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

For my parents, who believed before I did.

Preface

Every chemistry textbook tells you that water’s bond angle is 104.5° and waves at VSEPR theory. Every quantum computing textbook tells you that fermions need to be encoded as qubits and shows you the Jordan-Wigner transform. Very few resources connect these two worlds with enough detail that you could actually compute the bond angle from the quantum simulation, or understand why the encoding choice matters for the circuit you’ll run on a quantum computer.

This book fills that gap.

We walk through the complete pipeline — from molecular geometry to quantum circuit — with two running examples: the hydrogen molecule (H₂), because it is the simplest system that exhibits all the essential structure, and the water molecule (H₂O), because it is the most important molecule on Earth and rich enough to make the engineering trade-offs tangible.

Every formula in this book has a corresponding executable computation in the FockMap library, an open-source F# framework for symbolic Fock-space operator algebra. Every sign, every coefficient, every intermediate Pauli string is computed explicitly. Nothing is left as an exercise for the reader — though there are plenty of exercises at the end of each chapter for readers who want to deepen their understanding.

The book follows a deliberate pedagogical ordering: chemistry and physics first, mathematical formalism second, executable code third. We believe the question “why does this matter?” should always be answered before the question “how does this work?” — and both should be answered before “how do I compute it?”

Who This Book Is For

Graduate students starting in quantum chemistry simulation who need to understand the encoding layer between chemistry and circuits
Physicists crossing into quantum computing who know Hamiltonians but not encodings
Software engineers building quantum simulation pipelines who need the physics explained carefully
Lecturers building a course module on quantum simulation who want homework exercises with verifiable answers

What You Need to Know

We assume familiarity with linear algebra (vectors, matrices, eigenvalues) and introductory quantum mechanics (wavefunctions, bra-ket notation, the hydrogen atom). We do not assume prior knowledge of:

Second quantization or Fock space
Pauli algebra or qubit representations
Fermion-to-qubit encodings
F# or functional programming

All of these are developed from scratch within the book.

How to Read This Book

The 23 chapters are organized into seven stages, following the quantum simulation pipeline:

The Molecule (Chapters 1–3) — from molecules to integrals
The Machine (Chapter 4) — qubits, gates, and circuits
Encoding (Chapters 5–9) — from fermions to qubits
Tapering (Chapters 10–13) — removing redundant qubits
Circuits (Chapters 14–17) — Trotterization and cost analysis
The Pipeline (Chapters 18–21) — complete pipeline, bond angle, algorithms, export
Horizons (Chapters 22–23) — scaling and what comes next

You can read straight through (recommended for first reading), or jump to a specific stage if you already know the earlier material. Each chapter begins with “In This Chapter” learning objectives and ends with “Key Takeaways” and exercises.

The Companion Software

All computations in this book are reproducible using the FockMap library:

Source code: https://github.com/johnazariah/encodings
NuGet package: dotnet add package FockMap
Web documentation: https://johnazariah.github.io/encodings/

The library is open-source (MIT license), runs on Windows, macOS, and Linux via .NET 10, and has zero runtime dependencies.

Acknowledgements

This work grew out of research at the Centre for Quantum Software and Information at the University of Technology Sydney.

I am grateful to Dr Chris Ferrie for his patient guidance of my exploration, to Dr Guang Hao Low for sparking the journey, and to Dr Stephen Jordan and Dr Helmut Katzgraber for their continued encouragement.

And to my family, for bearing all my burdens.

Sydney, March 2026

About the Author

John S Azariah is a software engineer, language designer, and quantum computing researcher whose work sits at the intersection of chemistry, computation, and executable mathematics. He earned a BA in Computer Science and Chemistry from Dartmouth College, a combination that now finds a natural expression in quantum simulation.

Over more than three decades in professional software development, he has worked through successive technology shifts including desktop software, the internet, cloud computing, quantum programming, and AI. His career has included roles on Microsoft Excel and Microsoft Project, early SharePoint prototypes, Visual J#.NET, large-scale Azure systems, and principal engineering work across Azure Batch, Azure Kubernetes Service, and AI systems in APAC. He was also compiler lead and one of the language designers behind Microsoft Q#.

Alongside industry work, he has maintained a longstanding interest in functional programming, scientific computing, and software craft. He is currently a Principal AI SME at Microsoft and is pursuing a PhD in Quantum Computing at the University of Technology Sydney under Dr. Chris Ferrie.