Testing the stability of the holographic area law in quantum circuits with different connectivity topologies: a reproducible protocol and numerical results
This work presents a numerical proof-of-concept for an algorithmic pipeline designed to test the stability of the holographic area law in quantum circuits with different connectivity topologies (chain and complete). METHODOLOGY:- Randomized measurements protocol (Elben et al., PRL 2018) for estimating Rényi-2 entropy- Weighted linear regression of S₂ vs classical cut area- Bootstrap error estimation (B=200 iterations, 95% confidence intervals)- Statistical validation of scaling stability via p-value analysis KEY RESULTS:- Chain topology (nearest-neighbor): G_eff = 0.50 ± 0.05- Complete topology (all-to-all): G_eff = 0.26 ± 0.02- Ratio G_eff^chain / G_eff^complete ≈ 1.95 ± 0.25- Scaling stability confirmed for both topologies (p > 0.83) IMPORTANT NOTE ON ZNE:Zero-noise extrapolation (ZNE) is implemented in the algorithmic pipeline but was intentionally NOT applied to this simulator dataset, as no hardware noise is present. This ensures that the reported uncertainties reflect only the statistical uncertainty of the randomized measurements protocol. The ZNE module will be activated in future experiments on real quantum hardware. FILES INCLUDED:- Grav_Article_ENG.pdf: Full article text (6 pages, 3 tables)- main.tex: LaTeX source code of the article- raw_dataset_chain.json: Complete raw measurement data for chain topology (30 subsystem-depth combinations)- raw_dataset_complete.json: Complete raw measurement data for complete topology (30 subsystem-depth combinations)- README.md: Documentation of data format and reproduction instructions LIMITATIONS:- System size: N=4 qubits- Fixed ansatz parameters (θ=π/4, γ=π/4)- Idealized simulator environment (no hardware noise) This study establishes a reproducible baseline protocol. Experimental validation on real quantum hardware is planned for a follow-up publication.
This work presents a numerical proof-of-concept for an algorithmic pipeline designed to test the stability of the holographic area law in quantum circuits with different connectivity topologies (chain and complete). METHODOLOGY:- Randomized measurements protocol (Elben et al., PRL 2018) for estimating Rényi-2 entropy- Weighted linear regression of S₂ vs classical cut area- Bootstrap error estimation (B=200 iterations, 95% confidence intervals)- Statistical validation of scaling stability via p-value analysis KEY RESULTS:- Chain topology (nearest-neighbor): G_eff = 0.50 ± 0.05- Complete topology (all-to-all): G_eff = 0.26 ± 0.02- Ratio G_eff^chain / G_eff^complete ≈ 1.95 ± 0.25- Scaling stability confirmed for both topologies (p > 0.83) IMPORTANT NOTE ON ZNE:Zero-noise extrapolation (ZNE) is implemented in the algorithmic pipeline but was intentionally NOT applied to this simulator dataset, as no hardware noise is present. This ensures that the reported uncertainties reflect only the statistical uncertainty of the randomized measurements protocol. The ZNE module will be activated in future experiments on real quantum hardware. FILES INCLUDED:- Grav_Article_ENG.pdf: Full article text (6 pages, 3 tables)- main.tex: LaTeX source code of the article- raw_dataset_chain.json: Complete raw measurement data for chain topology (30 subsystem-depth combinations)- raw_dataset_complete.json: Complete raw measurement data for complete topology (30 subsystem-depth combinations)- README.md: Documentation of data format and reproduction instructions LIMITATIONS:- System size: N=4 qubits- Fixed ansatz parameters (θ=π/4, γ=π/4)- Idealized simulator environment (no hardware noise) This study establishes a reproducible baseline protocol. Experimental validation on real quantum hardware is planned for a follow-up publication.
This technical note documents the numerical stability of a two-peak (Z-level) interference pattern observed in a fixed 29-qubit quantum circuit. The analysis was performed using the Qiskit AerSimulator backend under controlled conditions, including variations in shot count and phase-parameter offsets. The results demonstrate that the observed interference structure is reproducible, robust against sampling fluctuations, and not dependent on fine-tuned parameter choices. This document does not present a theoretical proof or physical interpretation. Its sole purpose is to provide an empirical numerical reference for the stability and reproducibility of the observed interference pattern, serving as a supporting record for related theoretical and conceptual works by the author. In version v3, the numerical analysis is extended beyond 63 qubits up to 79 qubits by employing reduced statistical descriptors rather than full state-space representations. This extension demonstrates that the observed Z-level interference stability persists under scalable statistical evaluation, without requiring exponentially growing state-vector simulations. Additional clarification The reported “Z-level progress” refers to an internal, relative coverage metric within the author’s numerical stability framework. It is not a physical completeness measure, nor a claim of global optimality. Earlier high-percentage stability values (e.g. 95–98%) correspond to local robustness indicators such as sampling consistency, seed invariance, and phase-noise tolerance within fixed parameter windows. The lower percentage reported here reflects a stricter, global aggregation across multiple independent stability dimensions. This distinction is intentional and avoids conflating local robustness with global domain coverage. This work reports a numerical stability certificate for X–Z basis statistical divergence measures obtained in controlled simulations. A multiscale grid and zoom analysis identifies a non-trivial stable phase-offset interval (pj ∈ [0.038, 0.052]) in which total variation distance, Shannon entropy, and directed Kullback–Leibler divergences remain reproducible under changes in sampling depth, random seed, and numerical resolution. A persistent directional asymmetry between KL_XZ and KL_ZX is numerically observed across all tested scales. This document reports numerical results only and does not claim physical realization, theoretical proof, or universal validity. Interpretational note: The absence of a direct Lorentz-covariant or gauge-field embedding should not be interpreted as a deficiency of the numerical results, but as a limitation of applying classical spacetime questions to an emergent, resonance-based stability structure. The reported Z-level behavior constrains the space of viable physical interpretations rather than completing one.
Six quantum circuit experiments performed on ibm_marrakesh (IBM Quantum, 156 qubits, Heron R2 architecture) are reported, designed to test boundary signatures of the ER=EPR conjecture within the AdS₂/JT gravity framework, with the Sachdev-Ye-Kitaev (SYK) model as the holographic dual of the black hole horizon. The experiments were executed iteratively: each result raised new questions that directly motivated the next protocol. In order of execution they implement: tripartite entanglement certification via the Mermin operator (|⟨M⟩| = 3.885); a traversable wormhole baseline using a Bell-pair thermofield double (F(g=π/2) = 1.000); black hole evaporation unitarity via the Hayden-Preskill protocol (F_rec = 0.9983, with the decoder reduced to zero two-qubit gates by the compiler); a corrected traversable wormhole using genuine SYK scrambling and exact Majorana coupling (F(g=0) = 0.020, F(g=π/2) = 0.953, GJW gain = 0.933); quantum chaos via out-of-time-order correlators (λ_L = 1.975 ± 0.294, 54.8% of the Maldacena-Shenker-Stanford bound); and black hole entropy under information absorption, where the hardware reproduces the entropy growth S(t) of SYK formation on top of a noise floor and the energy response to Alice’s input (correlation 0.998 with simulation), while the predicted area growth consistent with the Bekenstein law is obtained in simulation but is not resolved above noise on hardware. Total QPU time across all experiments: 7 minutes 2 seconds. Five of the six experiments reproduce the boundary predictions of the ER=EPR framework on hardware; the sixth reproduces them in simulation and only partially on hardware. Version 3 corrects Experiments 2 and 3 following the original laboratory reports. Exp. 2 reports the raw F(g=π/2) = 1.01 ± 0.01 instead of the clipped 1.000. In Exp. 3 the fused U†U circuit is identity by construction; its fidelity (0.9983) is reported as a noise-floor control, not as hardware recovery. The hardware scrambling result (F_scr = 0.530 vs. 0.532 in simulation) is kept; recovery is demonstrated in simulation. Experiments 2b, 5 and 6 are unchanged. Code and data: https://github.com/horca17/er-epr-six-protocols
Attention: This version is preliminary and contains errors that will be corrected in subsequent publications. Within the FKGP (Fractal Quantum Holographic Paradigm) framework we propose and experimentally verify a scaling law for the second-order Rényi entanglement entropy S2(A) in graph states of varying topological dimension D. The key result is S2(A) = rank_GF2(Gamma) · ln 2, where the GF(2)-rank of the A-B boundary adjacency matrix increases by exactly one at each step of the dimensional ladder. On IBM Heron r2 (ibm_fez) with 65,536 shots per circuit, four steps are confirmed: D=2.00 (ΔSNR=76.9σ), D1=2.05 (34.3σ), D2=e (7.8σ), and D=3 (5.1σ). All detections exceed the 5σ threshold. Cosmological hypotheses are advanced: D1 = 2 + Ω_b (baryon fraction) and D2 = e (informational efficiency). This paper includes a complete reproducibility appendix (Batches 1–10): all QASM circuits, measured P₀ values, noise corrections, and ΔSNR calculations.
Standard quantum circuit simulation represents amplitudes as double-precision complex numbers, so quantities such as entanglement entropy are computed only approximately. We revisit two well-known facts about the stabilizer formalism — that entanglement entropy of a stabilizer state is always an integer, and that the distance of a stabilizer code is a combinatorial quantity — and implement both exactly, using binary linear algebra over F2 on the stabilizer tableau, with no floating-point step anywhere in the computation. We verify the entropy formula against Qiskit's floating-point von Neumann entropy on canonical states and on 30 random Clifford circuits (all matches exact), reproduce the area-law entanglement of the 1D cluster state, and verify the code-distance procedure against three canonical quantum error-correcting codes (Steane, the 5-qubit code, and Shor's code), recovering their published distances exactly. As an illustration of the exact toolkit in use, we report an empirical scaling law for the depth at which entanglement saturates in a specific random-circuit model. Finally, since every result above concerns exact mathematics or exact simulation of noiseless Clifford circuits, we complement it with a small, honestly-labelled real-hardware comparison: the GHZ state on 3–7 qubits, run on IBM Quantum superconducting hardware, measured against the exact noiseless prediction our formalism computes, together with a quantitative check against a naive noise model built from the device's own published calibration data — which the data systematically contradicts by 18–21 percentage points, a discrepancy we report along with our best physical explanation for it. A second, independent hardware experiment measures the cluster state's own stabilizer generators directly, and identifies a single miscalibrated physical qubit as correlated with elevated error; a follow-up control experiment, forcing the circuit onto verified low-error qubits instead, confirms the effect is causal but partial (leakage drops by 1.84 percentage points, roughly a third of the excess) rather than the whole explanation, which we report at its true, unexaggerated size. Applying the naive noise model to the control circuit itself reverses its bias (now optimistic rather than pessimistic) by an amount consistent, to order of magnitude, with ordinary decoherence over the circuit's execution time — a physical effect the naive model was never built to capture.
Replication Data and Code for: "Capturing the Universality Class of the 2D Ising Model via Symmetry-Corrected Machine Learning and Finite-Size Scaling" This repository contains the complete source code, simulation datasets, and analysis scripts required to reproduce all findings presented in the associated manuscript. The study introduces a symmetry-corrected machine learning pipeline (Gauge-Fixed PCA + Logistic Regression) to identify phase transitions and estimate critical temperatures with high precision, validated on the two-dimensional Ising model. Repository Contents: 01_generate_data.py — Generates 2D Ising model spin configurations using the Wolff cluster algorithm. Produces equilibrium datasets for lattice sizes L∈{32,48,64,96}L \in \{32, 48, 64, 96\} L∈{32,48,64,96} with 10,000 independent configurations per temperature point. 02_analysis_pipeline_v2.py — Core analysis script: performs gauge fixing (σ′=σ⋅sgn(m)\sigma' = \sigma \cdot \mathrm{sgn}(m) σ′=σ⋅sgn(m)), PCA dimensionality reduction (d=16d = 16 d=16, cross-validated), confusion-scheme classification with parabolic sub-grid interpolation, bootstrap uncertainty quantification (Nboot=50N_{\rm boot} = 50 Nboot=50), and pFMp_{\rm FM} pFM computation with isotonic calibration. Outputs summary_ml_v2.json. 03_plot_main_figures.py — Reproduces Figures 1–6 of the main manuscript. 04_plot_supplementary_v2.py — Reproduces Supplementary Figures S1–S5 and Table 1. ablation_preprocessing_S5.py — Standalone preprocessing ablation script: runs raw and ∣m∣|m| ∣m∣-only comparisons and generates Supplementary Figure S5. Does not modify main results. summary_ml_v2.json — Pre-computed results (gauge-fixed pipeline) for immediate reproduction of all main figures without re-running Monte Carlo simulations. summary_ml_v2_raw.json — Pre-computed ablation results: raw (no gauge fixing) preprocessing, L∈{32,64,96}L \in \{32, 64, 96\} L∈{32,64,96}. summary_ml_v2_absm.json — Pre-computed ablation results: ∣m∣|m| ∣m∣-only preprocessing, L∈{32,64,96}L \in \{32, 64, 96\} L∈{32,64,96}. Table1_Tc_Comparison.tex — LaTeX source for Table 1. requirements.txt — Python dependencies. README.txt — Step-by-step reproduction instructions. Methodology Highlights: System sizes: L∈{32,48,64,96}L \in \{32, 48, 64, 96\} L∈{32,48,64,96} 10,000 independent configurations per temperature point; Wolff cluster algorithm with 10410^4 104 thermalization steps Symmetry correction: global Z2\mathbb{Z}_2 Z2 gauge fixing prior to PCA Confusion scheme: validation cross-entropy minimization over T0∈[2.10,2.45]T_0 \in [2.10, 2.45] T0∈[2.10,2.45] (60 candidate values, band Δ=0.05\Delta = 0.05 Δ=0.05, 5-fold CV) Bootstrap: 50 configuration-level resamples; parabolic sub-grid interpolation for discretization bias reduction Key result: Tc≈2.2717(8)T_c \approx 2.2717(8) Tc≈2.2717(8) for L=96L = 96 L=96 (0.6%0.6\% 0.6% from Onsager); universality class confirmed via FSS data collapse with ν=1\nu = 1 ν=1
Experimental measurement data of the QITE formalism (Quantum-Inspired Thermal Equilibrium) on IBM quantum processors of the Heron R2 generation with 156 physical qubits. Altogether 75+ jobs on two different backends (ibm_kingston and ibm_fez), over 830,000 shots, carried out on 8, 10, and 11 April 2026. Seven experiment series: (1) Static phases (R=0.5 / 1.0 / 2.0): three clearly separated Hamming-weight distributions (16.3σ significance), positive ZZ correlations under QITE (+0.318 at R=2.0) versus negative ones in the null experiment (−0.061). Controlled comparison experiment with null hypothesis (identical circuit structure with random parameters). (2) Real-time feedback loop (8 cycles): two targeted perturbations (R=3.0 and R=0.3); the system returns autonomously to equilibrium within 2 cycles. First demonstration of dynamic controllability of a QITE quantum collective on a real quantum computer. (3) Cross-backend reproducibility: identical results on ibm_kingston and ibm_fez — deviations within a few percent. (4) Zero-noise extrapolation (ZNE): circuit folding (1×, 3×, 5×) with Richardson extrapolation. Result: coherence 0.474 → 0.474 extrapolated (improvement: −0.0005). The optimized circuit (depth 13, 120 CZ gates) already operates at the hardware noise floor — ZNE cannot improve it further because the circuit is already near-optimal. (5) Scaling comparison 27 vs. 156 qubits: the 27-qubit version on the same backend (ibm_kingston) shows ZZ correlation of 0.788 against 0.318 at 156 qubits (Δ = +0.470). Circuit depth drops from 13 to 6, CZ gates from 120 to 13. Confirms: the QITE signal scales with gate quality. (6) Extended feedback loop (20 cycles): four extreme perturbations — R=3.0 (excess), R=0.3 (deficit), R=5.0 (5-fold excess), R=0.1 (10-fold deficit). Maximum coherence C=0.785 (highest value ever measured). The system reorganizes itself after each perturbation within 2–4 cycles. Long-term stability demonstrated across 20 cycles and 436 seconds of total runtime. (7) 3-minute stability demonstration (20 cycles, 262.7 seconds): dedicated long-term test with two targeted shocks (R=4.0 and R=0.2). Result: ones_fraction remains between 0.491 and 0.499 over 4.4 minutes (maximum deviation from equilibrium: 0.009). Drift across the entire runtime: +0.003. Complete recovery after both shocks. 18/18 non-shock cycles stable. First demonstration of a self-correcting quantum field on a real quantum computer over several minutes. Circuit optimization relative to 8 April: reduction of circuit depth from 295 to 13 (factor 22×), two-qubit gates from 1,946 to 120, by exclusive use of native Heron R2 gates (CZ, SX, RZ, X) and direct lambda encoding via RY(θ) with θ = 2·arctan(λ₂/λ₁). The system demonstrably produces controllable, correlated, self-organizing quantum states on real hardware — no simulation, no theory, but physical measurement data of a concrete technical system. All 59 IBM job IDs are contained in the raw data and verifiable on quantum.ibm.com. German patent applications Az. 10 2025 003 906.9 (QITE) and Az. 10 2025 004 891.2 (QISE). Final document (12 April 2026) — 21 sections, 9 experiment series, 372 individual quantum measurements. Extensions compared to the previous version: Section 19: 9-minute experiment (209 cycles, 156 qubits, 4096 shots) Phase A: null test (23 cycles, R=1.0) — baseline control measurement Phase B: 8 shocks (R=0.1 to R=8.0) incl. double shock — complete recovery after each shock Section 20: overall summary of all 9 experiment series (8–12 April 2026) Section 21: raw-data appendix with SHA-256 checksums of all 9 JSON files Core results: 372 quantum measurements on IBM Heron R2 (ibm_kingston, ibm_fez) Dynamic range R=0.1 to R=8.0 (factor 80) with complete self-correction The null test proves: recovery is an active process, not a hardware artefact No drift, no collapse across 209 consecutive measurements Optimized circuit: depth 13, 120 CZ gates, at the hardware noise floor (ZNE-confirmed)