Sub-linear Power-Law Scaling of Information Processing Capacity in Noisy Physical Reservoirs

How does the computational capacity of a physical reservoir scale with the number of devices? This paper quantifies the dependence of information processing capacity (IPC) on device count under device noise identified from hardware measurements and reports three findings. (1)~IPC scales sub-linearly with device count $N$ as $N^{\\alpha}$, $\\alpha = 0.5$--$0.8$. Varying the noise level shifts the prefactor but leaves the exponent nearly unchanged. (2)~IPC retains 93\\% of its full-precision value at 2-bit weight precision; the noise floor sets an upper limit on precision requirements. (3)~In task performance (NARMA10), increasing $N$ by a factor of 16 yields almost no improvement, whereas a 20-stage digital delay line reduces error by more than half. A single delay stage is worth more than 75 analog devices. These results show that the upper bound $\\mathrm{IPC} \\leq N$ is extremely loose for ensembles of noisy analog devices and that augmenting digital memory is more effective than adding more devices.