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Efficiently and Reliably Measuring Information Processing Capacity in Dynamical Systems via Kernels

Alessio Benavoli, Felix Binder, Cangxiong Chen

arXiv:2610.03281v1quant-phcs.ITmath.DSmath.IT

TL;DR

Standard IPC estimates reconstruction of an infinite basis by finite, combinatorially costly calculations. This paper introduces KIPC, which uses kernels to estimate total capacity without enumerating basis functions and provides a single test for comparing systems. In an echo-state-network example, KIPC detects a capacity maximum at the spectral-radius boundary where total IPC underestimates capacity; the paper also applies it to a disordered quantum spin chain.

  • Problem

    Standard IPC requires truncating an infinite basis, and its computation grows combinatorially with polynomial degree and delay.

    Source: §I · p. 1, A. Information Processing Capacity · p. 3

  • Method

    KIPC uses kernels to compute total IPC without explicitly enumerating basis functions and provides a single null-hypothesis test for comparing systems.

    Source: §I · p. 2, §V · p. 12

  • Results

    As the spectral radius approaches s = 1, KIPC indicates capacity reaches its maximum there, while total IPC begins decreasing before s = 1 and underestimates capacity.

    Source: A. Echo-state network · p. 10

  • Takeaways & Limitations

    KIPC offers a way to estimate total capacity without degree truncation and is demonstrated on an echo state network and a disordered quantum spin chain reservoir.

    Source: §I · p. 2, §V · p. 12

Abstract

from arXiv · show

Information Processing Capacity (IPC) is a powerful system-agnostic metric for quantifying a dynamical system's computational capability and is widely used in quantum reservoir computing. However, standard IPC suffers from severe computational limitations. Computing IPC involves estimating how well the system can reconstruct, via linear regression, an (in principle) infinite family of orthogonal polynomial functions of delayed inputs. In practice, this evaluation must be made finite, so we need only to consider polynomials up to a chosen maximum degree. Moreover, the computational complexity for computing IPC grows combinatorially with the considered maximum degree and delay. Here, we introduce Kernel-IPC (KIPC), a scalable framework that uses kernel methods to compute the total IPC without explicitly enumerating basis functions. This allows us to account for the full (infinite) basis (without degree truncation). We also provide a null-hypothesis test for statistically comparing the KIPC of two dynamical systems. We validate KIPC on an echo state network and demonstrate its utility for quantum reservoir computing on a disordered spin chain.

I. INTRODUCTION

IPC measures how many linearly independent input functions a dynamical system can reconstruct from its states, providing a task-independent basis for comparing systems. Standard calculations truncate an infinite polynomial basis and can become combinatorially costly, motivating the kernel-based alternative KIPC.

A. Information Processing Capacity

  • IPC quantifies a system’s ability to reconstruct basis functions from its internal states, with total capacity interpreted as the number of independent input functions it can compute.The basis can be formed from orthogonal polynomials of delayed inputs; Legendre polynomials suit uniform inputs, while Hermite polynomials suit Gaussian inputs.Source: §I · p. 1, A. Information Processing Capacity · pp. 2–3

I. INTRODUCTION

  • IPC is widely used in quantum reservoir computing to compare architectures, input-embedding strategies, readouts, and measurement back-action..Source: §I · p. 1
  • KIPC uses kernels to estimate total IPC without enumerating basis functions or imposing a maximum polynomial degree, and adds a single null-hypothesis test for comparing systems.The standard approach instead tests each basis element separately, which can raise false-discovery concerns from spurious correlations.Source: §I · pp. 1–2

A. Information Processing Capacity

  • For τmax = 20 and d = 10, the basis contains 20,030,010 terms, making exhaustive IPC computation combinatorial and practically restrictive.The standard method therefore limits polynomial degree and maximum delay.Source: A. Information Processing Capacity · p. 3
  • Standard total IPC can assign equal capacity to reservoirs with different nonlinearities, so the total alone may not distinguish their computational profiles.The paper describes one reservoir as low-degree nonlinear and the other as highly nonlinear, with poorer performance on linear-memory tasks.Source: A. Information Processing Capacity · p. 3
  • Finite time series can create spurious correlations, while the standard thresholding heuristic lacks multiple-testing correction and can inflate total IPC at large delays and degrees.The input distribution also determines the appropriate orthogonal polynomial basis, which may be unknown or non-standard.Source: A. Information Processing Capacity · p. 3

III. COMPUTING IPC THROUGH KERNELS

Kernel-IPC uses kernels to compute total information processing capacity without enumerating basis functions or imposing a maximum polynomial degree. Its weighted infinite-degree formulation captures nonlinearities while supporting efficient estimation.

III. COMPUTING IPC THROUGH KERNELS

  • KIPC kernels encode tensor products of orthogonal-polynomial basis functions across delayed inputs, covering polynomials of any degree without explicit enumeration.The kernel-IPC trace formulation computes the total IPC over that infinite basis.Source: §III · pp. 4–5
  • The trace formula computes total IPC as an infinite sum with decreasing eigenvalue weights, which down-weight higher-degree polynomials.The authors connect this weighting to the standard practice of prioritizing lower-degree polynomials.Source: §III · p. 5
  • In the example, evaluating 1,916,797,310 individual IPC terms is computationally infeasible, so truncation estimates total IPC for X2 as 2; kernel IPC considers all possible degrees.Source: §III · p. 5
  • For finite nT, removing diagonal terms yields an unbiased U-statistic, computable without allocating the full projection and kernel matrices.The stated computational complexity is O(m nT^2), where m = τmax.Source: §III · p. 6
  • At nT = 10^4, unbiased KIPC reaches near-accurate estimates of ≈0.5543 (i=1) and ≈0.3745 (i=2), with the same computation time as Example 1.These are the stated theoretical values for the two examples.Source: §III · p. 6
  • KIPC separates linear and nonlinear contributions by computing the linear component with a finite-dimensional kernel and subtracting it from total KIPC.For both example systems, the theoretical linear KIPC is ρ/3 ≈0.316 because the only linear component is L1(U·1).Source: §III · p. 6
  • KIPC accounts for all nonlinearities without an upper degree bound and can do so in a fraction of a second for nT = 10^4.The authors also report that same-rank reservoirs can have different capacities when one includes a higher-degree polynomial term.Source: §III · p. 6

A. Hypothesis testing

The authors use a permutation-based test to assess whether two systems have different KIPC. In the example, the test distinguishes systems with different capacities while not rejecting comparisons with matched capacities.

A. Hypothesis testing

  • Under the null, the authors use random permutations to approximate the distribution of the capacity difference and test it with two-sided percentile cutoffs.The test uses the α/2 and 1 − α/2 percentiles of the permutation statistics.Source: A. Hypothesis testing · p. 7
  • Using α = 0.05 and nsamples = 300, the test cannot reject the null for X1 versus X3 and X2 versus X4, but rejects it for X1 versus X2 and X3 versus X4.The pairs X1/X3 and X2/X4 have KIPC values ≈0.5543 (i=1,3) and ≈0.3745 (i=2,4), respectively.Source: A. Hypothesis testing · p. 7
  • The joint KIPC test avoids separate tests for every IPC term, which would require multiple-comparison correction or risk inflating total IPC.The authors state that correction can severely impair statistical power, while omission can artificially inflate total IPC.Source: A. Hypothesis testing · p. 7

B. The choice of ρ

KIPC weights polynomial degree through ρ, but choosing ρ near 1 creates numerical instability that makes significance harder to assess. The section illustrates how sample size affects detection and notes that delay-specific weighting is not implemented.

B. The choice of ρ

  • For the Legendre kernel, degree i receives weight ρ^i/(2i + 1); with ρ = 0.95, degree 50 receives weight ≈0.000762.The authors use ρ = 0.95 to avoid instability as ρ approaches 1.Source: B. The choice of ρ · p. 7
  • As ρ approaches 1, numerical instability raises the hypothesis-test significance threshold, requiring a larger nT; the experiments therefore use ρ = 0.95.At ρ = 1, the Legendre kernel is infinite for ut = ut′ = 1.Source: B. The choice of ρ · p. 7
  • The test cannot reject equal capacity for X1 versus X3 or X2 versus X4, but rejects for X1 versus X2 and X3 versus X4.Source: Figure 1 · p. 8
  • At nT = 10^5, estimated significance 0.00008 is below the degree-50 weight ρ^50/101 ≈ 0.000762, making that nonlinearity detectable.At nT = 10^4, the test only just rejects the null; at 2.5 · 10^4, the difference is clearly significant.Source: B. The choice of ρ · pp. 7–8
  • Although delay-specific weights could decrease with delay, this work uses the same ρ for every delay.Source: B. The choice of ρ · p. 8

C. Gram matrix

The Gram-matrix formulation lets KIPC evaluate capacity with respect to an orthogonal basis, including when the input distribution differs from the reference measure used to define the kernel.

C. Gram matrix

  • For finite nT, the trace calculation can be expressed using the kernel Gram matrix’s eigenvalue-eigenvector decomposition.Source: C. Gram matrix · p. 8
  • KIPC evaluates information capacity with respect to an orthogonal basis even when the input distribution does not match the kernel’s reference measure.For example, the Legendre kernel’s reference measure is u ∼ Unif(−1, 1).Source: C. Gram matrix · p. 8
  • This orthogonal-basis property holds without requiring the input to follow the polynomial kernel’s reference measure.Source: C. Gram matrix · p. 8

IV. NUMERICAL EXPERIMENTS

The numerical experiments use a Legendre kernel to evaluate KIPC in an echo-state network and a disordered quantum spin-chain reservoir. A hypothesis test also rejects the null for each reported comparison between X1 and X5.

IV. NUMERICAL EXPERIMENTS

  • The experiments use the Legendre kernel to evaluate capacity in an echo-state network and a disordered quantum spin-chain reservoir.Source: §IV · p. 8
  • The experiments are designed to demonstrate KIPC’s reliability and computational efficiency relative to standard IPC.Source: §IV · p. 8
  • For every reported comparison between X1 and X5, the null hypothesis is rejected.Source: Figure 2 · p. 9

A. Echo-state network

In the echo-state network, KIPC captures capacity peaking at the edge of stability, where standard total-IPC underestimates it. It also avoids degree truncation and is reported to compute ten times faster, while thresholding and delay caps can distort total-IPC estimates.

A. Echo-state network

  • The comparison evaluates a 50-state ESN over spectral radii 0.1–1.7, using p = 1 for the standard network and p = 9 for a highly nonlinear input transformation.The p = 9 transformation is used to illustrate issues in computing total-IPC.Source: A. Echo-state network · p. 9
  • The simulations run for 210,000 steps, discard 10,000 warm-up steps, and repeat across 10 Monte Carlo runs.Source: A. Echo-state network · p. 9
  • Standard total-IPC evaluates degrees 1–10 with reduced maximum delays, whereas KIPC fixes maximum delay at 30 while accounting for all degrees.The reported upper bounds are 50 for total-IPC and 13.11 for KIPC.Source: A. Echo-state network · p. 9
  • For p = 9, threshold factor 3 leaves spurious correlations that push total-IPC above the ESN rank limit of 50; factor 5 also removes genuine terms and underestimates capacity.KIPC’s uncertainty is reported as very low in the stable-dynamics region s < 1.Source: A. Echo-state network · p. 10
  • KIPC values for p = 9 are lower than for p = 1, consistent with the weighting assumption; standard ESNs do not polynomially transform their inputs.Source: A. Echo-state network · p. 10
  • With τmax = 15, KIPC capacity begins decreasing at s = 0.9, chiefly in nonlinear terms, explaining total-IPC’s underestimation at s = 1.Source: Figure 3 · p. 10, A. Echo-state network · p. 10
  • 10-times faster: KIPC computation is reported to be faster than standard IPC and does not truncate the maximum polynomial degree.The comparison uses the same GPU and nT = 200,000.Source: Table I · p. 10, A. Echo-state network · p. 10

B. Disordered quantum spin chain reservoir

The study applies KIPC to a disordered transverse-field Ising spin chain, examining how field strength and disorder relate to its information-processing capacity. The disorder sweep reveals a divergence between the linear and nonlinear KIPC components.

B. Disordered quantum spin chain reservoir

  • The QRC uses a disordered all-to-all transverse-field Ising model, sweeping h at W=0 and W at fixed h=10.The authors chose the model for its varied dynamics as field strength and disorder change.Source: B. Disordered quantum spin chain… · pp. 10–11
  • At each step, the readout uses single-qubit expectations and distinct two-qubit correlations, calculated as idealized measurements without back-action.A physical implementation would estimate these statistics across repeated runs on identically prepared reservoir copies.Source: B. Disordered quantum spin chain… · p. 12
  • The simulation uses eight qubits and 52 output signals, with 30 Monte Carlo runs and 20,000 post-warm-up time steps for KIPC.The density-matrix simulation is computationally demanding; the 52-signal matrix rank is comparable to the ESN’s rank of 50.Source: B. Disordered quantum spin chain… · p. 12
  • For h>0.1, KIPC decreases from its large values at small h and eventually reaches a plateau, consistent with reported NARMA10 results.The same trend is reported for KIPC and its linear component, KIPClin.Source: B. Disordered quantum spin chain… · p. 12
  • For W>3, linear KIPC falls while nonlinear KIPC continues rising, and total KIPC peaks near W=5; this spin-chain behavior had not been reported.The authors caution that the exact maximizer may differ because of the experiment’s step size.Source: B. Disordered quantum spin chain… · p. 12

C. Software

The KIPC algorithm is implemented in Python, and its source code is available for computing the metric and replicating the experiments.

C. Software

  • Python source code is available for computing KIPC and reproducing the experiments.The code is hosted on GitHub.Source: C. Software · p. 12

V. CONCLUSIONS

The conclusions present KIPC as a kernel-based way to compute total information-processing capacity without degree truncation and to compare systems with one null-hypothesis test. Its QRC application identifies a mismatch between the peaks of linear and nonlinear capacity components.

V. CONCLUSIONS

  • KIPC computes information-processing capacity without an upper bound on polynomial degree and supports efficient computation and a single null-hypothesis test for comparing systems.The standard IPC instead requires a separate test for each basis function.Source: §V · p. 12
  • For the disordered quantum spin-chain reservoir, KIPC reveals that the linear and nonlinear KIPC components do not peak at the same disorder strength.The authors say this finding warrants further investigation.Source: §V · p. 12
  • The authors identify replacing permutation testing with an analytically computed parametric null approximation and exploring Bayesian significance testing as future work.They also propose investigating how KIPC might be tailored to QRC’s computational characteristics.Source: §V · pp. 12–13
  • In the QRC disorder sweep, linear KIPC decreases above W=3 while nonlinear KIPC continues increasing, producing a total-KIPC maximum at W=5.The exact maximizer may differ because of the experiment’s step size.Source: §V · p. 12, Figure 5 · p. 13

Appendix A: Appendixes

The appendix proofs use projection and kernel-matrix properties to establish results about KIPC estimators and bounds.

Appendix A: Appendixes

  • Theorem 1’s proof defines X as the stacked system states and U as the matrix of delayed inputs, then forms the kernel matrix K⊗(U,U).The proof examines its eigenvalues and eigenvectors and relates finite-sample eigenvalue estimates to kernel-function eigenvalues.Source: §1 · p. 14
  • The regression proof expresses the optimal prediction using projection Π=X(X⊤X)−1X⊤ and uses the idempotence property Π²=Π.Source: §1 · p. 14
  • Theorem 2’s bound follows by choosing the projection onto the ns eigenvectors of K⊗(U,U) with the largest eigenvalues.Source: §1 · p. 14
  • The Proposition 2 proof removes the diagonal term from a V-statistic to obtain an unbiased U-statistic; the remaining terms number nT(nT−1).Source: §1 · p. 14
  • The Corollary 2 proof follows from the definition of K(lin) and Proposition 2, simplifying (r+1) using r=ρ to establish result (11).Source: §1 · pp. 13–14

2. Computation of KIPC

KIPC computation uses blockwise kernel operations and an SVD-based reduction of the projection trace. The algorithm assumes the input length n is divisible by the block size B.

2. Computation of KIPC

  • The SVD of X reduces the projection trace to an r × r matrix, where r is the rank of X and r ≤ τ_max.Writing Π = V V⊤ gives Tr(ΠK⊗(U, U)) = Tr(V⊤K⊗(U, U)V).Source: §2 · p. 14
  • KIPC is computed efficiently using B × B blocks of the kernel matrix K⊗(U, U).Source: §2 · p. 14
  • The algorithm assumes n = kB for some integer k.Source: §2 · p. 14
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