This roadmap integrates quantum computing, deep learning, and physics-informed modeling, tailored for theoretical physics and computational science researchers.
📘 Phase 1: Mathematical & Physical Foundations¶
✅ Topics¶
- Linear Algebra: Hermitian operators, eigenvalues/eigenvectors
- Differential Equations: ODEs, PDEs (Schrödinger, diffusion, etc.)
- Functional Analysis and Variational Methods
- Quantum Mechanics: Time-dependent and independent Schrödinger Equation
- Numerical Methods: Finite difference, spectral methods
📚 Resources¶
- Mathematics for Physicists – Mary L. Boas
- Introduction to Quantum Mechanics – David J. Griffiths
- MIT OCW: Quantum Physics I
- Numerical Recipes (for PDE solvers)
🧠 Phase 2: Deep Learning & PINNs¶
✅ Topics¶
- Neural Networks (MLPs), activation functions, backpropagation
- Autograd and symbolic differentiation
- PINNs architecture: Loss = Data + PDE residual + Boundary/IC
- Implementation with PyTorch or TensorFlow
📚 Resources¶
- Raissi et al., “Physics-Informed Neural Networks”
- DeepXDE library
- YouTube: PINNs tutorials by Karniadakis Group or SciML
- Neural PDEs tutorial – Martin Krasser
⚛️ Phase 3: Quantum Computing Foundations¶
✅ Topics¶
- Qubits, Bloch sphere, superposition, entanglement
- Quantum gates and circuits
- Variational Quantum Algorithms (VQE, QAOA)
- Parameterized Quantum Circuits (PQCs)
📚 Resources¶
- Quantum Computation and Quantum Information – Nielsen & Chuang
- Qiskit Textbook
- PennyLane Tutorials
- MIT OCW: Quantum Computation
🧩 Phase 4: Hybrid Quantum-Classical PINNs¶
✅ Topics¶
- PQCs for function approximation or eigenvalue estimation
- Classical PINNs + Quantum subcircuits (e.g., solving quantum PDEs)
- Training with hybrid optimizers (e.g., ADAM + SPSA)
- Circuit Differentiation (parameter-shift rule)
📚 Tools¶
📚 Papers¶
- Marrero et al., 2020 — Quantum Neural Network Architectures
- Quantum PINNs – Quantum-assisted learning of quantum PDEs
- Ghosh et al. – Hybrid Quantum-Classical PINNs
🧪 Phase 5: Applications & Projects¶
🧠 Project Ideas¶
- Solve the time-dependent Schrödinger equation with hybrid quantum NN
- Use VQE to find the ground state energy in PINNs setup
- Infer unknown potentials or Hamiltonians from quantum data
📚 Tools¶
- Jupyter Notebook + Qiskit/PennyLane
- PyTorch with autograd for PINNs loss
- Hugging Face for model hosting
🚀 Phase 6: Publication & Contribution¶
✅ Goals¶
- Publish code on GitHub (with tutorials)
- Contribute to DeepXDE, PennyLane, or Qiskit open-source
- Write and host a Jupyter Book with HQPINNs demos
- Upload pretrained HQPINN models on Hugging Face 🤗
C. Albornoz, G. Alonso, M. Andrenkov, P. Angara, A. Asadi, A. Ballon, S. Bapat, L. Botelho, I. De Vlugt, O. Di Matteo, P. Downing, P. Finlay, A. Fumagalli, A. Gardhouse, J. Geoffrion, N. Girard, A. Hayes, J. Izaac, R. Janik, T. Kalajdzievski, A. Kanwar Singh, A. Khomchenko, N. Killoran, I. Kurečić, O. Landon-Cardinal, A. Martin, D. Nino, A. Otto, C. Pere, J. Pickering, K. Renaud, J. Soni, D. Wakeham, L. Young. PennyLane Codebook. 2024. https://
pennylane .ai /codebook