Exponential relaxation of Fisher path speed on the statistical manifold of learned transition matrices: numerical evidence from quantum processors (Grammar Fingerprinting continuation)
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·K^0.57 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 & Verrucchi (Phys. Rev. A 106, 042208, 2022) and explicitly distinguishes structural analogies from parametric identities (different manifold; Coppo’s A_C×τ_C 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/QuantumCircuit_Grammar_Research Sycamore raw readouts: Dryad 10.5061/dryad.k6t1rj8 (as in prior Grammar Fingerprinting deposits).
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·K^0.57 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 & Verrucchi (Phys. Rev. A 106, 042208, 2022) and explicitly distinguishes structural analogies from parametric identities (different manifold; Coppo’s A_C×τ_C 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/QuantumCircuit_Grammar_Research Sycamore raw readouts: Dryad 10.5061/dryad.k6t1rj8 (as in prior Grammar Fingerprinting deposits).
We present Grammar Fingerprinting, a model-free method for identifying quantum processor hardware topology directly from raw measurement noise. Using SAX encoding and a character-level LSTM, the method extracts learned transition matrices from readout sequences and clusters them to classify hardware topology blindly. Applied to 28 Google Sycamore quantum supremacy configurations, it achieves 84.5% mean accuracy with six independent controls confirming the signal. A sharp data-length threshold at 8,000 measurements reveals long-range temporal structure in multi-qubit noise correlations. Full code, results, and validation included.
This dataset and code repository accompanies the article of the same title (version 5, May 2026) and provides all simulation data, source code, and raw terminal output necessary to reproduce and verify the results. Study summary We investigate the critical dynamics of the two-dimensional Ising model at the exact critical point (Tc ≈ 2.269185) using the local Metropolis Monte Carlo algorithm. The central focus is the characterization and comparison of two widely used correlation time estimators: the exponential decay criterion τ₁/e and the integrated autocorrelation time τ_int, following the convention of Sokal (1997). Simulations were performed for seven system sizes L ∈ {16, 24, 32, 48, 64, 96, 128} with up to 50 independent replicas per system size, using both Python (NumPy/PCG64) and C++ (std::mt19937, compiled with -O2) implementations. The two implementations were developed independently, enabling cross-validation of results across languages. Main findings The coefficient of variation (CV = std/mean) of τ_int converges to a physical floor of approximately 0.44–0.52 regardless of the number of replicas (8 or 50) or programming language, confirming that this floor is determined by the physics of the estimator under local Metropolis dynamics at criticality — not by statistical sampling limitations. The distribution of τ₁/e is genuinely bimodal at the critical point: individual replicas alternate between low (~200–500 MC steps) and high (~3000–5000 MC steps) correlation times for the same system size, with mean-to-median ratios of up to 2.6× and bimodal ratios (mean of upper half / mean of lower half) of up to 23.5×. This bimodality does not disappear with more replicas, confirming it is a physical property of the estimator. No detectable power-law scaling τ ~ L^z is found for L ≤ 128 in any of the six independent runs (R² < 0.14 in all log-log fits using all 7 system sizes). The accepted asymptotic value z ≈ 2.17 (Li & Sokal, 1989) implies an expected τ_int of approximately 73,600 MC steps at L = 128 — 76× larger than the observed value of 972. This quantifies the dominance of finite-size corrections in the range studied, and justifies treating the effective dynamic exponent z_eff as a continuously varying quantity rather than a fixed constant. The C++ implementation achieves speedups of 98× to 782× relative to Python depending on system size, enabling the full 50-replica run across all seven system sizes to complete in approximately 33 minutes. Cross-validation caveat Cross-validation between Python and C++ is fully valid for τ₁/e. For τ_int, the Python scripts used a fixed equilibration of n_equil = 3000 steps for all L — documentably insufficient for L ≥ 24 where τ_int > 600 (n_equil/τ < 3–5τ). This generates systematic underestimation of up to 86% at L = 64. The valid cross-validation for τ_int is between the C++-8 and C++-50 runs, where CV agreement (~0.52 in both) confirms the physical floor is robust. Connection to the EIE/ZIE framework This study provides the empirical foundation for the EIE/ZIE adaptive meta-control framework (Patent BR 10 2025 024459 4), justifying three architectural decisions: (1) τ_int as the preferred estimator of the correlation time τ_corr; (2) the effective dynamic exponent z_eff treated as a continuously estimated dynamic variable rather than a fixed constant; and (3) the dynamic freezing time τ_freeze introduced as an observable independent of τ_corr, motivated by the bimodal behavior of τ₁/e which reveals that the system alternates between regimes of high and low correlation persistence — a phenomenon invisible to conventional autocorrelation-based diagnostics. Repository contents Cpp/ — C++ source code (3 scripts) and simulation data (6 CSV files + raw terminal output) Python/ — Python source code (2 scripts) and simulation data (4 CSV files + raw screen output) Paper/ — Article in English (PDF) and Portuguese (PDF) README.md — Full repository documentation including file descriptions, column definitions, reproduction instructions, and physical parameters LICENSE.txt — CC BY 4.0 for data; MIT for code; patent notice Joemerson da Silva Lima — Independent Researcher, Cuiabá, Mato Grosso, Brazil joemerson93lima@gmail.com Patent BR 10 2025 024459 4
Ziel dieser Arbeit ist es, den Begriff der Quanten-Kolmogorov-Komplexität formal zu definieren und seine wichtigsten Eigenschaften rigoros zu beweisen. Die klassische Kolmogorov-Komplexität ist ein bekanntes und nützliches Maß für die Zufälligkeit endlicher Wörter. In den letzten Jahren wurden unterschiedliche Quantenverallgemeinerungen der Kolmogorov-Komplexität vorgeschlagen. Die natürlichste Art der Verallgemeinerung stammt von Berthiaume u.a., die die Komplexität eines Quantenwortes definieren als die Länge der kürzesten Quanteneingabe für einen universellen Quantencomputer, die als Ausgabe das entsprechende Wort produziert. Abgesehen von kleinen Änderungen soll dieser Komplexitätsbegriff in der hier vorliegenden Arbeit untersucht werden. Zunächst untersuchen wir verschiedene Aspekte des zugrunde liegenden Modells der Quantenturingmaschine (QTM), und zwar mit größerer formaler Genauigkeit als in bisherigen Arbeiten. Anschließend wenden wir diese Resultate auf die Quanten-Kolmogorov-Komplexität an. Unser erstes Ergebnis, basierend auf der Arbeit von Bernstein und Vazirani, ist ein Beweis für die Existenz einer universellen QTM, die jede andere QTM für eine beliebige Anzahl von Zeitschritten simulieren kann, und dann selbst mit Wahrscheinlichkeit eins hält. Weiterhin zeigen wir, dass jede Eingabe, die eine QTM beinahe halten lässt, modifiziert werden kann, um eine Eingabe zu erhalten, die die universelle QTM vollständig halten lässt, wobei sich die Eingabelänge höchstens um eine konstante Anzahl von Qubits vergrößert. Daraus folgt, dass die Quanten-Kolmogorov-Komplexität die Invarianzeigenschaft besitzt, d.h. sie ist bis auf eine additive Konstante unabhängig von der Wahl der universellen QTM. Außerdem stimmt die Quantenkomplexität klassischer Wörter mit deren klassischer Komplexität überein, wieder bis auf eine additive Konstante. Die entsprechenden Beweise beruhen auf verschiedenen analytischen Abschätzungen. Weiterhin beweisen wir mehrere Sätze, die zeigen, dass nur wenige Quantenwörter kleine Quantenkomplexität besitzen können. Schließlich zeigen wir, dass bei ergodischen Quantendatenquellen Komplexitätsrate und Entropierate mit Wahrscheinlichkeit eins übereinstimmen. Den Abschluss der Arbeit bildet ein Ausblick auf eine mögliche Anwendung der Quanten-Kolmogorov-Komplexität in der statistischen Mechanik.
Come si può apprendere uno stato quantistico della luce? Come si può distribuire informazione quantistica attraverso un collegamento ottico? È possibile superare la sfida delle regolarizzazioni nella teoria di Shannon quantistica? Questa tesi affronta tutte e tre queste questioni. La prima parte della tesi studia l’intersezione tra due importanti ambiti di ricerca: la teoria dell’apprendimento quantistico e i sistemi a variabili continue (continuous-variable, CV). La teoria dell’apprendimento quantistico si occupa di come estrarre informazione classica da sistemi quantistici nel modo più efficiente possibile. I sistemi CV sono onnipresenti in natura e nelle tecnologie quantistiche, in quanto modellano sistemi bosonici e piattaforme di ottica quantistica. Questa intersezione solleva numerose domande aperte di grande interesse, alcune delle quali vengono affrontate in questa tesi. Una prima domanda naturale riguarda i limiti ultimi della tomografia per sistemi CV. Rispondiamo a questa domanda dimostrando che: (i) la tomografia di stati non gaussiani è estremamente inefficiente; (ii) la tomografia di stati gaussiani è efficiente; e (iii) la complessità campionaria della tomografia CV cresce esponenzialmente con il grado di non gaussianità dello stato incognito. Altre domande naturali affrontate in lavori successivi correlati includono l’apprendimento efficiente di processi gaussiani e il problema di testare in modo efficiente se uno stato CV incognito sia gaussiano oppure lontano dall’insieme degli stati gaussiani. Come sottoprodotto delle nostre analisi, otteniamo diversi limiti sulla distanza in traccia tra stati CV in funzione delle loro matrici di covarianza, che forniscono un utile insieme di strumenti per la ricerca nella teoria dell’apprendimento quantistico con sistemi CV e potenzialmente anche in altri contesti. La seconda parte della tesi si concentra sulla teoria di Shannon quantistica per sistemi CV. Uno degli obiettivi principali di questa disciplina è comprendere come trasmettere informazione classica o quantistica in modo efficiente attraverso canali quantistici. Iniziamo determinando il massimo livello di rumore compatibile con la distribuzione di chiavi quantistiche e con la distribuzione di entanglement su canali gaussiani bosonici a fase insensibile, che costituiscono modelli standard per fibre ottiche e collegamenti in spazio libero. Raggiungiamo questo obiettivo stabilendo limiti inferiori allo stato dell’arte per la capacità di chiave segreta e per la capacità quantistica a due vie nel regime di alto rumore. Studiamo poi la comunicazione quantistica bosonica sotto diversi vincoli realistici, inclusi l’effetto combinato di perdita di fotoni e decoerenza di fase, nonché la presenza di rumore non markoviano (con memoria); in lavori successivi correlati consideriamo inoltre il regime one-shot di un numero finito di usi del canale e tolleranza di errore non nulla. La parte finale della tesi si sposta sui sistemi a dimensione finita e affronta un ostacolo centrale nella teoria di Shannon quantistica: il problema delle regolarizzazioni, che rende computazionalmente intrattabile ottenere una comprensione quantitativa precisa dell’efficienza ultima di compiti operativi fondamentali quali la manipolazione dell’entanglement. In questo contesto, consideriamo il problema del calcolo del costo asintotico di entanglement per la preparazione esatta di stati quantistici rumorosi sotto operazioni PPT. Basandoci su una precedente caratterizzazione del costo di entanglement PPT in termini di una formula regolarizzata, costruiamo invece una gerarchia di programmi semidefiniti che aggira completamente il problema della regolarizzazione e converge esponenzialmente al vero valore asintotico del costo di entanglement. Per quanto a nostra conoscenza, questa è la prima volta in cui una misura di entanglement asintotica risulta essere computabile in modo efficiente nonostante l’assenza di una formula single-letter in forma chiusa.
Abstract This work presents an empirical characterization of long-time coherence decay in a trapped-ion qubit on the IonQ Forte-1 quantum processor using high-depth CPMG dynamical-decoupling sequences. Across 97 independent runs, the measured coherence envelope exhibits a shallow long-time curvature that a standard single-exponential model cannot capture. To summarize the behavior over a wide time range, we introduce an extended empirical envelope of the form C(t) = A exp(−t/τ) t⁻ᵅ , which significantly improves fit quality (AIC/BIC, RMSE, and residual autocorrelation) relative to a pure exponential decay. The extracted power-law exponents fall in the range 0.1 ≤ α ≤ 0.3 , indicating persistent long-time structure in the coherence decay. This extended envelope provides a compact descriptive model for benchmarking trapped-ion systems, evaluating dynamical-decoupling performance, and informing hardware-aware noise characterization. The empirical crossover timescale tₚ = ατ offers a simple indicator of when the power-law component becomes relevant within the experimental window. The methods include AWS Braket execution, IonQ device calibration reporting, nonlinear least-squares fitting, and statistical diagnostics. This dataset and analysis serve as a practical reference for long-time coherence studies in trapped-ion quantum hardware.