Reference shelf

Gate reference

Every single- and two-qubit gate the course builds, one row each: matrix, Bloch action, what it does, and the lesson that derives it.

Written by Claude Fable 5

Every gate the course actually builds, on one page. Come back to these cold and try to say, for each, what the matrix does to $∣ 0 ⟩$ and $∣ 1 ⟩$ before you read the last two columns. Every gate here is derived or defined in the lesson it links to.

The one rule behind every single-qubit gate

A one-qubit gate is a

2 \times 2

unitary matrix (

U^{†} U = I

). Unitarity makes it both reversible and length-preserving, so a state that summed to one still sums to one after the gate. A length-preserving map of the Bloch sphere is exactly a rotation, which is why every single-qubit gate below is a turn of the arrow.

Single-qubit gates

Each acts on the column $[α β]$ by ordinary matrix multiplication. Chaining gate $A$ then $B$ is the single matrix $B A$ , with the later gate on the left.

Gate	Matrix	Bloch action	What it does	From
$I$	$[1001]$	No rotation: the arrow stays put.	The do-nothing gate, the reference every other one is judged against. Two Hadamards compose to it ( $H H = I$ ).	Single-qubit gates
$X$	$[0110]$	Half-turn about the x-axis: swaps the two poles.	The bit flip, the quantum NOT. Swaps the amplitudes, so $X ∣ 0 ⟩ = ∣ 1 ⟩$ and $X ∣ 1 ⟩ = ∣ 0 ⟩$ .	Single-qubit gates
$Z$	$[10 0 - 1]$	Half-turn about the z-axis: fixes the poles, spins the equator by half a turn.	The phase flip. Leaves $∣ 0 ⟩$ alone and multiplies $∣ 1 ⟩$ by $- 1$ , touching no measurement probability at all.	Single-qubit gates
$H$	$\frac{1}{2} [11 1 - 1]$	Sends the north pole to the +x point of the equator, and back.	The Hadamard: makes and unmakes an equal superposition, $H ∣ 0 ⟩ = ∣ + ⟩$ . It is its own inverse, which is what lets you spread a state out and fold it back.	Single-qubit gates
$S$	$[10 0 i]$	Quarter-turn about the z-axis.	A gentler cousin of $Z$ : rotates the phase of $∣ 1 ⟩$ by a quarter turn instead of a half.	Single-qubit gates
$T$	$[10 0 e^{iπ /4}]$	Eighth-turn about the z-axis.	The gentlest phase cousin of $Z$ : rotates the phase of $∣ 1 ⟩$ by an eighth of a turn.	Single-qubit gates

Two identities worth keeping in hand

H H = I

(a Hadamard undoes itself) and

H Z H = X

(a phase flip between two Hadamards is a bit flip). Both fall straight out of multiplying the matrices, and both are the machinery interference runs on: spread a state out, let phases act, fold it back.

Two-qubit gates and read-out

Two qubits have four basis states, so a two-qubit gate is a $4 \times 4$ matrix. CNOT is the one the course leans on; Toffoli is its three-bit classical ancestor, and measurement is the only step here that is not a gate at all.

Gate	Matrix	Bloch action	What it does	From
CNOT	$1000010000010010$	Two qubits: no single-arrow picture. It entangles, so the pair has no separate states.	Controlled-NOT: flips the target exactly when the control is $∣ 1 ⟩$ . On a superposed control it builds a Bell state, not two copies.	Two-qubit gates and CNOT
Toffoli	$CCNOT: ∣ a, b, c ⟩ \to ∣ a, b, c \oplus ab ⟩$	Three classical bits: a reversible truth table, not a Bloch rotation.	Flips the third bit only when the first two are both $1$ . It is universal for reversible computing: any classical circuit can be rebuilt from Toffoli gates alone.	Classical bits and reversible gates
Measure	$not unitary$	Collapses the arrow onto a pole; irreversible, unlike every gate above.	The read-out. Samples one basis state by the Born rule, $Pr (0) = ∣ α ∣^{2}$ , and returns one classical bit per run.	Measurement and the Born rule

Notice how little is memorized. Four of the six single-qubit gates are pure phase turns about the same z-axis, differing only in how far they rotate $∣ 1 ⟩$ : half a turn for $Z$ , a quarter for $S$ , an eighth for $T$ . The whole zoo is a handful of rotations plus one gate, CNOT, that couples two wires.