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Project 3: Self-Consistent Discrete Hamiltonian Inference (SCF-Inspired Low-Fidelity PINN)🪶

This project is still in the initial phases of development.

Goal

Explore whether self-consistent field structure and weak physics contraints can recover meaningful Hamiltonian representations from noisy quantum observables.

Questions That Need Answering

🧙‍♂️ Q1: Can we infer the parameters of an unknown quantum Hamiltonian from noisy observables using a self-consistent field (SCF)-inspired learning loop where the model alternates between:

  • Solving for quantum states given a Hamiltonian.
  • Updating the Hamiltonian from observed/estimated structure.

This mirrors the fixed-point structure of Hartree-Fock / density functional theory (DFT) methods used in electronic structure theory (e.g., implementations such as PySCF).

🧙‍♂️ Q2: Can the neural network discover the parameterized generator of unitary flow that preserves symplectic structure?

Relation to Other Low-Fidelity PIML Projects

Project Description
Project 1 forward variational PINN (hamonic oscillator)
Project 2 inverse Schrödinger equation (semi-supervised learning)
Project 3 inverse Hamiltonian inference with SCF and geometric regularization

Model System: Discrete Qubit Hamiltonian

We consider a small transverse Ising-like Hamiltonian:

\[ H^\theta = \sum_{i}{h_i\sigma_i^2} + \sum_{i<j}{J_{ij}\sigma_i^2\sigma_j^2} + \sum_{i}{g_i\sigma_i^2} \]

Unkown Parameters:

\[\theta = \{h_i, J_{ij}, g_i\}\]

Observables

We train on noisy estimates of:

  • energy levels
  • expectation values \(\langle \sigma_i(t)\rangle\)
  • correlation functions
  • short-time dynamics

SCF-Inspired Training Loop

Each iteration consists of:

  1. Forward solve \(\implies\) diagonalization or time evolution of \(H^\theta\)
  2. Observation model \(\implies\) compute noisy measurement statistics
  3. Hamiltonian update \(\implies\) update parameters using mismatch between predicted and observed structure
  4. SCF regularization \(\implies\) encourages convergence twoard a fixed Hamiltonian

Loss Function

\[\mathcal{L}_\text{total} = \mathcal{L}_\text{data} + \lambda_1 \mathcal{L}_\text{phys} + \lambda_2 \mathcal{L}_\text{SCF} + \lambda_3 \mathcal{L}_\text{symplectic}\]

where

  • Data loss: mismatch in observables
  • Physics loss: Schrödinger consistency
  • SCF loss: stabilization of Hamiltonian updates
  • Symplectic loss (soft constraint): encourages geometric consistency of phase-space-like trajectories

Connection to PySCF / DFT

Self-consistent field methods in electronic structure theory (e.g., PySCF) solve:

  • Hamiltonian depends on electron density
  • electron density depends on eigenstates of Hamiltonian

This project mirrors that structure, but replaces:

DFT concept This project
electron density learned Hamiltonian parameters
Fock operator neural parameter update
SCF loop fixed-point Hamiltonian inference

POD-Based Diagnostics (Post-Hoc Analysis)

As in Project 2, we apply Proper Orthogonal Decomposition (POD) as a purely diagnostic tool to:

  • the learned eigenstates \(\psi_n^\theta\).
  • SCF iteration trajectories.
  • observable time series.

POD reveals:

  • dominant spectral modes of learned quantum states.
  • low-rank structure of the inferred state manifold.
  • convergence behavior of SCF learning dynamics.

References

  • PySCF documentation (SCF methods in quantum chemistry)
  • Hartree-Fock theory and density functional theory (DFT) literature
  • Physics-informed neural networks (Raissi et al., 2019)
  • Brunton & Kutz (data-driven scientific computing)