Reader’s guide
How to read this book
Work with the state or probability distribution on paper. Each chapter gives enough intermediate steps to check where its result comes from.
How each chapter is built
Every chapter has the same shape. It opens with a concrete question. A worked example carries the argument, with a figure generated from the same calculation. Longer derivations and general proofs sit in sections marked optional, closed by default; the main path is complete without them, and conditions on a result are never hidden in one. A closing section, “Where this leaves us”, answers the opening question and says why the next chapter follows. Two problems with worked answers end the chapter.
Background and reading order
You should be comfortable with complex conjugation, matrix multiplication, inner products, eigenvalues, and elementary probability. Chapter 1 introduces bras, kets, qubits, and tensor products. Chapter 5 also uses the Schrödinger equation. You do not need prior quantum computing or access to a particular textbook. For a slower introduction to quantum mechanics itself, the companion guide Quantum Mechanics, Explained is a recommended, not required, starting point; chapters link to its specific essays.
Read Chapters 1–3 before working with measurements or reduced states. Chapters 4–5 establish gate conventions and control. Chapters 6–9 develop entanglement and communication; Chapters 10–12 calculate algorithms. Chapters 13–15 cover noise, recovery, and information. Each chapter lists the prerequisites relevant to its calculation.
All fifteen chapters are written, with thirty problems and worked answers. Their length varies with the calculation: a protocol needs its register states, a bound needs its assumptions, and an algorithm needs its output distribution and resource count.
How the content is checked
Two kinds of checking are kept separate. Every worked number and every figure is recomputed by a test script, by a second method where possible: a figure computed from a formula is checked against a direct matrix simulation, and a circuit figure against the register and gate order used in the text. Passing those tests shows that the calculations are right. It does not show that the prose around them is right, so each chapter has also had a separate scientific review of its statements and their conditions. Neither replaces the other.
Notation used throughout
| Symbol | Meaning |
|---|---|
| $|\psi\rangle$, $\langle\psi|$ | Column state vector and its conjugate transpose. |
| $\langle\phi|\psi\rangle$ | Inner product; its absolute square is a projective measurement probability for normalised vectors. |
| $|\psi\rangle\langle\psi|$ | Outer product, the density matrix of a normalised pure state. |
| $AB|\psi\rangle$ | $B$ acts first, then $A$. |
| $|ab\rangle$ | $|a\rangle\otimes|b\rangle$. Two-qubit basis order is $00,01,10,11$. |
| $X,Y,Z$; $H$ | Pauli matrices; Hadamard gate. A Hamiltonian is written $\mathcal H$. The companion guide, which has no Hadamard gate, writes the Hamiltonian as $H$ and the Pauli matrices as $\sigma_x,\sigma_y,\sigma_z$; its spin up and down are $|0\rangle$ and $|1\rangle$ here. |
| $\mathbf r$ | Bloch vector $(\langle X\rangle,\langle Y\rangle,\langle Z\rangle)$. For spin ½ its angles are spatial angles; for linear photon polarisation they are twice the polariser angle (Chapter 5). |
| $\rho$; $\operatorname{Tr}_B$ | Density matrix; partial trace over subsystem $B$. |
| $E_m$; $M_m$ | Measurement effect; measurement operator. $E_m=M_m^\dagger M_m$ in the single-operator case. |
| $\oplus$; $x\cdot z$ | Addition modulo two; bitwise dot product modulo two. |
| $S(\rho)$; $H(X)$ | Von Neumann entropy; Shannon entropy. All information logarithms are base two. |
| $\|A\|$ | Operator norm: the largest value of $\|A|\psi\rangle\|$ on a unit vector. |
| $[A,B]$ | Commutator $AB-BA$. |
| $X_1$, $Z_1Z_2$ | A Pauli acting on the numbered qubit, identity elsewhere; products act on several qubits. |
| $|\Phi^+\rangle$; $|\beta_{ab}\rangle$ | $(|00\rangle+|11\rangle)/\sqrt2$; the four Bell states $(I\otimes X^bZ^a)|\Phi^+\rangle$. |
| GHZ state | $(|000\rangle\pm|111\rangle)/\sqrt2$, a superposition of the all-zero and all-one strings. |
| POVM; $\{E_y\}$ | A general measurement: effects $E_y\ge0$ with $\sum_yE_y=I$ (Chapter 2). |
| $K_j$; $\operatorname{Tr}_E$ | Kraus operators of a channel; partial trace over an environment (Chapter 13). |
| $\rho^{T_B}$ | Partial transpose: transpose the second tensor factor only. |
| $h_2(p)$ | Binary entropy $-p\log_2p-(1-p)\log_2(1-p)$. |
| $I(X{:}Y)$; $\chi$ | Mutual information; Holevo quantity $S(\bar\rho)-\sum_xp_xS(\rho_x)$ (Chapter 15). |
| $O(g)$, $\Theta(g)$ | Asymptotic upper bound; matching upper and lower bounds up to constants (Chapter 12). |
How to check your understanding
After a state transformation, check its norm. After a probability calculation, check positivity and that probabilities sum to one. For density matrices, check trace and eigenvalues. For a protocol, identify which party holds each register and which records they know. For an algorithm, distinguish the probability of an answer from the cost of obtaining it.
The problems change an input, a measurement, or an assumption from the worked example. Their answers are hidden until opened, and use intermediate calculations rather than a verdict alone. If an answer disagrees with yours, first check basis order, conjugation, and which matrix acts first.
Sources and proof boundaries
The main reference is Michael A. Nielsen and Isaac L. Chuang, Quantum Computation and Quantum Information, 10th Anniversary Edition (2010). Section pointers were checked against that edition’s contents. Sakurai and Napolitano’s Modern Quantum Mechanics provides further physical background where identified. Primary papers are linked in the relevant chapters.
References support the mathematics and provide longer treatments. The chapters are independent explanations, not a comparison of textbooks. No chapter needs to argue that a physics course missed an idea before introducing it.
The book derives small-system calculations in full and explicitly identifies the larger theorems it uses without proof, including universality, the unstructured-search lower bound, the general recovery criterion, and the Holevo bound. These references are the next reading step if you need those proofs. Source prose and textbook PDFs are not included in the published site.