What question did this study set out to answer?

The research aims to explore the exponential relaxation of Fisher path speed in learned transition matrices using quantum processor data.

April 13, 2026Open Access

Exponential relaxation of Fisher path speed on the statistical manifold of learned transition matrices: numerical evidence from quantum processors (Grammar Fingerprinting continuation)

Key Points

The research aims to explore the exponential relaxation of Fisher path speed in learned transition matrices using quantum processor data.
Analysis of Fisher-Rao summaries for 28 Google Sycamore readouts
Comparison of exponential vs. power-law tails using AIC
Statistical examination of τ and A×τ across row-stochastic transition matrices
Exploratory results on LLM entropy time series
Exponential decay of path speed is preferred in 25–26 out of 28 readouts across tested sizes
Median τ / N* is approximately 3.2 for K=7
Stable A×τ across different hardware topologies while maintaining qualitative patterns
Power law fit summarizes cross-K scaling with small residuals

Abstract

This deposit accompanies the preprint “Exponential relaxation of Fisher path speed on the statistical manifold of learned transition matrices” (April 2026), extending the Grammar Fingerprinting / Fisher-threshold line of work. What is included Fisher–Rao style summaries of learned row-stochastic transition matrices T (N) as a function of data length N for 28 Google Sycamore readout files, on a fixed training grid, for SAX alphabet sizes K ∈ 5, 7, 9 (420 stored transition matrices per K, plus CSV summaries). Fisher path speed along N, model comparison (exponential vs. power-law tail) by AIC, and reporting of τ, A, and A×τ together with the Fisher-trace grammar threshold N* (N-star) on the same N-grid. Exploratory parallel results on LLM entropy time series (same pipeline family; power-law preferred in the reported subset — treated as non-primary relative to the Sycamore scaling analysis). Main quantitative claims (Sycamore) Exponential decay of path speed is preferred on 25–26 / 28 readouts for each tested K (26/28 at K=5; 25/28 at K=7; 25/28 at K=9). Median τ / N* ≈ 3. 2 (K=7 narrative in the preprint). A×τ is comparatively stable across hardware topologies within fixed K (detailed at K=7; consistent qualitative pattern at K=5 and K=9) ; cross-K median scaling is summarized by a power law fit A×τ ≈ 21. 5·K⁰. 57 with small median residuals across the three K values. Theory comparison (non-parametric) The preprint compares the functional form (exponential relaxation and logarithmic threshold dependence) to the macroscopicity / classicality construction of Coppo, Pranzini Coppo’s AC×τC normalization does not carry over verbatim). Related deposits / continuity Grammar Fingerprinting (Zenodo 10. 5281/zenodo. 19158088) Fisher information threshold study on Sycamore (10. 5281/zenodo. 19391582) LLM entropy extension (10. 5281/zenodo. 19454201) Code & data Public code and committed results: https: //github. com/csaplard/QuantumCircuitGrammarResearch Sycamore raw readouts: Dryad 10. 5061/dryad. k6t1rj8 (as in prior Grammar Fingerprinting deposits).

Exponential relaxation of Fisher path speed on the statistical manifold of learned transition matrices: numerical evidence from quantum processors (Grammar Fingerprinting continuation)

Key Points

Abstract

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