for the present experiments, as mentioned in Section 2.2.While noise adds randomness to system output and reduces the echo state property, chaos reduces the echo state property due to high sensitivity to an initial state.The roles of the noise and chaos on IPC might be distinguished by evaluating temporal IPC (TIPC), [53,68] which is an extension of IPC to dynamical systems which show a dependence on an initial state.We will also keep an analysis of TIPC in future.
In summary, the total IPC of the spintronic oscillator can be varied by using the feedback gain to manipulate the dynamical state of the magnetization.The maximum total IPC reaches close to 5. The nonlinear components of the IPC appear with increasing feedback gain.These results indicate that the linear and nonlinear components can be tuned electrically.This point will be used in the next section, where we show that a functionapproximation task can be efficiently performed by tuning the components.In the Supporting Information, we describe  Similarly, "Rest of 3rd" includes capacities not only for ff3, dgg (d > 1) but also for ff2, additional experimental and numerical results, such as the IPC for various modulation amplitudes ΔV and delay times and the IPC estimated by using AðtÞ without averaging.

A Function-Approximation Task
The IPC evaluation is applicable not only to physical reservoirs but also to real-world and benchmark tasks. [53]Here, we show that incorporating an IPC analysis in a task provides us a way to perform the task efficiently.

Definition of NARMA2 Task and Its Reconstruction
We focused on a second-order nonlinear autoregressive moving average (NARMA2) task. [69]This task has been used for evaluating the computational capability of physical reservoirs. [10,11,31,70]t aims to predict the target data y N2 defined by the following second-order difference equations The parameters ðα, β, γ, δ, μ, σÞ were set to ð0.4, 0.4, 0.6, 0.1, 0.1, 0.05Þ. [11,38]The red line in Figure 5a is an example of the target data.
Next, we introduced the system output y STO,k to reproduce the target data from the oscillator output, A k,i (see also the Experimental Section).The blue line in Figure 5a is an example of the system output, where g fb ¼ À21 dB.The system output well reproduces the target data.The reproducibility is quantified by the normalized mean square error, and Figure 5b shows its dependence on g fb .The error in the absence of the feedback circuit was 4.0 Â 10 À7 .The error slightly decreases as g fb increases from À30 dB and it reaches a minimum, 3.8 Â 10 À7 , at À21 dB.

Relationship between Errors in Reconstruction and IPC
Here, we compute the IPCs of the NARMA2 task by using the method described in Experimental Section.Note that the procedure described here is applicable to any task, and so we will denote the target of the task by y tar,k for generality.For the NARMA2 task, it is y N2,k .Let e m be the mth coefficient of the expanded target.Accordingly, the target is represented as where ŷtar,k is normalized such that P k ŷtar,k ¼ 0 and P k ŷ2 tar,k ¼ 1.Similar to Equation ( 16), the target IPC is given by As the data obtained from Equation ( 5) are a 1D data, the total target IPC is 1 at maximum.
Figure 5c summarizes the IPCs of the NARMA2 task, where the parameter σ varies from 0.05 to 0.30.The task mainly consists of ff1, 0gg, ff1, 1gg, ff2, 0gg, and ff2, 1gg components and so reservoirs must have these capacities in order to perform the NARMA2 task efficiently.
The IPC of system output can be evaluated in a similar way (see the Experimental Section).The ff1, 0gg and ff1, 1gg components of the IPC of the system output are shown as solid circles and open squares in Figure 5d and the components of the target data of the NARMA2 task are shown as dotted lines.We can see that the error becomes small when IPCs of the system output and the target data match.The result indicates that a reservoir can solve a task efficiently, even if its total IPC is small, if the IPC of the task matches that of the reservoir.

Echo State Property
Finally, we analyzed the echo state property of the spintronic oscillator and examine its relation to the IPC.The echo state property is a necessary factor for physical reservoir computing, [2,71] wherein the dynamical state of the reservoir becomes independent of its initial state, which is often arbitrary, and is solely determined by the input data.Several factors, such as noise in the reservoir and/or chaotic dynamics, cause the echo state property to be lost.The total IPC is maximized only when the reservoir has the echo state property.While some studies [31,67] report that physical reservoirs work well when the reservoirs are at the edge of chaos, it has recently been asserted that the edge of chaos is not a necessary condition for enhancing the computational performance. [4,19,72]In addition, it is also argued that the edge in question in those papers is not that of chaos, but rather the edge of the echo state property. [4]In the following, we quantify the echo state property of the spintronic oscillator and compare it with the IPC.
Remind that we used the amplitude of the output voltage averaged over 60 times in Section 3 and 4. Here, let us denote the output signal AðtÞ for the αth trial (α ¼ 1, 2, • • • , N s ) as A α ðtÞ (N s ¼ 60).The index of the echo state property is defined as The standard deviation of the 60 trials is where is the amplitude averaged over trials and time steps.We discarded the first 280 of the 5280 time steps and set the number of sampling to N t ¼ 5000.The number of combinations of A α and The initial state of the oscillator is expected to be affected by thermal fluctuation, i.e., the initial value of A α differed for each trial.If the spintronic oscillator has the echo state property, however, the output voltage would solely be determined by the input data and become independent of the initial state after a long time passes.In such case, the index of the echo state property becomes zero.Accordingly, the index can be used as a measure of the echo state property.
Figure 6a shows examples of the output voltage for two trials and the random input signal, where g fb ¼ À40 dB.The two output voltages often overlap, which indicates the presence of the echo state property.On the other hand, the overlap disappears when the feedback gain increases to g fb ¼ 7 dB (Figure 6b). Figure 6c summarizes the index of the echo state property in 60 trials with various g fb .The total IPC from Figure 4a is shown for comparison.The index is relatively small and nearly constant when g fb is below approximately 0 dB.It slightly increases from g fb ¼ 0 dB and reaches a local maximum around g fb ¼ 10 dB.Above g fb ¼ 10 dB, it decreases until g fb ≃ 20 dB and then begins to increase again.This behavior corresponds to the power spectral density in Figure 1d, where the single peak turns into two peaks near g fb ¼ 0 dB and then into multiple occurs near 20 dB.Note that the autocorrelation function survives for a long time for g fb > 0 dB and the maximum amplitude of the dynamic trajectory increases for g fb > 20 dB, as shown in Figure 2. As can be seen from a comparison between Figure 1d, 2, and 6c, the change in the echo state property relates to the change in the dynamical state.In addition, there are correspondences between the index of the echo state property and the total IPC.For example, the total IPC reaches a local maximum around a feedback gain of 7 dB, near which the index starts to show a visible increase.The maximum total IPC occurs around 20 dB, near the edge of the echo state property, where the index shows a local minimum.

Conclusion
In summary, we experimentally evaluated the IPC of a spintronic oscillator for physical reservoir computing.We found that the total IPC and its linear and nonlinear components could be varied by changing the feedback gain with a circuit attached to the oscillator.We also found that the feedback circuit could be used to control the dynamical state of the magnetization, which was analyzed from the power spectral density, the dynamical trajectory in the embedding space, and the index of the echo state property.On the basis of these analyses, we concluded that the total IPC was maximized at the edge of the echo state property.A trade-off between the linear and nonlinear components of the IPC was also found, which might be due to the completeness property of the total IPC.These results indicate that the IPC in the spintronic oscillator can be tuned with the feedback gain.The spintronic oscillator could perform several tasks by matching its IPC to that of a task, even though the maximum total IPC (5.6)  was smaller than that of other reservoirs.As an example, we performed a function approximation task and proved that the error was minimized when the IPCs of the oscillator and task matched.

Experimental Section
Materials and Size of Spintronic Oscillator: The spintronic oscillator used in this work consisted of substrate/buffer/IrMn(5)/CoFe(2.5)/Ru(0.9)/CoFeB(2.1)/CoFe(0.44)/MgO(1.0)/FeB(5)/MgO(1)/Ta(5)/Ru(5)(thickness in nm) with a diameter of 425 nm.Here, CoFe(0.44)/MgO(1.0)/FeB(5) corresponds to the three layers schematically shown in Figure 1a.The resistance of the oscillator depends on the relative angle between the magnetizations in the CoFe reference layer and FeB free layer separated by MgO insulator.The free layer has a magnetic vortex owing to the shape magnetic anisotropy.The sample preparation procedure and the dynamical properties of the oscillator were almost the same as those in our previous reports. [35,73]An external magnetic field of 0.6 T is applied to the direction normal to the film plane.The magnetic vortex shows an auto-oscillation through spin-transfer effect when a direct voltage above a threshold value (150 mV in the present experiment) is applied.
Definition of Target Data: The target data for the evaluation of the IPC are defined by Equation (1).For example, there are three kinds of target data that have The case of n m,d ¼ 0 for all d values (i.e., N m ¼ 0) is excluded from the target data because its target data are constant.
To precisely compute the IPCs, we used suitable methods for the obtained time-series.The length of the uniform random-input series (5280) is relatively short, so its distribution is slightly different from the theoretically uniform one.This fact results in nonorthogonal Legendre polynomials of the input.To compute the IPCs with a short time-series, we used Gram-Schmidt chaos, [53] which refer to orthogonal polynomials obtained via Gram-Schmidt orthogonalization based on the given input stream.The univariate polynomials P n ðu tÀd Þ ðn ¼ 0, 1, : : : Þ in Equation (1) are computed as follows where T is number of the input data.Note that P 0 ðuÞ ¼ 1.Using Eq. ( 11) and ( 12), we can derive the orthogonal polynomials even for short input time-series.For example, the firstand second-order polynomials are written as follows where ).In addition, the IPCs of the NARMA2 model were also computed using the same type of polynomial chaos.To calculate the IPC of the spintronic oscillator, we set the maximum delay τ max for each degree of nonlinearity n as ffn, τ max gg ¼ ff1, 280g, f2, 50g, f3, 15g, f4, 10g, f5, 8gg.As a result, M ¼ 4711 orthogonalpolynomial combinations were tested to calculate the capacity.
The obtained IPCs included numerical errors due to the finite lengths of the time series, which were added to the total capacity.The accumulated error was reduced as follows. [53]First, we prepared 200 surrogates C sur,i ði ¼ 1, : : : , 200Þ that were input series shuffled in the time direction and computed IPCs using the surrogates to obtain 200 capacities (see Figure 7a).As the shuffled inputs are not held in the reservoir state, the surrogates represent the numerical error capacities caused by the short lengths.
Furthermore, we set the significance level be αð¼ 1Þ% and chose the original IPC, which exceeded 1.5 times the value in the top ðα=2Þ% of the 200 capacities (see the dotted line in Figure 7a).We performed the above operation for each polynomial and obtained significant IPCs. Figure 7b shows an example of the first-order component-wise capacity with g fb ¼ À21 dB.As the delay increased, the value of the capacity decreased, indicating that the spin torque oscillator certainly held the processed inputs.These capacities were precisely calculated by the truncation with 1.5 Â max i C sur,i ¼ 0.023.The threshold successfully removed the numerical error capacities included in the distribution of surrogates.
Another Perspective of IPC: It will be useful to have another perspective of IPC.The IPC can be regarded as the coefficient magnitude of states expanded by orthogonal IPC targets. [53,57]Let us denote the reservoir state at time k as x k for generality; for the present case, it corresponds to the amplitude of the oscillator's output, A k,i .From this viewpoint, the (a) (b) coefficient of an expanded reservoir state expresses an amount of processed input.Using singular value decomposition, we obtain the normalized reservoir state x k as [53] xk ¼ X where c m is a coefficient vector for the mth target polynomial and is related to the IPC by Equation ( 16) was used for the evaluation of IPC in the tasks discussed in Section 4.
IPC of System Output in NARMA2 Task: Here, we describe the evaluation method of the IPC of the system output in NARMA2 task.The method is based on the perspective of the IPC in Experimental Section.Here, the system output is defined similar to that in Section 3.2 but its value is different because the target data are different.That is, we set y STO,k ¼ P i w i A k,i so that P k ðy N2,k À y STO,k Þ 2 would be minimized.In other words, the weight w i was different from the one estimated in Section 3.2.The system output ŷSTO,k , which is the linear sum of the reservoir states, is generated from the existing polynomials in the reservoir state as follows where ŷSTO,k is normalized such that P k ŷSTO,k ¼ 0 and P k ŷ2 STO,k ¼ 1, while w is the readout weight vector of linear regression.The mth IPC for the system output is, similar to Equation ( 16) The IPC of the reservoir states indicates which type of memory the reservoir originally holds, and the IPC of the system output represents how the reservoir states are used to adjust the memories to the task.Note that the IPC defined by Equation ( 18) is different from the one given by Equation (3) because the target data, as well as the system output, are different.The IPC in Section 3.4 is estimated using y m IPC,k as the target data and is task-independent, while the IPC evaluated here depends on task because the target data y N2,k are defined from the NARMA2 task.Comparing the IPCs of the target and system output, we can analyze what balance of polynomials realizes the target emulation.

Figure 1 .
Figure 1.a) Schematic illustration of spintronic oscillator with delayed-feedback circuit.The main structure consists of, from top to bottom, a ferromagnetic free layer, thin insulating layer, and ferromagnetic reference layer.The arrows in the ferromagnets represent magnetization directions.Direct voltage drives the magnetization dynamics in the free layer.The output voltage from the oscillator is measured by oscilloscope.The output power from the oscillator is sent to the feedback circuit and generates a radiofrequency magnetic field.The feedback gain g fb is controlled by a tunable attenuator.b) Examples of random input signal, output voltage from the spintronic oscillator, and c) its power spectral density at a feedback gain of À40 dB.d) Power spectral density as a function of feedback gain.

Figure 2 .
Figure 2. a) Autocorrelation functions and b) dynamical trajectory in an embedding space for various feedback gains.

Figure 5 .
Figure 5. a) Examples of target data (red) and system output (blue) in NARMA2 task for a feedback gain of À21 dB.b) Normalized mean square error (NMSE) versus the feedback gain.c) Total capacities of NARMA2 model versus input range σ. d) IPCs of the system output (solid circles and open squares) and the target data (dotted lines) versus feedback gain.

Figure 6 .
Figure 6.Examples of output voltages of two trials with feedback gains g fb of a) À40 dB and b) 7 dB.c) Echo state property (ESP) index for various g fb averaged over 60 trials.Green color represents the distribution of the ESP index between trials.The total IPC from Figure 4a is shown for comparison.

Figure 7 .
Figure 7. a) Distributions of 200 surrogates C sur,i (red circles) threshold 1.5 Â max i C sur,i (red dotted line).b) First-order capacities as a function of delay τ, where black-open and blue-solid circles are the original and the truncated capacities, respectively.The feedback gain is g fb of À21 dB.The original capacities are truncated to be zero when the values are below the threshold shown by the red dotted line.The inset shows an enlarged view of the capacities near the threshold.