# Real-Example Results

<!-- AUTO-GENERATED by scripts/extract_notebook_plots.py. -->
<!-- Re-run after executing notebooks; do not edit ledgers by hand. -->

This generated page displays embedded setup schematics, diagnostic plots, and text outputs from every real-example notebook.

## Current Status

- Source notebooks: `notebooks/real_examples/`
- Notebooks displayed: `30`
- Embedded plot artefacts displayed: `55`
- Plain-text notebook results displayed: `80`
- Plot manifest: [`results/tables/real_examples_plot_manifest.csv`](../../results/tables/real_examples_plot_manifest.csv)

## Related Pages

- [Results summary](results.md)
- [Notebook index](notebooks.md)
- [Tutorial notebook outputs](tutorial_results.md)
- [Benchmark notebook outputs](benchmark_results.md)

## Regeneration

Execute notebooks, extract their embedded outputs, and refresh this page with:

```bash
python scripts/extract_notebook_plots.py --preset real-examples --execute --write-docs
```

## Notebook Results

### `01_ground_state_filtering.ipynb`

Source: [`notebooks/real_examples/01_ground_state_filtering.ipynb`](../../notebooks/real_examples/01_ground_state_filtering.ipynb)

```{image} ../../results/plots/real_examples/01_ground_state_filtering-plot-01.png
:alt: Ground-State Filtering for a Toy Hamiltonian plot 1
:width: 760px
```

```{image} ../../results/plots/real_examples/01_ground_state_filtering-plot-02.png
:alt: Ground-State Filtering for a Toy Hamiltonian plot 2
:width: 520px
```

Output 1 (cell 6):

```text
Energies [model energy units]: [-1.7205 -1.      1.      1.7205]
```

Output 2 (cell 10):

```text
Cutoff [model energy units]: -1.3602325267042625
Scale: 3.0806975801127883
Scaled Energies [model energy units]: [-1.     -0.7661 -0.1169  0.1169]
```

Output 3 (cell 12):

```text
Projector Error: 0.13103683490762716
Idempotence Error: 0.11947057773046423
```

Output 4 (cell 14):

```text
Initial Ground Overlap [probability]: 0.21077170866211486
Filtered Ground Overlap [probability]: 0.9974954270617575
```

### `02_tight_binding_band_filter.ipynb`

Source: [`notebooks/real_examples/02_tight_binding_band_filter.ipynb`](../../notebooks/real_examples/02_tight_binding_band_filter.ipynb)

```{image} ../../results/plots/real_examples/02_tight_binding_band_filter-plot-01.png
:alt: Band Filtering in a Tight-Binding Chain plot 1
:width: 760px
```

```{image} ../../results/plots/real_examples/02_tight_binding_band_filter-plot-02.png
:alt: Band Filtering in a Tight-Binding Chain plot 2
:width: 760px
```

Output 1 (cell 6):

```text
Energies [hopping units]: [-1.8794 -1.5321 -1.     -0.3473  0.3473  1.      1.5321  1.8794]
```

Output 2 (cell 10):

```text
Band Weights [probability]: [-0.      0.0083  0.1822  0.9885  0.9885  0.1822  0.0083 -0.    ]
```

### `03_imaginary_time_filtering.ipynb`

Source: [`notebooks/real_examples/03_imaginary_time_filtering.ipynb`](../../notebooks/real_examples/03_imaginary_time_filtering.ipynb)

```{image} ../../results/plots/real_examples/03_imaginary_time_filtering-plot-01.png
:alt: Imaginary-Time Filtering with an Exponential Polynomial plot 1
:width: 760px
```

Output 1 (cell 6):

```text
Energies [model energy units]: [0.2    0.8193 1.5306]
```

Output 2 (cell 8):

```text
Scaled Energies [scaled model energy units]: [-1.      0.0691  1.    ]
```

Output 3 (cell 10):

```text
Operator Error: 1.852552687200035e-11
```

Output 4 (cell 12):

```text
Initial Ground Weight [probability]: 0.018745945993631295
Cooled Ground Weight [probability]: 0.9930808704605139
Exact Ground Weight [probability]: 0.9930808704691421
```

### `04_heat_equation_pde.ipynb`

Source: [`notebooks/real_examples/04_heat_equation_pde.ipynb`](../../notebooks/real_examples/04_heat_equation_pde.ipynb)

```{image} ../../results/plots/real_examples/04_heat_equation_pde-plot-01.png
:alt: Solving a Real Physics PDE: The 1D Heat Equation plot 1
:width: 760px
```

```{image} ../../results/plots/real_examples/04_heat_equation_pde-plot-02.png
:alt: Solving a Real Physics PDE: The 1D Heat Equation plot 2
:width: 520px
```

```{image} ../../results/plots/real_examples/04_heat_equation_pde-plot-03.png
:alt: Solving a Real Physics PDE: The 1D Heat Equation plot 3
:width: 760px
```

Output 1 (cell 6):

```text
First 5 Eigenvalues [inverse grid-length units]: [  9.8566  39.271   87.7794 154.6166 238.7288]
Last Eigenvalue [inverse grid-length units]: 2490.1433766430973
```

Output 2 (cell 12):

```text
First Eigenvalue [model spectral units]: -0.9999999999999999
Last Eigenvalue [model spectral units]: 1.0000000000000002
Beta: 2.232258077957575
Prefactor: 0.9824145391823961
```

Output 3 (cell 14):

```text
Relative Error: 1.2039058554241987e-15
```

### `05_poisson_equation_pde.ipynb`

Source: [`notebooks/real_examples/05_poisson_equation_pde.ipynb`](../../notebooks/real_examples/05_poisson_equation_pde.ipynb)

```{image} ../../results/plots/real_examples/05_poisson_equation_pde-plot-01.png
:alt: Solving Real Physics PDEs: 1D and 2D Poisson Equations plot 1
:width: 760px
```

```{image} ../../results/plots/real_examples/05_poisson_equation_pde-plot-02.png
:alt: Solving Real Physics PDEs: 1D and 2D Poisson Equations plot 2
:width: 760px
```

```{image} ../../results/plots/real_examples/05_poisson_equation_pde-plot-03.png
:alt: Solving Real Physics PDEs: 1D and 2D Poisson Equations plot 3
:width: 760px
```

Output 1 (cell 6):

```text
First Eigenvalue [inverse grid-length units]: 9.705050945562935
Last Eigenvalue [inverse grid-length units]: 186.29494905443704
```

Output 2 (cell 10):

```text
Gamma: 0.05209508360168709
Condition Number: 19.1956693580892
```

Output 3 (cell 12):

```text
Relative Error: 0.014150625457763643
```

Output 4 (cell 15):

```text
Gamma: 0.05209508360168687
Condition Number: 19.195669358089283
Matrix Shape [rows, cols]: (36, 36)
```

Output 5 (cell 17):

```text
Relative Error: 0.034962641317254864
```

Output 6 (cell 21):

```text
1D condition_number: 19.196
1D relative_error: 1.415e-02
2D condition_number: 19.196
2D relative_error: 3.496e-02
validation: passed
```

### `06_hamiltonian_simulation_schrodinger_dynamics.ipynb`

Source: [`notebooks/real_examples/06_hamiltonian_simulation_schrodinger_dynamics.ipynb`](../../notebooks/real_examples/06_hamiltonian_simulation_schrodinger_dynamics.ipynb)

```{image} ../../results/plots/real_examples/06_hamiltonian_simulation_schrodinger_dynamics-plot-01.png
:alt: Hamiltonian Simulation: Real-Time Dynamics plot 1
:width: 760px
```

Output 1 (cell 5):

```text
State Error: 3.324978121461713e-13
```

### `07_quantum_chemistry_h2_toy_solver.ipynb`

Source: [`notebooks/real_examples/07_quantum_chemistry_h2_toy_solver.ipynb`](../../notebooks/real_examples/07_quantum_chemistry_h2_toy_solver.ipynb)

```{image} ../../results/plots/real_examples/07_quantum_chemistry_h2_toy_solver-plot-01.png
:alt: Quantum Chemistry: Toy H2 Hamiltonian Solver plot 1
:width: 520px
```

```{image} ../../results/plots/real_examples/07_quantum_chemistry_h2_toy_solver-plot-02.png
:alt: Quantum Chemistry: Toy H2 Hamiltonian Solver plot 2
:width: 520px
```

Output 1 (cell 4):

```text
Ground Energy [hartree]: -1.85727503020238
```

Output 2 (cell 7):

```text
Initial Overlap [probability]: 3.4911941432755493e-35
Filtered Overlap [probability]: 6.719832005838039e-33
Energy Estimate [hartree]: -0.9424088608987823
```

### `08_greens_function_response.ipynb`

Source: [`notebooks/real_examples/08_greens_function_response.ipynb`](../../notebooks/real_examples/08_greens_function_response.ipynb)

```{image} ../../results/plots/real_examples/08_greens_function_response-plot-01.png
:alt: Green's Functions: Resolvent Response Solver plot 1
:width: 520px
```

Output 1 (cell 4):

```text
Response Error: 0.001236443161688136
```

### `09_spectral_density_estimation.ipynb`

Source: [`notebooks/real_examples/09_spectral_density_estimation.ipynb`](../../notebooks/real_examples/09_spectral_density_estimation.ipynb)

```{image} ../../results/plots/real_examples/09_spectral_density_estimation-plot-01.png
:alt: Spectral Density Estimation plot 1
:width: 520px
```

Output 1 (cell 4):

```text
Curve Error: 0.0019122824182919095
```

### `10_gibbs_state_thermal_weights.ipynb`

Source: [`notebooks/real_examples/10_gibbs_state_thermal_weights.ipynb`](../../notebooks/real_examples/10_gibbs_state_thermal_weights.ipynb)

```{image} ../../results/plots/real_examples/10_gibbs_state_thermal_weights-plot-01.png
:alt: Gibbs States: Thermal Weight Solver plot 1
:width: 520px
```

Output 1 (cell 4):

```text
Rho Error: 1.8102065129092896e-13
```

### `11_transport_physics_landauer_chain.ipynb`

Source: [`notebooks/real_examples/11_transport_physics_landauer_chain.ipynb`](../../notebooks/real_examples/11_transport_physics_landauer_chain.ipynb)

```{image} ../../results/plots/real_examples/11_transport_physics_landauer_chain-plot-01.png
:alt: Transport Physics: Landauer-Style Chain Transmission plot 1
:width: 760px
```

```{image} ../../results/plots/real_examples/11_transport_physics_landauer_chain-plot-02.png
:alt: Transport Physics: Landauer-Style Chain Transmission plot 2
:width: 520px
```

Output 1 (cell 4):

```text
Curve Error: 0.00625639231073604
```

### `12_tensor_network_hybrid_filtering.ipynb`

Source: [`notebooks/real_examples/12_tensor_network_hybrid_filtering.ipynb`](../../notebooks/real_examples/12_tensor_network_hybrid_filtering.ipynb)

```{image} ../../results/plots/real_examples/12_tensor_network_hybrid_filtering-plot-01.png
:alt: Tensor-Network Hybrid: Product-State Energy Filtering plot 1
:width: 760px
```

Output 1 (cell 4):

```text
Initial Energy [model energy units]: -1.6606649134095948
Filtered Energy [model energy units]: -2.3777913739261485
Initial Overlap [probability]: 0.651393401185215
Filtered Overlap [probability]: 0.8772155174329324
```

### `13_heat_equation_2d_pde.ipynb`

Source: [`notebooks/real_examples/13_heat_equation_2d_pde.ipynb`](../../notebooks/real_examples/13_heat_equation_2d_pde.ipynb)

```{image} ../../results/plots/real_examples/13_heat_equation_2d_pde-plot-01.png
:alt: Solving a Real Physics PDE: The 2D Heat Equation plot 1
:width: 520px
```

```{image} ../../results/plots/real_examples/13_heat_equation_2d_pde-plot-02.png
:alt: Solving a Real Physics PDE: The 2D Heat Equation plot 2
:width: 520px
```

```{image} ../../results/plots/real_examples/13_heat_equation_2d_pde-plot-03.png
:alt: Solving a Real Physics PDE: The 2D Heat Equation plot 3
:width: 760px
```

```{image} ../../results/plots/real_examples/13_heat_equation_2d_pde-plot-04.png
:alt: Solving a Real Physics PDE: The 2D Heat Equation plot 4
:width: 520px
```

Output 1 (cell 6):

```text
First 5 Eigenvalues [inverse grid-length units]: [19.6054 48.2193 48.2193 76.8333 93.3264]
Last Eigenvalue [inverse grid-length units]: 948.3945992294168
Matrix Shape [rows, cols]: (100, 100)
```

Output 2 (cell 12):

```text
First Eigenvalue [model spectral units]: -1.0000000000000009
Last Eigenvalue [model spectral units]: 1.000000000000001
Beta: 0.27863675953765005
Prefactor: 0.9883056759592374
```

Output 3 (cell 14):

```text
Relative Error: 4.054746215837559e-13
```

Output 4 (cell 18):

```text
Maximum Approximation Error: 4.127809205556332e-13
```

Output 5 (cell 20):

```text
relative_error: 4.055e-13
max_abs_difference [field units]: 3.149e-13
validation: passed
```

### `14_advection_diffusion_pde.ipynb`

Source: [`notebooks/real_examples/14_advection_diffusion_pde.ipynb`](../../notebooks/real_examples/14_advection_diffusion_pde.ipynb)

```{image} ../../results/plots/real_examples/14_advection_diffusion_pde-plot-01.png
:alt: Advection-Diffusion as a Non-Hermitian PDE Example plot 1
:width: 760px
```

```{image} ../../results/plots/real_examples/14_advection_diffusion_pde-plot-02.png
:alt: Advection-Diffusion as a Non-Hermitian PDE Example plot 2
:width: 520px
```

Output 1 (cell 5):

```text
Non-normality [operator-norm units]: 24.743080487279663
```

Output 2 (cell 9):

```text
Gamma: 0.057449326656195406
1.0 / Gamma: 17.406644397844456
Relative Error: 0.22813071877482616
```

Output 3 (cell 12):

```text
non_normality [operator-norm units]: 2.474e+01
normal_equation_condition_number: 17.407
relative_error: 2.281e-01
validation: passed
```

### `15_wave_equation_dynamics.ipynb`

Source: [`notebooks/real_examples/15_wave_equation_dynamics.ipynb`](../../notebooks/real_examples/15_wave_equation_dynamics.ipynb)

```{image} ../../results/plots/real_examples/15_wave_equation_dynamics-plot-01.png
:alt: Wave Equation Dynamics from Spectral Polynomials plot 1
:width: 760px
```

```{image} ../../results/plots/real_examples/15_wave_equation_dynamics-plot-02.png
:alt: Wave Equation Dynamics from Spectral Polynomials plot 2
:width: 760px
```

Output 1 (cell 5):

```text
First Eigenvalue [model spectral units]: -0.9828268973302636
Last Eigenvalue [model spectral units]: 0.9999999999999998
Beta: 4.265725211247927
```

Output 2 (cell 9):

```text
Relative Error: 2.942102758155022e-15
```

Output 3 (cell 12):

```text
beta: 4.266
relative_error: 2.942e-15
validation: passed
```

### `16_helmholtz_equation_pde.ipynb`

Source: [`notebooks/real_examples/16_helmholtz_equation_pde.ipynb`](../../notebooks/real_examples/16_helmholtz_equation_pde.ipynb)

```{image} ../../results/plots/real_examples/16_helmholtz_equation_pde-plot-01.png
:alt: Helmholtz Equation with an Indefinite Spectrum plot 1
:width: 760px
```

```{image} ../../results/plots/real_examples/16_helmholtz_equation_pde-plot-02.png
:alt: Helmholtz Equation with an Indefinite Spectrum plot 2
:width: 760px
```

Output 1 (cell 5):

```text
First 4 Eigenvalues [inverse grid-length units]: [-51.1675 -22.5535  22.5535  80.4994]
Last Eigenvalue [inverse grid-length units]: 413.22712589466073
Gap [model energy units]: 0.05457901297346752
```

Output 2 (cell 9):

```text
Relative Error: 0.16873834910647584
Maximum Fit Error: 0.3658574904383274
```

Output 3 (cell 12):

```text
spectral_gap [model energy units]: 5.458e-02
max_fit_error: 3.659e-01
relative_error: 1.687e-01
validation: passed
```

### `17_quantum_walk_search_toy.ipynb`

Source: [`notebooks/real_examples/17_quantum_walk_search_toy.ipynb`](../../notebooks/real_examples/17_quantum_walk_search_toy.ipynb)

```{image} ../../results/plots/real_examples/17_quantum_walk_search_toy-plot-01.png
:alt: Quantum Walk Search Toy Model plot 1
:width: 520px
```

```{image} ../../results/plots/real_examples/17_quantum_walk_search_toy-plot-02.png
:alt: Quantum Walk Search Toy Model plot 2
:width: 760px
```

Output 1 (cell 5):

```text
Eigenvalues [oracle energy units]: [-1.2286 -0.5214  0.125   0.125   0.125   0.125   0.125   0.125 ]
Eigenvalues of A: [-1.  1.]
```

Output 2 (cell 9):

```text
Best Time [inverse energy units]: 4.452830188679245
Best Probability [probability]: 0.9999891776297675
```

Output 3 (cell 11):

```text
State Error: 1.0120998885188465e-13
Polynomial Probability [probability]: 0.9999891776298157
```

Output 4 (cell 14):

```text
best_time [inverse energy units]: 4.453
best_probability [probability]: 0.999989
poly_probability [probability]: 0.999989
state_error: 1.012e-13
validation: passed
```

### `18_ssh_chain_edge_state_filtering.ipynb`

Source: [`notebooks/real_examples/18_ssh_chain_edge_state_filtering.ipynb`](../../notebooks/real_examples/18_ssh_chain_edge_state_filtering.ipynb)

```{image} ../../results/plots/real_examples/18_ssh_chain_edge_state_filtering-plot-01.png
:alt: SSH Chain Edge-State Filtering plot 1
:width: 760px
```

```{image} ../../results/plots/real_examples/18_ssh_chain_edge_state_filtering-plot-02.png
:alt: SSH Chain Edge-State Filtering plot 2
:width: 760px
```

Output 1 (cell 6):

```text
Near-Zero Eigenvalues [hopping units]: [-0.5577 -0.0058  0.0058  0.5577]
```

Output 2 (cell 10):

```text
Edge Weight Fraction [probability]: 0.6880404379008599
Trace of Soft Edge Projector [states]: 2.0327822807342835
```

Output 3 (cell 13):

```text
near_zero_eigenvalues [hopping units]: [-0.55766 -0.00584  0.00584  0.55766]
edge_weight_fraction [probability]: 0.688
projector_trace [states]: 2.033
validation: passed
```

### `19_anderson_localization.ipynb`

Source: [`notebooks/real_examples/19_anderson_localization.ipynb`](../../notebooks/real_examples/19_anderson_localization.ipynb)

```{image} ../../results/plots/real_examples/19_anderson_localization-plot-01.png
:alt: Anderson Localization in a Disordered Tight-Binding Chain plot 1
:width: 760px
```

Output 1 (cell 5):

```text
Localized IPR: 0.6321472059662562
Maximum Clean IPR: 0.045454545454545796
```

Output 2 (cell 7):

```text
Peak Site [site index]: 14
Filtered Weight at Peak Site [probability]: 0.10072442254521352
```

Output 3 (cell 10):

```text
localized_energy [hopping units]: 1.6901
localized_ipr: 0.6321
clean_max_ipr: 0.0455
peak_site_filter_weight [probability]: 0.1007
validation: passed
```

### `20_schrodinger_bound_states.ipynb`

Source: [`notebooks/real_examples/20_schrodinger_bound_states.ipynb`](../../notebooks/real_examples/20_schrodinger_bound_states.ipynb)

```{image} ../../results/plots/real_examples/20_schrodinger_bound_states-plot-01.png
:alt: Schrödinger Bound States in a Potential Well plot 1
:width: 760px
```

```{image} ../../results/plots/real_examples/20_schrodinger_bound_states-plot-02.png
:alt: Schrödinger Bound States in a Potential Well plot 2
:width: 760px
```

Output 1 (cell 5):

```text
Eigenvalues [model energy units]: [-6.7968 -4.5635 -2.6903 -1.2181 -0.2361  0.1906]
Number of Bound States [states]: 5
```

Output 2 (cell 9):

```text
Ground State Overlap [probability]: 0.9505375069652005
Filtered Energy [model energy units]: -6.586964339925885
```

Output 3 (cell 12):

```text
lowest_energies [model energy units]: [-6.7968 -4.5635 -2.6903 -1.2181 -0.2361  0.1906]
n_bound [states]: 5
ground_overlap [probability]: 0.9505
filtered_energy [model energy units]: -6.5870
validation: passed
```

### `21_quantum_harmonic_oscillator_grid.ipynb`

Source: [`notebooks/real_examples/21_quantum_harmonic_oscillator_grid.ipynb`](../../notebooks/real_examples/21_quantum_harmonic_oscillator_grid.ipynb)

```{image} ../../results/plots/real_examples/21_quantum_harmonic_oscillator_grid-plot-01.png
:alt: Quantum Harmonic Oscillator on a Grid plot 1
:width: 760px
```

Output 1 (cell 5):

```text
Eigenvalues [model energy units]: [0.4991 1.4953 2.4878 3.4765 4.4614 5.4424]
Spectrum Error [model energy units]: 0.02350013331927281
```

Output 2 (cell 6):

```text
Ground State Overlap [probability]: 0.9969945579767748
```

Output 3 (cell 9):

```text
finite_difference_energies [model energy units]: [0.4991 1.4953 2.4878 3.4765 4.4614 5.4424]
analytic_energies [model energy units]: [0.5 1.5 2.5 3.5 4.5 5.5]
spectrum_error_first_four [model energy units]: 2.3500e-02
ground_overlap [probability]: 0.9970
validation: passed
```

### `22_electrostatic_green_function_poisson.ipynb`

Source: [`notebooks/real_examples/22_electrostatic_green_function_poisson.ipynb`](../../notebooks/real_examples/22_electrostatic_green_function_poisson.ipynb)

```{image} ../../results/plots/real_examples/22_electrostatic_green_function_poisson-plot-01.png
:alt: Electrostatic Green's Function from a 2D Poisson Solve plot 1
:width: 760px
```

Output 1 (cell 5):

```text
Potential at Positive Charge [potential units]: 0.0037023041787180174
Potential at Negative Charge [potential units]: -0.0037023041787180165
Sum of Charges [charge units]: 0.0
```

Output 2 (cell 7):

```text
Relative Error: 0.2679780308568579
Selected Degree [polynomial degree]: 29
Condition Number: 48.37415007870855
```

Output 3 (cell 10):

```text
condition_number: 48.374
positive_charge_potential [potential units]: 3.7023e-03
negative_charge_potential [potential units]: -3.7023e-03
selected_degree [polynomial degree]: 29
relative_error: 2.680e-01
validation: passed
```

### `23_coupled_oscillator_normal_modes.ipynb`

Source: [`notebooks/real_examples/23_coupled_oscillator_normal_modes.ipynb`](../../notebooks/real_examples/23_coupled_oscillator_normal_modes.ipynb)

```{image} ../../results/plots/real_examples/23_coupled_oscillator_normal_modes-plot-01.png
:alt: Coupled Oscillator Normal Modes plot 1
:width: 760px
```

```{image} ../../results/plots/real_examples/23_coupled_oscillator_normal_modes-plot-02.png
:alt: Coupled Oscillator Normal Modes plot 2
:width: 760px
```

Output 1 (cell 5):

```text
Frequencies [angular frequency units]: [0.2072 0.4426 0.7012 0.9629 1.2121]
```

Output 2 (cell 9):

```text
Slow Mode Overlap [probability]: 0.9147264843650682
```

Output 3 (cell 12):

```text
frequencies [angular frequency units]: [0.2072 0.4426 0.7012 0.9629 1.2121]
lowest_stiffness [angular frequency squared units]: 4.2916e-02
slow_mode_overlap [probability]: 0.9147
validation: passed
```

### `24_ising_phase_transition_filtering.ipynb`

Source: [`notebooks/real_examples/24_ising_phase_transition_filtering.ipynb`](../../notebooks/real_examples/24_ising_phase_transition_filtering.ipynb)

```{image} ../../results/plots/real_examples/24_ising_phase_transition_filtering-plot-01.png
:alt: Transverse-Field Ising Phase Transition Filtering plot 1
:width: 760px
```

```{image} ../../results/plots/real_examples/24_ising_phase_transition_filtering-plot-02.png
:alt: Transverse-Field Ising Phase Transition Filtering plot 2
:width: 760px
```

Output 1 (cell 5):

```text
Ising sweep diagnostics
-----------------------
Minimum gap field [coupling ratio h/J]       : 0.6
Minimum gap [coupling units]             : 1.394
Maximum magnetization field [coupling ratio h/J] : 0.2
Maximum magnetization [magnetization squared]       : 0.9818
```

Output 2 (cell 9):

```text
Projector eigenweights [probability]: [0.7761 0.2239 0.0087 0.1109 0.1143]
Projector error: 0.5286773050862152
```

Output 3 (cell 11):

```text
minimum_doublet_gap_field [coupling ratio h/J]: 0.600
magnetization_drop [magnetization squared]: 0.653
projector_error: 0.529
validation: passed
```

### `25_diffusion_heat_treatment_slab.ipynb`

Source: [`notebooks/real_examples/25_diffusion_heat_treatment_slab.ipynb`](../../notebooks/real_examples/25_diffusion_heat_treatment_slab.ipynb)

```{image} ../../results/plots/real_examples/25_diffusion_heat_treatment_slab-plot-01.png
:alt: Diffusion-Limited Heat Treatment in a Slab plot 1
:width: 760px
```

```{image} ../../results/plots/real_examples/25_diffusion_heat_treatment_slab-plot-02.png
:alt: Diffusion-Limited Heat Treatment in a Slab plot 2
:width: 760px
```

Output 1 (cell 5):

```text
Relative Error: 2.8910082648989737e-15
```

Output 2 (cell 9):

```text
relative_temperature_error: 0.0000
initial_norm [temperature units]: 2.7653
cooled_norm [temperature units]: 2.7152
validation: passed
```

### `26_graphene_nanoribbon_density_of_states.ipynb`

Source: [`notebooks/real_examples/26_graphene_nanoribbon_density_of_states.ipynb`](../../notebooks/real_examples/26_graphene_nanoribbon_density_of_states.ipynb)

```{image} ../../results/plots/real_examples/26_graphene_nanoribbon_density_of_states-plot-01.png
:alt: Graphene Nanoribbon Density of States plot 1
:width: 520px
```

```{image} ../../results/plots/real_examples/26_graphene_nanoribbon_density_of_states-plot-02.png
:alt: Graphene Nanoribbon Density of States plot 2
:width: 760px
```

Output 1 (cell 5):

```text
Edge Fraction [probability]: 0.6643744409979471
```

Output 2 (cell 9):

```text
near_zero_window_weight: 5.032
edge_fraction_of_near_zero_ldos [probability]: 0.664
validation: passed
```

### `27_fermi_dirac_electronic_occupations.ipynb`

Source: [`notebooks/real_examples/27_fermi_dirac_electronic_occupations.ipynb`](../../notebooks/real_examples/27_fermi_dirac_electronic_occupations.ipynb)

```{image} ../../results/plots/real_examples/27_fermi_dirac_electronic_occupations-plot-01.png
:alt: Fermi-Dirac Electronic Occupations plot 1
:width: 760px
```

Output 1 (cell 5):

```text
Occupation Error: 7.804084211610466e-05
Exact Particle Number [electrons]: 9.385723227081577
Polynomial Particle Number [electrons]: 9.385329756322626
```

Output 2 (cell 7):

```text
relative_density_matrix_error: 0.0001
exact_particle_number [electrons]: 9.386
polynomial_particle_number [electrons]: 9.385
validation: passed
```

### `28_photonic_crystal_band_gap_filtering.ipynb`

Source: [`notebooks/real_examples/28_photonic_crystal_band_gap_filtering.ipynb`](../../notebooks/real_examples/28_photonic_crystal_band_gap_filtering.ipynb)

```{image} ../../results/plots/real_examples/28_photonic_crystal_band_gap_filtering-plot-01.png
:alt: Photonic Crystal Band-Gap Filtering plot 1
:width: 760px
```

```{image} ../../results/plots/real_examples/28_photonic_crystal_band_gap_filtering-plot-02.png
:alt: Photonic Crystal Band-Gap Filtering plot 2
:width: 760px
```

```{image} ../../results/plots/real_examples/28_photonic_crystal_band_gap_filtering-plot-03.png
:alt: Photonic Crystal Band-Gap Filtering plot 3
:width: 760px
```

Output 1 (cell 5):

```text
Gap Size [model frequency units]: 500.45893808166556
Target Energy [model frequency units]: 2718.4282533951346
Maximum Mode Weight: 0.8135833929662374
```

Output 2 (cell 10):

```text
selected_gap_index [index]: 18
gap_size [model frequency units]: 500.459
max_window_weight: 0.814
validation: passed
```

### `29_topological_band_projector_chern_marker.ipynb`

Source: [`notebooks/real_examples/29_topological_band_projector_chern_marker.ipynb`](../../notebooks/real_examples/29_topological_band_projector_chern_marker.ipynb)

```{image} ../../results/plots/real_examples/29_topological_band_projector_chern_marker-plot-01.png
:alt: Topological Band Projector and Chern Marker plot 1
:width: 520px
```

```{image} ../../results/plots/real_examples/29_topological_band_projector_chern_marker-plot-02.png
:alt: Topological Band Projector and Chern Marker plot 2
:width: 760px
```

Output 1 (cell 6):

```text
Dimension [states]: 50
Spectral range [model energy units]: (-2.800243765865765, 2.8002437658657637)
Gap around zero [model energy units]: 0.13742301418061872
```

Output 2 (cell 10):

```text
Scaled gap: 0.04907537545686846
Projector relative error: 0.10636398990180881
```

Output 3 (cell 12):

```text
Marker relative error: 0.45526864295151465
Bulk exact marker [Chern marker]: 0.9583268200049554
Bulk polynomial marker [Chern marker]: 0.6430191290198954
```

### `30_block_encoded_laplacian_smoothing.ipynb`

Source: [`notebooks/real_examples/30_block_encoded_laplacian_smoothing.ipynb`](../../notebooks/real_examples/30_block_encoded_laplacian_smoothing.ipynb)

```{image} ../../results/plots/real_examples/30_block_encoded_laplacian_smoothing-plot-01.png
:alt: Block-Encoded QSVT Laplacian Smoothing plot 1
:width: 760px
```

```{image} ../../results/plots/real_examples/30_block_encoded_laplacian_smoothing-plot-02.png
:alt: Block-Encoded QSVT Laplacian Smoothing plot 2
:width: 760px
```

Output 1 (cell 6):

```text
Block-Encoding Alpha: 392.788
Logical Dimension: 8
Unitary Dimension: 16
Block Error: 0.000e+00
Unitarity Error: 3.819e-15
Operator Relative Error: 1.001e-12
State Relative Error: 1.001e-12
```

Output 2 (cell 8):

```text
QNode Execution Kind: pennylane-qnode-statevector-qsvt-execution
QNode Gate Types: {'StatePrep': 1, 'QSVT': 1}
QNode Logical Success Probability: 0.959851950246
QNode Real Logical Error: 9.746e-13
QNode Max Imaginary Logical Amplitude: 4.517e-02
```

Output 3 (cell 14):

```text
validation: passed
```