Abstract
The dynamical behavior of complex quantum systems can be harnessed for information processing. With this aim, quantum reservoir computing (QRC) with Ising spin networks was recently introduced as a quantum version of classical reservoir computing. In turn, reservoir computing is a neuro-inspired machine learning technique that consists in exploiting dynamical systems to solve nonlinear and temporal tasks. We characterize the performance of the spin-based QRC model with the Information Processing Capacity (IPC), which allows to quantify the computational capabilities of a dynamical system beyond specific tasks. The influence on the IPC of the input injection frequency, time multiplexing, and different measured observables encompassing local spin measurements as well as correlations is addressed. We find conditions for an optimum input driving and provide different alternatives for the choice of the output variables used for the readout. This work establishes a clear picture of the computational capabilities of a quantum network of spins for reservoir computing. Our results pave the way to future research on QRC both from the theoretical and experimental points of view.
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Funding
This study was financially supported by the Spanish State Research Agency, through the Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R&D (MDM-2017-0711), CSIC extension to QuaResC (PID2019-109094GB), and CSIC Quantum Technologies PTI-001. The work of RMP and MCS has been supported by MICINN, AEI, FEDER and the University of the Balearic Islands through a predoctoral fellowship (MDM-2017-0711-18-1), and a “Ramon y Cajal” Fellowship (RYC-2015-18140), respectively. GLG acknowledges funding from the CAIB postdoctoral program.
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Appendices
Appendix 1
Here, we motivate the choice of the parameters h and Js of the model presented in Eq. (5). Our choice is based on the numerical results shown in Fig. 8, in which the IPC is computed for different values of h and Js. Other relevant system parameters are set to Δt = 10 and N = 5, which have been our benchmark in the simulations according to the results presented in the main text. We have explored four orders of magnitude for both h and Js to observe the evolution of the normalized capacity together with the distribution of the linear and nonlinear contributions to the IPC. For small values of h, the total capacity does not saturate and the main contribution is the linear memory, being the only contribution for small values of Js. For higher values of either h or Js, the profile of memory capacities changes towards a higher presence of nonlinear contributions, while keeping the bound of the total capacity. Thus, the decision of choosing h = 1 and Js = 1 as our benchmark (with Δt = 10 and N = 5) is based on the fact that it is a well established operational point, with a saturated total capacity, and a good presence of nonlinear contributions.
IPC versus (a) natural frequency of the spins h and (b) coupling strength Js. The parameters in (a) are Δt = 10, Js = 1 and N = 5, while in (b) we have used Δt = 10, h = 1 and N = 5. Notice that the normalization factor in this plot is N. The error bars of the plots correspond to the standard deviation over 10 realizations
Appendix 2
In this appendix, we explain in more detail how the bars of the IPC are computed. Contributions to the IPC are usually shown according to the degree of the polynomial we want to reproduce. For each degree, we need to sum up the contributions coming from different delays. By delay we mean how far in the past we consider the influence of the input into the system. This influence is represented in Eq. (11) by taking the inputs sk−i, where i is the delay respect to present time k.
In the main text, we have only shown the sum of the capacities over the delays. To deepen in our characterization we include here an illustration of the role of the delay for the reproduction of polynomials of degree 1, i.e., linear memory. The name of linear memory comes from the fact that we are computing the capacity of reproducing or “remembering” targets of the form \(\bar {y}_{k}=s_{k-i}\). Figure 9 represents the bare capacity of Eq. (10) with respect to delay i for such a linear memory.
Linear memory of the spin-based QRC for parameters Δt = 10, h = 1, Js = 1 and N = 5 as a function of the delay in the input
The area under the curve of Fig. 9 is what we have plotted as IPC of degree d = 1 in the bar’s plots across this work. In some cases, e.g., Fig. 8 (b) (Js = 0.01), the influence of the input extends to delays longer than 100 past inputs and care needs to be taken to not disregard small non-vanishing contributions in the computation of the IPC. It is a straightforward procedure to represent linear memory, but nonlinear contributions are harder to untangle visually. Representing second- and third-order contributions is still possible with 2D and 3D heatmaps of the capacities respect to the multiple delays. However, such visual representations of the memory as a function of the delay find a limit when we go to nonlinearities of degree d ≥ 4. Therefore, although we are aware that the summation over delay contributions is somehow hiding the information regarding the distribution of the memory for different delays, it provides a compact and readable representation.
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Martínez-Peña, R., Nokkala, J., Giorgi, G.L. et al. Information Processing Capacity of Spin-Based Quantum Reservoir Computing Systems. Cogn Comput 15, 1440–1451 (2023). https://doi.org/10.1007/s12559-020-09772-y
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DOI: https://doi.org/10.1007/s12559-020-09772-y