Real-Example Results

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Notebook Results

01_ground_state_filtering.ipynb

Source: notebooks/real_examples/01_ground_state_filtering.ipynb

Ground-State Filtering for a Toy Hamiltonian plot 1 Ground-State Filtering for a Toy Hamiltonian plot 2

Output 1 (cell 6):

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

Output 2 (cell 10):

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):

Projector Error: 0.13103683490762716
Idempotence Error: 0.11947057773046423

Output 4 (cell 14):

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

Band Filtering in a Tight-Binding Chain plot 1 Band Filtering in a Tight-Binding Chain plot 2

Output 1 (cell 6):

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

Output 2 (cell 10):

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

Imaginary-Time Filtering with an Exponential Polynomial plot 1

Output 1 (cell 6):

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

Output 2 (cell 8):

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

Output 3 (cell 10):

Operator Error: 1.852552687200035e-11

Output 4 (cell 12):

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

Solving a Real Physics PDE: The 1D Heat Equation plot 1 Solving a Real Physics PDE: The 1D Heat Equation plot 2 Solving a Real Physics PDE: The 1D Heat Equation plot 3

Output 1 (cell 6):

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):

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

Output 3 (cell 14):

Relative Error: 1.2039058554241987e-15

05_poisson_equation_pde.ipynb

Source: notebooks/real_examples/05_poisson_equation_pde.ipynb

Solving Real Physics PDEs: 1D and 2D Poisson Equations plot 1 Solving Real Physics PDEs: 1D and 2D Poisson Equations plot 2 Solving Real Physics PDEs: 1D and 2D Poisson Equations plot 3

Output 1 (cell 6):

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

Output 2 (cell 10):

Gamma: 0.05209508360168709
Condition Number: 19.1956693580892

Output 3 (cell 12):

Relative Error: 0.014150625457763643

Output 4 (cell 15):

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

Output 5 (cell 17):

Relative Error: 0.034962641317254864

Output 6 (cell 21):

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

Hamiltonian Simulation: Real-Time Dynamics plot 1

Output 1 (cell 5):

State Error: 3.324978121461713e-13

07_quantum_chemistry_h2_toy_solver.ipynb

Source: notebooks/real_examples/07_quantum_chemistry_h2_toy_solver.ipynb

Quantum Chemistry: Toy H2 Hamiltonian Solver plot 1 Quantum Chemistry: Toy H2 Hamiltonian Solver plot 2

Output 1 (cell 4):

Ground Energy [hartree]: -1.85727503020238

Output 2 (cell 7):

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

Green's Functions: Resolvent Response Solver plot 1

Output 1 (cell 4):

Response Error: 0.001236443161688136

09_spectral_density_estimation.ipynb

Source: notebooks/real_examples/09_spectral_density_estimation.ipynb

Spectral Density Estimation plot 1

Output 1 (cell 4):

Curve Error: 0.0019122824182919095

10_gibbs_state_thermal_weights.ipynb

Source: notebooks/real_examples/10_gibbs_state_thermal_weights.ipynb

Gibbs States: Thermal Weight Solver plot 1

Output 1 (cell 4):

Rho Error: 1.8102065129092896e-13

11_transport_physics_landauer_chain.ipynb

Source: notebooks/real_examples/11_transport_physics_landauer_chain.ipynb

Transport Physics: Landauer-Style Chain Transmission plot 1 Transport Physics: Landauer-Style Chain Transmission plot 2

Output 1 (cell 4):

Curve Error: 0.00625639231073604

12_tensor_network_hybrid_filtering.ipynb

Source: notebooks/real_examples/12_tensor_network_hybrid_filtering.ipynb

Tensor-Network Hybrid: Product-State Energy Filtering plot 1

Output 1 (cell 4):

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

Solving a Real Physics PDE: The 2D Heat Equation plot 1 Solving a Real Physics PDE: The 2D Heat Equation plot 2 Solving a Real Physics PDE: The 2D Heat Equation plot 3 Solving a Real Physics PDE: The 2D Heat Equation plot 4

Output 1 (cell 6):

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):

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

Output 3 (cell 14):

Relative Error: 4.054746215837559e-13

Output 4 (cell 18):

Maximum Approximation Error: 4.127809205556332e-13

Output 5 (cell 20):

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

Advection-Diffusion as a Non-Hermitian PDE Example plot 1 Advection-Diffusion as a Non-Hermitian PDE Example plot 2

Output 1 (cell 5):

Non-normality [operator-norm units]: 24.743080487279663

Output 2 (cell 9):

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

Output 3 (cell 12):

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

Wave Equation Dynamics from Spectral Polynomials plot 1 Wave Equation Dynamics from Spectral Polynomials plot 2

Output 1 (cell 5):

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

Output 2 (cell 9):

Relative Error: 2.942102758155022e-15

Output 3 (cell 12):

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

16_helmholtz_equation_pde.ipynb

Source: notebooks/real_examples/16_helmholtz_equation_pde.ipynb

Helmholtz Equation with an Indefinite Spectrum plot 1 Helmholtz Equation with an Indefinite Spectrum plot 2

Output 1 (cell 5):

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):

Relative Error: 0.16873834910647584
Maximum Fit Error: 0.3658574904383274

Output 3 (cell 12):

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

Quantum Walk Search Toy Model plot 1 Quantum Walk Search Toy Model plot 2

Output 1 (cell 5):

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):

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

Output 3 (cell 11):

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

Output 4 (cell 14):

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

SSH Chain Edge-State Filtering plot 1 SSH Chain Edge-State Filtering plot 2

Output 1 (cell 6):

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

Output 2 (cell 10):

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

Output 3 (cell 13):

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

Anderson Localization in a Disordered Tight-Binding Chain plot 1

Output 1 (cell 5):

Localized IPR: 0.6321472059662562
Maximum Clean IPR: 0.045454545454545796

Output 2 (cell 7):

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

Output 3 (cell 10):

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

Schrödinger Bound States in a Potential Well plot 1 Schrödinger Bound States in a Potential Well plot 2

Output 1 (cell 5):

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):

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

Output 3 (cell 12):

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

Quantum Harmonic Oscillator on a Grid plot 1

Output 1 (cell 5):

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):

Ground State Overlap [probability]: 0.9969945579767748

Output 3 (cell 9):

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

Electrostatic Green's Function from a 2D Poisson Solve plot 1

Output 1 (cell 5):

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):

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

Output 3 (cell 10):

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

Coupled Oscillator Normal Modes plot 1 Coupled Oscillator Normal Modes plot 2

Output 1 (cell 5):

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

Output 2 (cell 9):

Slow Mode Overlap [probability]: 0.9147264843650682

Output 3 (cell 12):

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

Transverse-Field Ising Phase Transition Filtering plot 1 Transverse-Field Ising Phase Transition Filtering plot 2

Output 1 (cell 5):

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):

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

Output 3 (cell 11):

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

Diffusion-Limited Heat Treatment in a Slab plot 1 Diffusion-Limited Heat Treatment in a Slab plot 2

Output 1 (cell 5):

Relative Error: 2.8910082648989737e-15

Output 2 (cell 9):

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

Graphene Nanoribbon Density of States plot 1 Graphene Nanoribbon Density of States plot 2

Output 1 (cell 5):

Edge Fraction [probability]: 0.6643744409979471

Output 2 (cell 9):

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

Fermi-Dirac Electronic Occupations plot 1

Output 1 (cell 5):

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

Output 2 (cell 7):

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

Photonic Crystal Band-Gap Filtering plot 1 Photonic Crystal Band-Gap Filtering plot 2 Photonic Crystal Band-Gap Filtering plot 3

Output 1 (cell 5):

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

Output 2 (cell 10):

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

Topological Band Projector and Chern Marker plot 1 Topological Band Projector and Chern Marker plot 2

Output 1 (cell 6):

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

Output 2 (cell 10):

Scaled gap: 0.04907537545686846
Projector relative error: 0.10636398990180881

Output 3 (cell 12):

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

Block-Encoded QSVT Laplacian Smoothing plot 1 Block-Encoded QSVT Laplacian Smoothing plot 2

Output 1 (cell 6):

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):

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):

validation: passed