From Molecules to Quantum Circuits

A Computational Guide to Fermion-to-Qubit Encodings

John S Azariah Centre for Quantum Software and Information, University of Technology Sydney

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About This Book

This is a self-contained, 23-chapter computational tutorial covering the complete pipeline from molecular electronic structure to quantum circuit compilation for quantum simulation.

Starting from the one-body and two-body integrals of hydrogen (H₂), we construct the qubit Hamiltonian explicitly under six fermion-to-qubit encodings, verify spectral equivalence, reduce qubit count via Z₂ symmetry tapering, decompose the tapered Hamiltonian into Trotter circuits with explicit CNOT gate counts, and export the result to OpenQASM 3.0 and Q#.

Every formula has a corresponding executable computation in the companion FockMap library.

Contents

Part I — The Molecule

  1. The Electronic Structure Problem
  2. The Notation Minefield
  3. From Spatial to Spin-Orbital Integrals

Part II — The Machine

  1. The Quantum Computer’s Vocabulary

Part III — Encoding

  1. A Visual Guide to Encodings
  2. Building the Qubit Hamiltonian
  3. Six Encodings, One Interface
  4. Building a Tree Encoding
  5. Checking Our Answer

Part IV — Tapering

  1. Why Tapering?
  2. Diagonal Z₂ Symmetries
  3. General Clifford Tapering
  4. Tapering Benchmarks

Part V — Circuits

  1. From Hamiltonian to Time Evolution
  2. Trotterization in Practice
  3. The CNOT Staircase
  4. Cost Analysis Across Encodings

Part VI — The Pipeline

  1. The Question We Can Now Answer
  2. The Water Bond Angle
  3. Algorithms — VQE and QPE
  4. Speaking the Hardware’s Language

Part VII — Horizons

  1. Scaling — From H₂ to FeMo-co
  2. What Comes Next

Appendices

Citation

@misc{azariah2026molecules,
  author    = {Azariah, John S},
  title     = {From Molecules to Quantum Circuits: A Computational Guide to Fermion-to-Qubit Encodings},
  year      = {2026},
  doi       = {10.5281/zenodo.18917465},
  url       = {https://github.com/johnazariah/encodings-book}
}

License

Code: MIT. Manuscript text: © 2026 John S Azariah. All rights reserved.