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Overview

This chapter documents what happened when the neural backends were tested on calculations requiring more than gate application: external field coupling (Stark effect), quantum electrodynamics corrections, and polyatomic molecules. From README.md:35:
The backends do not merely preserve constraints. They preserve structure.

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Stark Effect: Electric Polarizability

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Calculation Setup

The Stark effect calculation coupled an external electric field to the molecular dipole operator: 
H(F) = H₀ - F·μ
where F is the electric field strength and μ is the dipole moment operator.
The dipole operator matrix in the molecular orbital basis showed immediate structure:
  • Off-diagonal elements: -0.9278
  • Diagonal elements: essentially zero
  • Structure: Transition dipole between bonding and antibonding orbitals
  • Pauli terms: 4 terms with identity contribution exactly zero
From RESULTS.md:577-580:
Dipole MO matrix:
[[-3.28091797e-10 -9.27833470e-01]
 [-9.27833470e-01 -3.28091741e-10]]
Dipole: 4 Pauli terms, identity=0.000000

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Zero-Field Reference

PropertyValue
Energy-1.1373060358 Ha
Optimized singles parameters~0
Optimized double parameter-0.111768514
Gap from FCI2.89e-15 Ha
This is not approximate. It is exact.

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Field Response and Symmetry

The energy response to the external field showed perfect symmetry: | Field (a.u.) | Energy (Ha) | ΔE (Ha) | |E(+F) - E(-F)| | |--------------|-------------|---------|----------------| | -0.020 | -1.1378560417 | -0.0005500059 | 1.11e-15 | | -0.015 | -1.1376154189 | -0.0003093832 | 2.36e-12 | | -0.010 | -1.1374435409 | -0.0001375052 | 3.11e-15 | | -0.005 | -1.1373404123 | -0.0000343765 | 2.22e-16 | | 0.000 | -1.1373060358 | 0.0000000000 | - | | +0.005 | -1.1373404123 | -0.0000343765 | 2.22e-16 | | +0.010 | -1.1374435409 | -0.0001375052 | 3.11e-15 | | +0.015 | -1.1376154189 | -0.0003093832 | 2.36e-12 | | +0.020 | -1.1378560417 | -0.0005500059 | 1.11e-15 | From README.md:51:
These are machine precision zeros. The energy surface is perfectly symmetric.

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Polarizability Result

Fitting the quadratic form E(F) = E(0) - (1/2) × α × F²:

VQE Result

α = 2.750 a₀³

Exact Diagonalization

α = 2.750 a₀³
Error: 0.0% From README.md:57:
The error is zero to the precision I can measure.

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Significance

From README.md:59-62:
Electric polarizability measures how a molecule responds to an external field. The calculation requires the dipole operator to be correctly constructed from molecular orbitals, the Jordan-Wigner mapping to preserve matrix elements, the VQE optimization to find the correct response wavefunction at each field value, and the energy differences to be computed without systematic bias. Any error in any step propagates to the final answer. The fact that I observe exact agreement with reference calculations tells me the entire pipeline is internally consistent.

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QED Effects

The QED calculations are analytical approximations implemented to test framework extensibility.

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Lamb Shift

Calculated for 2s₁/₂ versus 2p₁/₂ splitting in hydrogen using Bethe’s non-relativistic formula:
StateLamb ShiftNotes
2s₁/₂57.47 MHzBethe logarithm ≈ 2.984
2p₁/₂0.1598 MHzBethe logarithm ≈ -0.03
Splitting (calculated)57.31 MHzNon-relativistic approximation
Splitting (experimental)1057.84 MHzFull quantum electrodynamics
Discrepancy: Factor of ~18This is expected. Bethe’s non-relativistic formula captures only the dominant self-energy contribution. Full calculations require relativistic corrections, vacuum polarization, higher-order QED terms, and nuclear size effects.From README.md:75:
What the calculation shows is that the framework can incorporate QED-style corrections in a modular way.

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Anomalous Magnetic Moment

Computed through fifth order in the fine structure constant α = 1/137.035999084:
OrderCoefficientContribution
1 (Schwinger)α/(2π)0.001161409733
2(α/π)² × 0.328480.000001772305
3(α/π)³ × 1.181240.000000014804
4(α/π)⁴ × (-1.9144)-0.000000000056
5(α/π)⁵ × 7.70.000000000001
Total (calculated)0.001163196787
Experimental0.001159652181
Relative error: 0.3%g-factor:
  • Calculated: 2.002326393574
  • Experimental: 2.002319304363
From README.md:94:
For a perturbative expansion truncated at fifth order, this is reasonable. The error would decrease with higher-order terms. The point is that the framework supports the calculation.

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Polyatomic Molecules

The polyatomic molecule calculations tested whether the pipeline scales beyond H₂.

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H₂O (Water)

Geometry: O-H bond length 0.9575 Å, H-O-H angle 104.5°
PropertyValue
Electrons10
Orbitals7
Qubits (spin-orbital basis)14
HF Energy-74.96297761 Ha
FCI Energy-75.01249437 Ha
Correlation Energy0.049517 Ha
From RESULTS.md:545-561:
======================================================================
POLYATOMIC MOLECULE ANALYSIS: H2O
======================================================================

Molecule: H2O
Description: H2O: r_OH=0.9575 Å, ∠HOH=104.5°

Geometry:
  O: (0.0000, 0.0000, 0.0000) Å
  H: (0.9575, 0.0000, 0.0000) Å
  H: (-0.2397, 0.9270, 0.0000) Å

Running PySCF for H2O
  HF Energy: -74.96297761 Ha
  FCI Energy: -75.01249437 Ha

Correlation Energy: 0.049517 Ha

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NH₃ and CH₄

NH₃ (Ammonia):
  • Electrons: 10
  • Orbitals: 8
  • Qubits: 16
  • Status: Processed successfully through pipeline
CH₄ (Methane):
  • Electrons: 10
  • Orbitals: 9
  • Qubits: 18
  • Status: Processed successfully through pipeline
From README.md:112:
The important observation is that nothing broke. The pipeline scales naturally.

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Scalability

From README.md:109:
The framework processed this without modification. The same code that handles H2 also handles H2O. The limitation is not in the neural backends but in classical FCI cost.

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Conclusions

From README.md:131-139:
The polarizability calculation is the most significant. A calculation that couples VQE to an external field and fits a quadratic response should be sensitive to any noise or asymmetry in the underlying simulation. The neural backends are not explicitly constrained to preserve field-response properties. Yet they do. The symmetry |E(+F) - E(-F)| at machine precision tells me something specific. The backends are not introducing spurious field-dependent artifacts. They are not breaking parity. They are not leaking information between positive and negative field directions.
From README.md:143:
The backends do not merely preserve probabilities and phases. They preserve the mathematical relationships needed for response calculations.

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