# PF r34 ISAES gallery: Floquet states and classical analysis

This study runs the released six-qubit Floquet workflow and the released ISAES analyzers on a fixed set of actual simulation outputs. It supplies the full density matrix, all nine reduced-state checkpoints, four classical-analysis panels, configurations, raw results, native reports, plotting sources and independent numerical checks. The retained observations are deterministic numerical simulations, not hardware measurements or experimental data.

## What was executed

The source is `PF_Simulation_Standalone_and_SaaS_r34_private_workflows.zip.xz`, SHA-256:

`1987acf5712a7d2311744089bfb350885501bd925e18a2c31ee43447329d6a74`

The unmodified importable workflow is `documentation/workflow_examples/qiskit_floquet_isaes_v2.json`, SHA-256:

`61bc7f80e0f9201ee782a7c1ee52811955b1fe36842e85f500ea2978da1a47e4`

The gallery harness calls that bundle's `main.py:run`, followed by the released `MultivariateAnalyzer`, `DensityAnalyzer`, `AnomalyAnalyzer` and `PredictiveAnalyzer` local Python APIs. It also builds and renders the baseline through the released ISAES reporting engine. The baseline report completed with 16 native figures and no reported issues. This is a fresh local API execution; it does not exercise SaaS login, production process isolation or the Workbench GUI.

The environment uses Python 3.12.14, Qiskit 2.5.2, NumPy 2.3.5, SciPy 1.17.0, scikit-learn 1.8.0 and Matplotlib 3.10.8. Exact versions, source-module hashes, elapsed times and per-case artifact hashes are retained in `environment.json`, `provenance.json` and each `execution.json`.

## Baseline configuration and physical model

| Setting | Value | Meaning |
| --- | --- | --- |
| Qubits | 6 | Full density matrix has 64 rows and 64 columns |
| Completed cycles | 8 | State preparation and cycles 1–8 are retained |
| Ising coupling J | 0.34 rad | RZZ(2J) on each ring edge |
| Transverse rotation h | 0.71 rad | RX(2h) on each qubit |
| Phase-flip probability p | 0.025 | Independent channel on each qubit after every unitary cycle |
| Boundary | Ring | Six nearest-neighbor edges, each applied once |
| Initial state | Néel | q0=0, q1=1, q2=0, q3=1, q4=0, q5=1 |

Qubit q0 is the least-significant bit. Basis index 42 is therefore the initial state `|101010⟩` in the order `|q5 q4 q3 q2 q1 q0⟩`.

Every cycle applies all ZZ gates, then all X rotations. The ideal reference evolves under the same unitary U. The noisy density matrix first evolves as `U ρ U†`, followed on every qubit q by the exact channel `(1−p)ρ + p Zq ρ Zq`. The implementation uses Qiskit density-matrix evolution and Kraus operators. There is no shot sampling, invented point cloud or hand-assigned target curve in this study.

The calculated outputs are fidelity relative to the ideal evolved state, purity, mean Z magnetization, mean nearest-neighbor ZZ correlation, entropy of the low-index three-qubit subsystem and mutual information between the two halves. Entropies use base-2 logarithms and are expressed in bits. Other displayed observables are dimensionless. Cycle number is the time coordinate; no physical duration per cycle is asserted.

## Read the full-state and checkpoint figures

`figures/full-state-real.png` shows every real entry of the final 64 × 64 density matrix after eight cycles. The matching imaginary figure and complete complex matrix are retained. Diagonal entries are basis populations. Off-diagonal entries are coherences in this declared basis; neither a single color nor a visually prominent entry is itself an entanglement measure.

Both full-state component figures use the same symmetric linear color range, determined from the maximum absolute component across the retained real and imaginary matrices. Each pixel cell represents one retained matrix element. No interpolation, smoothing, thresholding or altered matrix values is applied.

`figures/q0-cycle-0.png` through `q0-cycle-8.png` show the real component of the reduced state after tracing out q1–q5. The website starts at cycle 3. All nine frames share the same symmetric range −1 to +1; the selector changes only which already calculated checkpoint is displayed. It does not run a new simulation or interpolate intermediate states. The JSON and CSV retain both complex components at full stored precision; the labels printed inside each image are rounded to six decimals for readability.

At cycle 3, the q0 populations are approximately 0.407038 and 0.592962, reduced-state purity is 0.634318, and coherence magnitude `|ρ01|` is 0.241903. The real off-diagonal entry is approximately −0.014031, while its imaginary component is approximately −0.241496. The real-component heatmap alone therefore does not show the full coherence magnitude.

The baseline ends with fidelity **0.328747735**, purity **0.125175878**, mean ZZ correlation **−0.271016509**, half-chain entropy **2.717278417 bits** and mutual information **0.946440932 bits**. Mean Z magnetization is approximately zero (−3.30 × 10⁻¹⁷). These are outputs of the stated noisy model, not software pass/fail thresholds or evidence of a privileged-frame advantage.

The gallery matrix figures call the unmodified r34 `pf_reporting.rendering.draw` on exact retained native figure specifications. Presentation changes add square cells, explicit basis labels, shared color ranges and checkpoint annotations. The original unstyled native exports remain in `baseline/native-report/figures`: `figure-012` is the full real state, `figure-013` its imaginary component and `figure-015` the retained q0 frame sequence. The HTML report provides r34's native interactive exploration.

## The 64-case analysis grid

All grid cases keep six qubits, eight cycles, ring boundary and Néel preparation. The fixed Cartesian product is:

| Parameter | Values |
| --- | --- |
| J, rad | 0.15, 0.34, 0.60, 0.90 |
| h, rad | 0.25, 0.50, 0.71, 1.00 |
| p | 0, 0.025, 0.075, 0.15 |

There are **64 distinct configured cases**. The baseline is executed separately and repeats `case-025`; it is not a 65th distinct configuration. The baseline plus grid therefore contain 65 workflow executions and 585 retained cycle records, including repeated baseline records. These deterministic grid cases are not independent random replicates. The grid and analysis choices are fixed in `configs/grid.json`; no points were selected after examining the results.

The four-panel figure analyzes the 64 final-cycle grid rows. Every row in `grid/summary.csv` maps a `case_id` to its configuration and exact result under `grid/cases/<case_id>/`. All nine cycle records and final states are preserved for every case.

### Panel A: principal components and descriptive groups

The inputs are fidelity, purity, half-chain entropy, mutual information and mean ZZ correlation. The released analyzer standardizes each feature by its retained mean and population standard deviation, computes PCA by singular-value decomposition and fits three K-means groups with seed 19. K-means uses 20 initializations; three additional restarts characterize optimization sensitivity.

The first two components explain approximately **78.375%** and **14.506%** of standardized variance. The plot coordinates, loadings, scaling, labels, centers and restart scores are saved in `ml/pca.json`. Group colors describe this grid under these chosen features. They do not establish distinct physical phases or statistically replicated populations.

### Panel B: density groups and explicit noise

The released DBSCAN analyzer uses purity and half-chain entropy with declared distance scales 1 and 3, respectively. These reflect the fixed physical upper bounds of purity and three-qubit entropy. Radius is 0.12 in the scaled feature space and `min_samples` is 4, including the observation itself.

The result has **three occupied groups and six noise-labelled rows**. A −1 noise label means that a point is not density-connected under this declared radius and scaling. It does not mean a failed simulation, an abnormal quantum apparatus or proof of a physical phase boundary. The graph axes retain purity and entropy in their original units; clustering operates in scaled coordinates. Exact labels and point IDs are retained in `ml/density.json`.

### Panel C: distance from a low-noise reference set

The 32 runs with p=0 or 0.025 form the reference set. The 32 different runs with p=0.075 or 0.15 are evaluated against it. The five features match Panel A. Their distance scales are population standard deviations computed from the reference rows only. The graph plots the released analyzer's Euclidean distance to each query's third nearest reference neighbor.

The horizontal coordinate is the retained query-row index, not time or a fitted trend. `figures/ml-anomaly-points.csv` maps each plotted index to its exact case ID. `ml/anomaly.json` also retains reference leave-one-out distances, reference identities and descriptive percentiles. These percentiles are not anomaly probabilities, p-values or fault diagnoses, and a low-noise reference is a supplied comparison choice rather than a proven normal operating distribution.

The released `AnomalyAnalyzer` returns these distances but does not emit a native figure. This panel is an explicitly declared gallery postprocessor of those exact results; no application code was changed to create it.

### Panel D: held-out purity prediction

The features are J, h and p; the target is final purity. A group split holds together all four noise settings belonging to each J/h pair. Twelve parameter groups (48 rows) train the model and four complete groups (16 rows) are held out. The held-out groups are J1-h3, J2-h1, J2-h2 and J3-h3, where the zero-based indices refer to the ordered parameter values above.

The released random forest uses its unchanged defaults: 100 trees, maximum depth 12, at most 32 leaves, minimum leaf size 2, seed 19 and one worker. Preprocessing and fitting use training rows only. No parameters are tuned using the test rows. `ml/prediction.json` retains the split, scaler, every tree, predictions, residuals, source hashes and metric definitions.

| Held-out measure | Result |
| --- | --- |
| Mean absolute error | 0.010947033 |
| Root mean squared error | 0.017638274 |
| Mean-training-target baseline RMSE | 0.412789909 |
| R² | 0.998133634 |

This is a single fixed holdout within a small deterministic grid, and final purity is strongly controlled by the declared dephasing parameter. The high score is not evidence of broad quantum-model prediction capability. It provides no uncertainty interval, extrapolation guarantee, hardware validation or justification for replacing the solver outside this evaluated grid.

The four-panel layout itself is composed with Matplotlib by `render_gallery.py`, using the native PCA, density and prediction figure arrays and the retained anomaly distances. Its purpose is to compare the four analyses on the same actual simulation study. It is not represented as a four-panel figure generated automatically by the Workbench interface.

## Independent numerical checks

A separately written NumPy implementation, with no Qiskit or PF imports, recomputed the full state evolution for all 65 executions. It compared the final complex density matrices and ideal statevectors, all nine q0 checkpoints and every retained trajectory value for all six observables. Absolute tolerance was 10⁻¹⁰. All 65 case audits passed.

| Comparison | Maximum absolute discrepancy |
| --- | --- |
| Final complex density matrix | 3.33 × 10⁻¹⁵ |
| Retained q0 checkpoints | 3.90 × 10⁻¹⁵ |
| Retained trajectory observables | 7.51 × 10⁻¹⁴ |

The checks also verified unit trace, Hermiticity, positivity within floating-point tolerance and consistency of the final partial trace. All 16 zero-noise grid cases retain unit purity and ideal-state fidelity within tolerance. A separate classical-analysis audit checked data identity, feature scaling, PCA coordinates, density labels, reference distances, grouped split separation, stored-tree prediction and held-out metrics against the retained inputs.

This validates numerical implementation agreement and evidence integrity for the stated cases. It does not establish experimental accuracy, Windows execution, researcher usability or production/BYOS qualification. The gallery study's execution and audit counts are reported separately from the site's cumulative release-validation totals; they are not added as an arbitrary number of new pytest cases.

## Local performance measurements

The timings below were recorded during this one execution in the documented Linux x86-64 environment. Numerical thread settings were one for OpenMP, OpenBLAS and MKL. They describe this small workload and runtime, rather than a cross-machine or production throughput benchmark.

| Timed operation | Elapsed time |
| --- | --- |
| 65 workflow calls, including the separately executed baseline | 5.0370 s total |
| 64 grid workflow calls | 0.0711 s median; 0.0585–0.2186 s range |
| Baseline report construction and export, 17 findings and 16 figures | 12.5734 s |
| PCA and K-means analyzer | 1.9268 s |
| DBSCAN analyzer | 0.0044 s |
| Reference-distance analyzer | 0.0014 s |
| Held-out random-forest analyzer | 0.2370 s |

Workflow measurements bracket the direct `main.py:run` calls, including artifact work performed inside those calls. They exclude interpreter startup, workflow import, SaaS scheduling, network transfer and GUI rendering, and they are not end-to-end user response times. The first PCA call also includes its lazy scientific-library imports, so it should not be compared as a steady-state algorithm-only timing with the later analyzers. Report timing includes both construction and native exports. These are single-run observations, without repeated timing trials or a hardware-normalized speed claim; full precision and per-case values remain in the execution records.

## Files and reproduction

| Material | Location in the study bundle |
| --- | --- |
| Fixed baseline and full grid protocol | `configs/baseline.json`, `configs/grid.json` |
| Importable workflow and unmodified embedded Python files | `workflow/` |
| Baseline raw output, complex states, trajectory CSV and execution hashes | `baseline/` |
| Native HTML, PDF, PowerPoint and exact figure exports | `baseline/native-report/` |
| All 64 configurations and raw execution results | `grid/cases/` |
| Analysis table | `grid/summary.csv`, `grid/summary.json` |
| Native analyzer inputs, findings, figures and fitted model record | `ml/` |
| Gallery PNG, SVG, PDF, exact plot tables and provenance | `figures/` |
| Runtime and source identities | `environment.json`, `provenance.json` |
| Execution instructions and dependency versions | `README-reproduction.md`, `requirements-reproduction.txt` |

Use the exact frozen release and recorded dependency versions. To perform a fully fresh run, copy `run_study.py` and `render_gallery.py` into a new empty study directory; these scripts write outputs relative to their own location. From that directory:

```bash
python run_study.py --source /path/to/privileged-frame-simulation --archive /path/to/PF_Simulation_Standalone_and_SaaS_r34_private_workflows.zip.xz
python render_gallery.py --pf-root /path/to/privileged-frame-simulation
```

In a directory containing this downloaded study, the execution harness deliberately reuses only checkpoints whose configurations, workflow identity and artifact hashes still match. It then regenerates the analyses and native report. Preserve the downloaded evidence before experimenting with a new protocol. The figure script performs no simulation or fitting and requires the retained native report and analyzer results.

Open `baseline/native-report/report.html` for native interactive inspection, `report.pdf` for the report, and the `figures/*.pdf` or SVG files for vector graphics. Use the raw JSON/CSV for analysis; numbers printed in plots and on the website are rounded display values. The reproducibility archive includes the corresponding independent audit reports and scripts.