Amplitudes, Measurement, and Gates

Declare the vector before touching a gate#

Fix basis order, normalization, and phase conventions.

A quantum state is not a list of probabilities; it is a vector of amplitudes. Gates transform the amplitudes. Measurement then converts the state into classical data, with probabilities computed from the squared magnitudes. Every quantum program, whatever its size, runs the same loop:

1. Prepare an initial state.
2. Apply gates that transform the amplitudes.
3. Arrange interference so useful outcomes grow and useless ones cancel.
4. Measure, producing one classical record.
5. Interpret the samples, usually over many repeated shots.

Confuse amplitudes with probabilities and almost every quantum algorithm becomes mysterious. Track amplitudes carefully and the mystery turns into linear algebra.

Work in the computational basis, $\lvert0\rangle=\begin{pmatrix}1\\0\end{pmatrix}$ and $\lvert1\rangle=\begin{pmatrix}0\\1\end{pmatrix}$. A general pure state is $\lvert\psi\rangle=\alpha\lvert0\rangle+\beta\lvert1\rangle=\begin{pmatrix}\alpha\\\beta\end{pmatrix}$, subject to the normalization condition $\lvert\alpha\rvert^2+\lvert\beta\rvert^2=1$. Measurement in this basis returns 0 with probability $\lvert\alpha\rvert^2$ and 1 with probability $\lvert\beta\rvert^2$.

Because amplitudes are complex, the squared magnitude deserves one line of care: for $z=a+bi$, $\lvert z\rvert^2=a^2+b^2$, equivalently $\lvert z\rvert^2=z^\ast z$ where $z^\ast$ is the complex conjugate. Normalize wrong and every probability you compute afterwards is wrong too.

Evidence boundary. A pure one-qubit state is a normalized vector in a two-dimensional complex Hilbert space. [John Watrous] [Michael A. Nielsen]

Four matrices, one execution order#

Derive X, Z, H, and S actions on the basis states.

X gate#

It swaps the basis states: $X\lvert0\rangle=\lvert1\rangle$ and $X\lvert1\rangle=\lvert0\rangle$. On the computational basis this is the quantum analogue of NOT.

Z gate#

It leaves $\lvert0\rangle$ alone and flips the sign of $\lvert1\rangle$: $Z\lvert1\rangle=-\lvert1\rangle$. Measure immediately and you see nothing; apply another gate and the flipped phase changes what interferes with what. Z is the purest phase gate in the set.

H gate#

It maps $\lvert0\rangle$ to $(\lvert0\rangle+\lvert1\rangle)/\sqrt2=\lvert+\rangle$ and $\lvert1\rangle$ to $(\lvert0\rangle-\lvert1\rangle)/\sqrt2=\lvert-\rangle$. Hadamard is the simplest gate that creates superposition, which makes it the best gate for learning interference.

S gate#

It leaves $\lvert0\rangle$ alone and multiplies the $\lvert1\rangle$ component by $i$; a quarter-turn of phase where Z applies a half-turn. Four gates, and you can already build every phenomenon this guide discusses.

Take the state $\lvert+\rangle=(\lvert0\rangle+\lvert1\rangle)/\sqrt2$. A computational-basis measurement gives 0 and 1 with equal probability. But apply $H$ first and then measure, and $H\lvert+\rangle=\lvert0\rangle$; the outcome is now 0 with certainty.

Same state, different statistics, because the effective measurement changed. This is why "the measurement probability" is never a complete description of a state: you must specify the basis, or equivalently the circuit that runs before the meter. A claim about a quantum state that does not say how it was measured is an incomplete claim.

Evidence boundary. X, Z, H, and phase gates are unitary matrices whose order of multiplication determines the result. [Michael A. Nielsen] [John Preskill]

The HZH phase detector#

Trace an imaginary or negative phase into a measurable outcome.

Two states that differ only by an overall phase; $\lvert\psi\rangle$ versus $-\lvert\psi\rangle$, or $\lvert\psi\rangle$ versus $i\lvert\psi\rangle$; make identical predictions for every possible measurement. Global phase is unphysical, a residue of the notation.

Relative phase is different. The states $(\lvert0\rangle+\lvert1\rangle)/\sqrt2$ and $(\lvert0\rangle-\lvert1\rangle)/\sqrt2$ differ by a sign on one component only, and as the Hadamard example showed, that sign decides whether a later gate produces $\lvert0\rangle$ or $\lvert1\rangle$. The quick test: if you cannot factor the phase out in front of the whole state, it is relative, and it matters.

Interference is amplitude addition, nothing more. When two computational paths contribute amplitudes to the same outcome, those amplitudes add. Aligned, they reinforce; $1/\sqrt2+1/\sqrt2=\sqrt2$ grows. Opposed, they cancel; $1/\sqrt2-1/\sqrt2=0$.

A quantum algorithm earns its keep when its circuit steers amplitudes so that wrong or irrelevant outputs cancel and useful outputs survive or grow. Probability can never do this; probabilities only ever add upward. The ability to subtract is the whole game.

Evidence boundary. Measurement probabilities depend on the selected basis and are computed from squared amplitude magnitudes. [John Watrous] [Michael A. Nielsen]

Changing the question changes the statistics#

Express X-basis measurement as a basis rotation followed by Z measurement.

A small statevector simulator is the reference implementation of everything in this guide:

state: complex vector
gate: unitary matrix
operation: state <- gate * state
measurement: sample according to squared magnitudes

Four ideas, each one line. This approach will not scale; the vector doubles with every qubit; but it is the right way to build early correctness, and Part V has you write one from scratch.

On a real device there are no matrices; there are pulses, fields, voltages, lasers, and optical elements, each engineered to approximate a target unitary. A symbolic $H$ in your circuit becomes hardware-specific control, and the approximation is never exact. The questions that determine whether the math survives contact with the machine:

• What gate set is native to the device?
• How long does each gate take?
• What is the measured error rate per gate?
• How many extra gates does compilation add?
• Does limited connectivity force swap operations?
• Does calibration drift change behavior over hours or days?

A one-qubit calculation protocol#

Turn the arithmetic into a reusable checklist and diagnostic.

1. Normalize $[\sqrt2,0]$, $[1,-1]$, and $[\sqrt2,\sqrt2 i]$.
2. Apply $X$, $Z$, $H$, and $S$ to both $\lvert0\rangle$ and $\lvert1\rangle$.
3. Compute $HZH\lvert0\rangle$ and identify the familiar gate you just built.
4. Show that $X^2=I$ and $Z^2=I$.
5. Write down two states with identical computational-basis probabilities but different outcomes after a Hadamard.
6. Write pseudocode for a one-qubit statevector simulator.
7. Explain why a shot histogram is not the same thing as a quantum state.

Part II of the book turns this working session into a complete toolkit:

• Complex Numbers for Quantum Computing; the arithmetic of phase, done properly.
• Vectors, Bases, and Amplitudes; the state space as a vector space.
• Matrices as Gates; unitary transformations beyond one qubit's worth of intuition.
• Interference as Engine of Quantum Algorithms; the engine, examined under load.

Then continue to Multi-Qubit Circuits, Entanglement, and No-Hype Protocols, where a second qubit changes the game entirely.