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READING GUIDE

# How to use this book

This book develops quantum computing from probability and complex numbers to algorithms, noisy devices, and fault-tolerant systems. It assumes ordinary algebra, a little trigonometry, and a willingness to calculate. Programming experience and a prior course in quantum mechanics are helpful but unnecessary. The mathematical ideas that do real work are introduced where they first become useful.

The central question is operational: what can we prepare, transform, measure, and reliably infer? A list of amplitudes is not a list of readable answers. An entangled state is not a communication channel by itself. A quantum algorithm is not a speedup until its input, output, accuracy, and classical comparison have been specified. These distinctions connect the mathematics to honest claims about technology.

## The two editions

The PDF is a continuous course with worked calculations, exercises, and solutions. The web edition contains the same chapter text and numbered exercises. Laboratory panels appear where experimentation helps. Each laboratory has a permanent identifier, and the PDF links to its location. Changing an input recomputes the mathematical model; the displays are not prerecorded demonstrations.

Begin by predicting a result, then run an experiment, then explain any discrepancy. For finite-shot experiments, distinguish disagreement caused by sampling from a wrong model. Record the preparation, basis, shot count, and random seed. A screenshot without those details is rarely a reproducible experiment.

The laboratories run classical simulations in the browser. They do not send circuits to quantum hardware. Some calculate exact state vectors, some evolve density matrices, and some use an explicitly stated reduced model. Their limits are part of the lesson: a small exact simulation, a repetition-code memory, and a hardware resource estimate answer different questions.

## Routes through the material

First encounter. Read Chapters 1–8 in order. Work the projections in Chapter 3 by hand, including complex conjugation. Use the circuit laboratory before attempting the algorithm chapters.

Algorithms. Chapters 9–15 move from oracle interference to period finding, Hamiltonian simulation, and variational methods. The oracle examples expose mechanisms; small black-box problems are not industrial applications.

Reliable systems. Chapters 16–21 develop simulation, coding, fault tolerance, resource estimates, and physical hardware. Chapter 22 connects information protocols to networking and cryptography. Chapters 23–25 cover software, research evidence, and integrated projects.

A two-semester course can cover the text with laboratories and projects. A one-semester course can emphasize Chapters 1–13, 17–19, 21, and 24. This is a suggested teaching arrangement, not a promise that every beginner will master all chapters at the same speed.

## Notation and arithmetic

The computational basis is {|0⟩,|1⟩}\{|0\rangle,|1\rangle\}. The X basis is

|+⟩=|0⟩+|1⟩2,|−⟩=|0⟩−|1⟩2.|+\rangle=\frac{|0\rangle+|1\rangle}{\sqrt2}, \qquad |-\rangle=\frac{|0\rangle-|1\rangle}{\sqrt2}.

The Y basis is

|+i⟩=|0⟩+i|1⟩2,|−i⟩=|0⟩−i|1⟩2.|+i\rangle=\frac{|0\rangle+i|1\rangle}{\sqrt2}, \qquad |-i\rangle=\frac{|0\rangle-i|1\rangle}{\sqrt2}.

A bra is the conjugate transpose of a ket. In particular, ⟨+i|=(⟨0|−i⟨1|)/2\langle+i|=(\langle0|-i\langle1|)/\sqrt2. This minus sign is not optional. The symbol i i satisfies i 2=−1 i^2=-1; indices j,k j,k are integers. Information logarithms are base two unless stated otherwise.

Throughout the book and browser simulator, q 0 q_0 is the leftmost, most significant qubit:

|q 0 q 1⋯q n−1⟩,x=∑j=0 n−1 q j 2 n−1−j.|q_0q_1\cdots q_{n-1}\rangle,\qquad x=\sum_{j=0}^{n-1}q_j2^{n-1-j}.

Many SDKs, including Qiskit, give qubit 0 the least significant weight in a state-vector index. Chapter 23 gives the conversion explicitly. Neither convention is more quantum; silently mixing them produces wrong answers.

Exact fractions and radicals are retained when they reveal cancellation. Laboratory decimals are rounded for display; calculations use double precision. A displayed zero may be smaller than the display precision. Numerical checks use tolerances and do not substitute for proofs.

## Evidence and the research boundary

Research cutoff: 5 September 2026. Dated statements describe sources available by this boundary. An issue year, online publication date, preprint date, and correction date can differ. Chapter 24 records these distinctions. The evidence selection is a researched introduction, not a systematic census of every laboratory or paper.

Label | Meaning
--- | ---
Foundation | Mathematical result or established physical framework, with assumptions stated.
Experiment | Measurements on physical hardware, with task and metric identified.
Theory / estimate | Analysis or simulation under assumptions; not a built machine.
Preprint | Public research not treated here as a verified journal result.
Company claim | A vendor’s stated capability or goal, distinguished from independent validation.
Open question | A question without a settled answer in the scope considered.

Peer review strengthens scrutiny; it does not remove uncertainty. A logical memory experiment does not establish a general-purpose fault-tolerant computer. Postselection can improve conditional fidelity while discarding many runs. A benchmark advantage does not automatically imply useful economic advantage.

## Provenance and independent authorship

Thomas G. Wong’s Introduction to Classical and Quantum Computing (2022) is the supplied curriculum reference [1]. Its accessible progression from classical information to qubits, linear algebra, multiple qubits, protocols, and algorithms motivated the entry level. This book uses independently written explanations, newly organized chapters, new exercises, original diagrams, and contemporary research. It is a separate educational work, not an authorized revision or publication by Wong. No substantial prose or figures from the source book are reproduced.

The emphasis differs: density operators and channels are foundational; error correction connects to logical operations and resource budgets; software follows explicit circuit semantics and current interfaces; and dated hardware evidence is separated from timeless theory.

The text and software were prepared with OpenAI Codex for this project. Mathematical identities and implemented models have reproducible checks. This production process is not an independent external peer review. References lead to underlying research, particularly where the introduction stops short of graduate-level detail.

## Working with exercises

Every chapter includes problems and solutions or guidance. In the web edition, try a problem before revealing its solution. The measurement-practice laboratory generates new numerical questions, checks a probability, and reveals its projection in stages. A correct decimal alone does not demonstrate correct conjugation.

Keep a notebook of mistakes: normalization, conjugation, tensor order, gate order, omitted conditioning, wrong noise convention, and inappropriate complexity comparison. These categories become a debugging method as circuits grow.

IN THIS CHAPTER